From c81838154a68b22a311b316df888b10d762ee344 Mon Sep 17 00:00:00 2001 From: Riccardo Finotello Date: Tue, 29 Sep 2020 18:48:42 +0200 Subject: [PATCH] Progress on fermions with defects Signed-off-by: Riccardo Finotello --- sciencestuff.sty | 1008 +++++++++++++++++++++++++- sec/app/parameters.tex | 6 +- sec/app/reflection.tex | 95 +++ sec/part1/dbranes.tex | 204 +++--- sec/part1/fermions.tex | 1378 +++++++++++++++++++++++++++++++++--- sec/part1/introduction.tex | 248 +++---- thesis.cls | 8 +- thesis.tex | 66 +- 8 files changed, 2609 insertions(+), 404 deletions(-) create mode 100644 sec/app/reflection.tex diff --git a/sciencestuff.sty b/sciencestuff.sty index a95c35b..a99d152 100644 --- a/sciencestuff.sty +++ b/sciencestuff.sty @@ -45,46 +45,213 @@ \numberwithin{figure}{section} \numberwithin{table}{section} +%---- abbreviations +\providecommand{\sm}{\textsc{sm}\xspace} +\providecommand{\eom}{\textsc{e.o.m.}\xspace} +\providecommand{\cft}{\textsc{CFT}\xspace} +\providecommand{\qft}{\textsc{QFT}\xspace} +\providecommand{\qed}{\textsc{QED}\xspace} +\providecommand{\qcd}{\textsc{QCD}\xspace} +\providecommand{\ope}{\textsc{o.p.e.}\xspace} +\providecommand{\cy}{\textsc{CY}\xspace} +\providecommand{\ap}{\ensuremath{\alpha'}\xspace} + %---- remap greek letters -\renewcommand{\alpha}{\upalpha\xspace} -\renewcommand{\beta}{\upbeta\xspace} -\renewcommand{\gamma}{\upgamma\xspace} -\renewcommand{\delta}{\updelta\xspace} -\renewcommand{\epsilon}{\upepsilon\xspace} -\renewcommand{\zeta}{\upzeta\xspace} -\renewcommand{\eta}{\upeta\xspace} -\renewcommand{\theta}{\uptheta\xspace} -\renewcommand{\iota}{\upiota\xspace} -\renewcommand{\kappa}{\upkappa\xspace} -\renewcommand{\lambda}{\uplambda\xspace} -\renewcommand{\mu}{\upmu\xspace} -\renewcommand{\nu}{\upnu\xspace} -\renewcommand{\xi}{\upxi\xspace} -\renewcommand{\pi}{\uppi\xspace} -\renewcommand{\rho}{\uprho\xspace} -\renewcommand{\sigma}{\upsigma\xspace} -\renewcommand{\tau}{\uptau\xspace} -\renewcommand{\upsilon}{\upupsilon\xspace} -\renewcommand{\phi}{\upphi\xspace} -\renewcommand{\chi}{\upchi\xspace} -\renewcommand{\psi}{\uppsi\xspace} -\renewcommand{\omega}{\upomega\xspace} -\renewcommand{\varepsilon}{\upvarepsilon\xspace} -\renewcommand{\vartheta}{\upvartheta\xspace} -\renewcommand{\varpi}{\upvarpi\xspace} -\renewcommand{\varphi}{\upvarphi\xspace} -\renewcommand{\Gamma}{\Upgamma\xspace} -\renewcommand{\Delta}{\Updelta\xspace} -\renewcommand{\Theta}{\Uptheta\xspace} -\renewcommand{\Lambda}{\Uplambda\xspace} -\renewcommand{\Xi}{\Upxi\xspace} -\renewcommand{\Pi}{\Uppi\xspace} -\renewcommand{\Sigma}{\Upsigma\xspace} -\renewcommand{\Upsilon}{\Upupsilon\xspace} -\renewcommand{\Phi}{\Upphi\xspace} -\renewcommand{\Psi}{\Uppsi\xspace} -\renewcommand{\Omega}{\Upomega\xspace} +\renewcommand{\alpha}{\ensuremath{\upalpha}\xspace} +\renewcommand{\beta}{\ensuremath{\upbeta}\xspace} +\renewcommand{\gamma}{\ensuremath{\upgamma}\xspace} +\renewcommand{\delta}{\ensuremath{\updelta}\xspace} +\renewcommand{\epsilon}{\ensuremath{\upepsilon}\xspace} +\renewcommand{\zeta}{\ensuremath{\upzeta}\xspace} +\renewcommand{\eta}{\ensuremath{\upeta}\xspace} +\renewcommand{\theta}{\ensuremath{\uptheta}\xspace} +\renewcommand{\iota}{\ensuremath{\upiota}\xspace} +\renewcommand{\kappa}{\ensuremath{\upkappa}\xspace} +\renewcommand{\lambda}{\ensuremath{\uplambda}\xspace} +\renewcommand{\mu}{\ensuremath{\upmu}\xspace} +\renewcommand{\nu}{\ensuremath{\upnu}\xspace} +\renewcommand{\xi}{\ensuremath{\upxi}\xspace} +\renewcommand{\pi}{\ensuremath{\uppi}\xspace} +\renewcommand{\rho}{\ensuremath{\uprho}\xspace} +\renewcommand{\sigma}{\ensuremath{\upsigma}\xspace} +\renewcommand{\tau}{\ensuremath{\uptau}\xspace} +\renewcommand{\upsilon}{\ensuremath{\upupsilon}\xspace} +\renewcommand{\phi}{\ensuremath{\upphi}\xspace} +\renewcommand{\chi}{\ensuremath{\upchi}\xspace} +\renewcommand{\psi}{\ensuremath{\uppsi}\xspace} +\renewcommand{\omega}{\ensuremath{\upomega}\xspace} +\renewcommand{\varepsilon}{\ensuremath{\upvarepsilon}\xspace} +\renewcommand{\vartheta}{\ensuremath{\upvartheta}\xspace} +\renewcommand{\varpi}{\ensuremath{\upvarpi}\xspace} +\renewcommand{\varphi}{\ensuremath{\upvarphi}\xspace} +\renewcommand{\Gamma}{\ensuremath{\Upgamma}\xspace} +\renewcommand{\Delta}{\ensuremath{\Updelta}\xspace} +\renewcommand{\Theta}{\ensuremath{\Uptheta}\xspace} +\renewcommand{\Lambda}{\ensuremath{\Uplambda}\xspace} +\renewcommand{\Xi}{\ensuremath{\Upxi}\xspace} +\renewcommand{\Pi}{\ensuremath{\Uppi}\xspace} +\renewcommand{\Sigma}{\ensuremath{\Upsigma}\xspace} +\renewcommand{\Upsilon}{\ensuremath{\Upupsilon}\xspace} +\renewcommand{\Phi}{\ensuremath{\Upphi}\xspace} +\renewcommand{\Psi}{\ensuremath{\Uppsi}\xspace} +\renewcommand{\Omega}{\ensuremath{\Upomega}\xspace} + +\providecommand{\dalpha}{\ensuremath{\dot{\upalpha}}\xspace} +\providecommand{\dbeta}{\ensuremath{\dot{\upbeta}}\xspace} +\providecommand{\dgamma}{\ensuremath{\dot{\upgamma}}\xspace} +\providecommand{\ddelta}{\ensuremath{\dot{\updelta}}\xspace} +\providecommand{\depsilon}{\ensuremath{\dot{\upepsilon}}\xspace} +\providecommand{\dzeta}{\ensuremath{\dot{\upzeta}}\xspace} +\providecommand{\deta}{\ensuremath{\dot{\upeta}}\xspace} +\providecommand{\dtheta}{\ensuremath{\dot{\uptheta}}\xspace} +\providecommand{\diota}{\ensuremath{\dot{\upiota}}\xspace} +\providecommand{\dkappa}{\ensuremath{\dot{\upkappa}}\xspace} +\providecommand{\dlambda}{\ensuremath{\dot{\uplambda}}\xspace} +\providecommand{\dmu}{\ensuremath{\dot{\upmu}}\xspace} +\providecommand{\dnu}{\ensuremath{\dot{\upnu}}\xspace} +\providecommand{\dxi}{\ensuremath{\dot{\upxi}}\xspace} +\providecommand{\dpi}{\ensuremath{\dot{\uppi}}\xspace} +\providecommand{\drho}{\ensuremath{\dot{\uprho}}\xspace} +\providecommand{\dsigma}{\ensuremath{\dot{\upsigma}}\xspace} +\providecommand{\dtau}{\ensuremath{\dot{\uptau}}\xspace} +\providecommand{\dupsilon}{\ensuremath{\dot{\upupsilon}}\xspace} +\providecommand{\dphi}{\ensuremath{\dot{\upphi}}\xspace} +\providecommand{\dchi}{\ensuremath{\dot{\upchi}}\xspace} +\providecommand{\dpsi}{\ensuremath{\dot{\uppsi}}\xspace} +\providecommand{\domega}{\ensuremath{\dot{\upomega}}\xspace} +\providecommand{\dvarepsilon}{\ensuremath{\dot{\upvarepsilon}}\xspace} +\providecommand{\dvartheta}{\ensuremath{\dot{\upvartheta}}\xspace} +\providecommand{\dvarpi}{\ensuremath{\dot{\upvarpi}}\xspace} +\providecommand{\dvarphi}{\ensuremath{\dot{\upvarphi}}\xspace} +\providecommand{\dGamma}{\ensuremath{\dot{\Upgamma}}\xspace} +\providecommand{\dDelta}{\ensuremath{\dot{\Updelta}}\xspace} +\providecommand{\dTheta}{\ensuremath{\dot{\Uptheta}}\xspace} +\providecommand{\dLambda}{\ensuremath{\dot{\Uplambda}}\xspace} +\providecommand{\dXi}{\ensuremath{\dot{\Upxi}}\xspace} +\providecommand{\dPi}{\ensuremath{\dot{\Uppi}}\xspace} +\providecommand{\dSigma}{\ensuremath{\dot{\Upsigma}}\xspace} +\providecommand{\dUpsilon}{\ensuremath{\dot{\Upupsilon}}\xspace} +\providecommand{\dPhi}{\ensuremath{\dot{\Upphi}}\xspace} +\providecommand{\dPsi}{\ensuremath{\dot{\Uppsi}}\xspace} +\providecommand{\dOmega}{\ensuremath{\dot{\Upomega}}\xspace} + +\providecommand{\balpha}{\ensuremath{\overline{\upalpha}}\xspace} +\providecommand{\bbeta}{\ensuremath{\overline{\upbeta}}\xspace} +\providecommand{\bgamma}{\ensuremath{\overline{\upgamma}}\xspace} +\providecommand{\bdelta}{\ensuremath{\overline{\updelta}}\xspace} +\providecommand{\bepsilon}{\ensuremath{\overline{\upepsilon}}\xspace} +\providecommand{\bzeta}{\ensuremath{\overline{\upzeta}}\xspace} +\providecommand{\beta}{\ensuremath{\overline{\upeta}}\xspace} +\providecommand{\btheta}{\ensuremath{\overline{\uptheta}}\xspace} +\providecommand{\biota}{\ensuremath{\overline{\upiota}}\xspace} +\providecommand{\bkappa}{\ensuremath{\overline{\upkappa}}\xspace} +\providecommand{\blambda}{\ensuremath{\overline{\uplambda}}\xspace} +\providecommand{\bmu}{\ensuremath{\overline{\upmu}}\xspace} +\providecommand{\bnu}{\ensuremath{\overline{\upnu}}\xspace} +\providecommand{\bxi}{\ensuremath{\overline{\upxi}}\xspace} +\providecommand{\bpi}{\ensuremath{\overline{\uppi}}\xspace} +\providecommand{\brho}{\ensuremath{\overline{\uprho}}\xspace} +\providecommand{\bsigma}{\ensuremath{\overline{\upsigma}}\xspace} +\providecommand{\btau}{\ensuremath{\overline{\uptau}}\xspace} +\providecommand{\bupsilon}{\ensuremath{\overline{\upupsilon}}\xspace} +\providecommand{\bphi}{\ensuremath{\overline{\upphi}}\xspace} +\providecommand{\bchi}{\ensuremath{\overline{\upchi}}\xspace} +\providecommand{\bpsi}{\ensuremath{\overline{\uppsi}}\xspace} +\providecommand{\bomega}{\ensuremath{\overline{\upomega}}\xspace} +\providecommand{\bvarepsilon}{\ensuremath{\overline{\upvarepsilon}}\xspace} +\providecommand{\bvartheta}{\ensuremath{\overline{\upvartheta}}\xspace} +\providecommand{\bvarpi}{\ensuremath{\overline{\upvarpi}}\xspace} +\providecommand{\bvarphi}{\ensuremath{\overline{\upvarphi}}\xspace} +\providecommand{\bGamma}{\ensuremath{\overline{\Upgamma}}\xspace} +\providecommand{\bDelta}{\ensuremath{\overline{\Updelta}}\xspace} +\providecommand{\bTheta}{\ensuremath{\overline{\Uptheta}}\xspace} +\providecommand{\bLambda}{\ensuremath{\overline{\Uplambda}}\xspace} +\providecommand{\bXi}{\ensuremath{\overline{\Upxi}}\xspace} +\providecommand{\bPi}{\ensuremath{\overline{\Uppi}}\xspace} +\providecommand{\bSigma}{\ensuremath{\overline{\Upsigma}}\xspace} +\providecommand{\bUpsilon}{\ensuremath{\overline{\Upupsilon}}\xspace} +\providecommand{\bPhi}{\ensuremath{\overline{\Upphi}}\xspace} +\providecommand{\bPsi}{\ensuremath{\overline{\Uppsi}}\xspace} +\providecommand{\bOmega}{\ensuremath{\overline{\Upomega}}\xspace} + +\providecommand{\talpha}{\ensuremath{\widetilde{\upalpha}}\xspace} +\providecommand{\tbeta}{\ensuremath{\widetilde{\upbeta}}\xspace} +\providecommand{\tgamma}{\ensuremath{\widetilde{\upgamma}}\xspace} +\providecommand{\tdelta}{\ensuremath{\widetilde{\updelta}}\xspace} +\providecommand{\tepsilon}{\ensuremath{\widetilde{\upepsilon}}\xspace} +\providecommand{\tzeta}{\ensuremath{\widetilde{\upzeta}}\xspace} +\providecommand{\teta}{\ensuremath{\widetilde{\upeta}}\xspace} +\providecommand{\ttheta}{\ensuremath{\widetilde{\uptheta}}\xspace} +\providecommand{\tiota}{\ensuremath{\widetilde{\upiota}}\xspace} +\providecommand{\tkappa}{\ensuremath{\widetilde{\upkappa}}\xspace} +\providecommand{\tlambda}{\ensuremath{\widetilde{\uplambda}}\xspace} +\providecommand{\tmu}{\ensuremath{\widetilde{\upmu}}\xspace} +\providecommand{\tnu}{\ensuremath{\widetilde{\upnu}}\xspace} +\providecommand{\txi}{\ensuremath{\widetilde{\upxi}}\xspace} +\providecommand{\tpi}{\ensuremath{\widetilde{\uppi}}\xspace} +\providecommand{\trho}{\ensuremath{\widetilde{\uprho}}\xspace} +\providecommand{\tsigma}{\ensuremath{\widetilde{\upsigma}}\xspace} +\providecommand{\ttau}{\ensuremath{\widetilde{\uptau}}\xspace} +\providecommand{\tupsilon}{\ensuremath{\widetilde{\upupsilon}}\xspace} +\providecommand{\tphi}{\ensuremath{\widetilde{\upphi}}\xspace} +\providecommand{\tchi}{\ensuremath{\widetilde{\upchi}}\xspace} +\providecommand{\tpsi}{\ensuremath{\widetilde{\uppsi}}\xspace} +\providecommand{\tomega}{\ensuremath{\widetilde{\upomega}}\xspace} +\providecommand{\tvarepsilon}{\ensuremath{\widetilde{\upvarepsilon}}\xspace} +\providecommand{\tvartheta}{\ensuremath{\widetilde{\upvartheta}}\xspace} +\providecommand{\tvarpi}{\ensuremath{\widetilde{\upvarpi}}\xspace} +\providecommand{\tvarphi}{\ensuremath{\widetilde{\upvarphi}}\xspace} +\providecommand{\tGamma}{\ensuremath{\widetilde{\Upgamma}}\xspace} +\providecommand{\tDelta}{\ensuremath{\widetilde{\Updelta}}\xspace} +\providecommand{\tTheta}{\ensuremath{\widetilde{\Uptheta}}\xspace} +\providecommand{\tLambda}{\ensuremath{\widetilde{\Uplambda}}\xspace} +\providecommand{\tXi}{\ensuremath{\widetilde{\Upxi}}\xspace} +\providecommand{\tPi}{\ensuremath{\widetilde{\Uppi}}\xspace} +\providecommand{\tSigma}{\ensuremath{\widetilde{\Upsigma}}\xspace} +\providecommand{\tUpsilon}{\ensuremath{\widetilde{\Upupsilon}}\xspace} +\providecommand{\tPhi}{\ensuremath{\widetilde{\Upphi}}\xspace} +\providecommand{\tPsi}{\ensuremath{\widetilde{\Uppsi}}\xspace} +\providecommand{\tOmega}{\ensuremath{\widetilde{\Upomega}}\xspace} + +\providecommand{\halpha}{\ensuremath{\widehat{\upalpha}}\xspace} +\providecommand{\hbeta}{\ensuremath{\widehat{\upbeta}}\xspace} +\providecommand{\hgamma}{\ensuremath{\widehat{\upgamma}}\xspace} +\providecommand{\hdelta}{\ensuremath{\widehat{\updelta}}\xspace} +\providecommand{\hepsilon}{\ensuremath{\widehat{\upepsilon}}\xspace} +\providecommand{\hzeta}{\ensuremath{\widehat{\upzeta}}\xspace} +\providecommand{\heta}{\ensuremath{\widehat{\upeta}}\xspace} +\providecommand{\htheta}{\ensuremath{\widehat{\uptheta}}\xspace} +\providecommand{\hiota}{\ensuremath{\widehat{\upiota}}\xspace} +\providecommand{\hkappa}{\ensuremath{\widehat{\upkappa}}\xspace} +\providecommand{\hlambda}{\ensuremath{\widehat{\uplambda}}\xspace} +\providecommand{\hmu}{\ensuremath{\widehat{\upmu}}\xspace} +\providecommand{\hnu}{\ensuremath{\widehat{\upnu}}\xspace} +\providecommand{\hxi}{\ensuremath{\widehat{\upxi}}\xspace} +\providecommand{\hpi}{\ensuremath{\widehat{\uppi}}\xspace} +\providecommand{\hrho}{\ensuremath{\widehat{\uprho}}\xspace} +\providecommand{\hsigma}{\ensuremath{\widehat{\upsigma}}\xspace} +\providecommand{\htau}{\ensuremath{\widehat{\uptau}}\xspace} +\providecommand{\hupsilon}{\ensuremath{\widehat{\upupsilon}}\xspace} +\providecommand{\hphi}{\ensuremath{\widehat{\upphi}}\xspace} +\providecommand{\hchi}{\ensuremath{\widehat{\upchi}}\xspace} +\providecommand{\hpsi}{\ensuremath{\widehat{\uppsi}}\xspace} +\providecommand{\homega}{\ensuremath{\widehat{\upomega}}\xspace} +\providecommand{\hvarepsilon}{\ensuremath{\widehat{\upvarepsilon}}\xspace} +\providecommand{\hvartheta}{\ensuremath{\widehat{\upvartheta}}\xspace} +\providecommand{\hvarpi}{\ensuremath{\widehat{\upvarpi}}\xspace} +\providecommand{\hvarphi}{\ensuremath{\widehat{\upvarphi}}\xspace} +\providecommand{\hGamma}{\ensuremath{\widehat{\Upgamma}}\xspace} +\providecommand{\hDelta}{\ensuremath{\widehat{\Updelta}}\xspace} +\providecommand{\hTheta}{\ensuremath{\widehat{\Uptheta}}\xspace} +\providecommand{\hLambda}{\ensuremath{\widehat{\Uplambda}}\xspace} +\providecommand{\hXi}{\ensuremath{\widehat{\Upxi}}\xspace} +\providecommand{\hPi}{\ensuremath{\widehat{\Uppi}}\xspace} +\providecommand{\hSigma}{\ensuremath{\widehat{\Upsigma}}\xspace} +\providecommand{\hUpsilon}{\ensuremath{\widehat{\Upupsilon}}\xspace} +\providecommand{\hPhi}{\ensuremath{\widehat{\Upphi}}\xspace} +\providecommand{\hPsi}{\ensuremath{\widehat{\Uppsi}}\xspace} +\providecommand{\hOmega}{\ensuremath{\widehat{\Upomega}}\xspace} %---- numerical sets @@ -95,6 +262,220 @@ \providecommand{\Z}{\ensuremath{\mathds{Z}}\xspace} \providecommand{\C}{\ensuremath{\mathds{C}}\xspace} +%---- dot, bar, tilde, hat + +\providecommand{\da}{\ensuremath{\dot{a}}\xspace} +\providecommand{\db}{\ensuremath{\dot{b}}\xspace} +\providecommand{\dc}{\ensuremath{\dot{c}}\xspace} +\providecommand{\dd}{\ensuremath{\dot{d}}\xspace} +\providecommand{\de}{\ensuremath{\dot{e}}\xspace} +\providecommand{\df}{\ensuremath{\dot{f}}\xspace} +\providecommand{\dg}{\ensuremath{\dot{g}}\xspace} +\providecommand{\dh}{\ensuremath{\dot{h}}\xspace} +\providecommand{\di}{\ensuremath{\dot{i}}\xspace} +\providecommand{\dj}{\ensuremath{\dot{j}}\xspace} +\providecommand{\dk}{\ensuremath{\dot{k}}\xspace} +\providecommand{\dl}{\ensuremath{\dot{l}}\xspace} +\providecommand{\dm}{\ensuremath{\dot{m}}\xspace} +\providecommand{\dn}{\ensuremath{\dot{n}}\xspace} +\providecommand{\do}{\ensuremath{\dot{o}}\xspace} +\providecommand{\dp}{\ensuremath{\dot{p}}\xspace} +\providecommand{\dq}{\ensuremath{\dot{q}}\xspace} +\providecommand{\dr}{\ensuremath{\dot{r}}\xspace} +\providecommand{\ds}{\ensuremath{\dot{s}}\xspace} +\providecommand{\dt}{\ensuremath{\dot{t}}\xspace} +\providecommand{\du}{\ensuremath{\dot{u}}\xspace} +\providecommand{\dv}{\ensuremath{\dot{v}}\xspace} +\providecommand{\dw}{\ensuremath{\dot{w}}\xspace} +\providecommand{\dx}{\ensuremath{\dot{x}}\xspace} +\providecommand{\dy}{\ensuremath{\dot{y}}\xspace} +\providecommand{\dz}{\ensuremath{\dot{z}}\xspace} +\providecommand{\dA}{\ensuremath{\dot{A}}\xspace} +\providecommand{\dB}{\ensuremath{\dot{B}}\xspace} +\providecommand{\dC}{\ensuremath{\dot{C}}\xspace} +\providecommand{\dD}{\ensuremath{\dot{D}}\xspace} +\providecommand{\dE}{\ensuremath{\dot{E}}\xspace} +\providecommand{\dF}{\ensuremath{\dot{F}}\xspace} +\providecommand{\dG}{\ensuremath{\dot{G}}\xspace} +\providecommand{\dH}{\ensuremath{\dot{H}}\xspace} +\providecommand{\dI}{\ensuremath{\dot{I}}\xspace} +\providecommand{\dJ}{\ensuremath{\dot{J}}\xspace} +\providecommand{\dK}{\ensuremath{\dot{K}}\xspace} +\providecommand{\dL}{\ensuremath{\dot{L}}\xspace} +\providecommand{\dM}{\ensuremath{\dot{M}}\xspace} +\providecommand{\dN}{\ensuremath{\dot{N}}\xspace} +\providecommand{\dO}{\ensuremath{\dot{O}}\xspace} +\providecommand{\dP}{\ensuremath{\dot{P}}\xspace} +\providecommand{\dQ}{\ensuremath{\dot{Q}}\xspace} +\providecommand{\dR}{\ensuremath{\dot{R}}\xspace} +\providecommand{\dS}{\ensuremath{\dot{S}}\xspace} +\providecommand{\dT}{\ensuremath{\dot{T}}\xspace} +\providecommand{\dU}{\ensuremath{\dot{U}}\xspace} +\providecommand{\dV}{\ensuremath{\dot{V}}\xspace} +\providecommand{\dW}{\ensuremath{\dot{W}}\xspace} +\providecommand{\dX}{\ensuremath{\dot{X}}\xspace} +\providecommand{\dY}{\ensuremath{\dot{Y}}\xspace} +\providecommand{\dZ}{\ensuremath{\dot{Z}}\xspace} + +\providecommand{\bara}{\ensuremath{\overline{a}}\xspace} +\providecommand{\barb}{\ensuremath{\overline{b}}\xspace} +\providecommand{\barc}{\ensuremath{\overline{c}}\xspace} +\providecommand{\bard}{\ensuremath{\overline{d}}\xspace} +\providecommand{\bare}{\ensuremath{\overline{e}}\xspace} +\providecommand{\barf}{\ensuremath{\overline{f}}\xspace} +\providecommand{\barg}{\ensuremath{\overline{g}}\xspace} +\providecommand{\barh}{\ensuremath{\overline{h}}\xspace} +\providecommand{\bari}{\ensuremath{\overline{i}}\xspace} +\providecommand{\barj}{\ensuremath{\overline{j}}\xspace} +\providecommand{\bark}{\ensuremath{\overline{k}}\xspace} +\providecommand{\barl}{\ensuremath{\overline{l}}\xspace} +\providecommand{\barm}{\ensuremath{\overline{m}}\xspace} +\providecommand{\barn}{\ensuremath{\overline{n}}\xspace} +\providecommand{\baro}{\ensuremath{\overline{o}}\xspace} +\providecommand{\barp}{\ensuremath{\overline{p}}\xspace} +\providecommand{\barq}{\ensuremath{\overline{q}}\xspace} +\providecommand{\barr}{\ensuremath{\overline{r}}\xspace} +\providecommand{\bars}{\ensuremath{\overline{s}}\xspace} +\providecommand{\bart}{\ensuremath{\overline{t}}\xspace} +\providecommand{\baru}{\ensuremath{\overline{u}}\xspace} +\providecommand{\barv}{\ensuremath{\overline{v}}\xspace} +\providecommand{\barw}{\ensuremath{\overline{w}}\xspace} +\providecommand{\barx}{\ensuremath{\overline{x}}\xspace} +\providecommand{\bary}{\ensuremath{\overline{y}}\xspace} +\providecommand{\barz}{\ensuremath{\overline{z}}\xspace} +\providecommand{\barA}{\ensuremath{\overline{A}}\xspace} +\providecommand{\barB}{\ensuremath{\overline{B}}\xspace} +\providecommand{\barC}{\ensuremath{\overline{C}}\xspace} +\providecommand{\barD}{\ensuremath{\overline{D}}\xspace} +\providecommand{\barE}{\ensuremath{\overline{E}}\xspace} +\providecommand{\barF}{\ensuremath{\overline{F}}\xspace} +\providecommand{\barG}{\ensuremath{\overline{G}}\xspace} +\providecommand{\barH}{\ensuremath{\overline{H}}\xspace} +\providecommand{\barI}{\ensuremath{\overline{I}}\xspace} +\providecommand{\barJ}{\ensuremath{\overline{J}}\xspace} +\providecommand{\barK}{\ensuremath{\overline{K}}\xspace} +\providecommand{\barL}{\ensuremath{\overline{L}}\xspace} +\providecommand{\barM}{\ensuremath{\overline{M}}\xspace} +\providecommand{\barN}{\ensuremath{\overline{N}}\xspace} +\providecommand{\barO}{\ensuremath{\overline{O}}\xspace} +\providecommand{\barP}{\ensuremath{\overline{P}}\xspace} +\providecommand{\barQ}{\ensuremath{\overline{Q}}\xspace} +\providecommand{\barR}{\ensuremath{\overline{R}}\xspace} +\providecommand{\barS}{\ensuremath{\overline{S}}\xspace} +\providecommand{\barT}{\ensuremath{\overline{T}}\xspace} +\providecommand{\barU}{\ensuremath{\overline{U}}\xspace} +\providecommand{\barV}{\ensuremath{\overline{V}}\xspace} +\providecommand{\barW}{\ensuremath{\overline{W}}\xspace} +\providecommand{\barX}{\ensuremath{\overline{X}}\xspace} +\providecommand{\barY}{\ensuremath{\overline{Y}}\xspace} +\providecommand{\barZ}{\ensuremath{\overline{Z}}\xspace} + +\providecommand{\tildea}{\ensuremath{\widetilde{a}}\xspace} +\providecommand{\tildeb}{\ensuremath{\widetilde{b}}\xspace} +\providecommand{\tildec}{\ensuremath{\widetilde{c}}\xspace} +\providecommand{\tilded}{\ensuremath{\widetilde{d}}\xspace} +\providecommand{\tildee}{\ensuremath{\widetilde{e}}\xspace} +\providecommand{\tildef}{\ensuremath{\widetilde{f}}\xspace} +\providecommand{\tildeg}{\ensuremath{\widetilde{g}}\xspace} +\providecommand{\tildeh}{\ensuremath{\widetilde{h}}\xspace} +\providecommand{\tildei}{\ensuremath{\widetilde{i}}\xspace} +\providecommand{\tildej}{\ensuremath{\widetilde{j}}\xspace} +\providecommand{\tildek}{\ensuremath{\widetilde{k}}\xspace} +\providecommand{\tildel}{\ensuremath{\widetilde{l}}\xspace} +\providecommand{\tildem}{\ensuremath{\widetilde{m}}\xspace} +\providecommand{\tilden}{\ensuremath{\widetilde{n}}\xspace} +\providecommand{\tildeo}{\ensuremath{\widetilde{o}}\xspace} +\providecommand{\tildep}{\ensuremath{\widetilde{p}}\xspace} +\providecommand{\tildeq}{\ensuremath{\widetilde{q}}\xspace} +\providecommand{\tilder}{\ensuremath{\widetilde{r}}\xspace} +\providecommand{\tildes}{\ensuremath{\widetilde{s}}\xspace} +\providecommand{\tildet}{\ensuremath{\widetilde{t}}\xspace} +\providecommand{\tildeu}{\ensuremath{\widetilde{u}}\xspace} +\providecommand{\tildev}{\ensuremath{\widetilde{v}}\xspace} +\providecommand{\tildew}{\ensuremath{\widetilde{w}}\xspace} +\providecommand{\tildex}{\ensuremath{\widetilde{x}}\xspace} +\providecommand{\tildey}{\ensuremath{\widetilde{y}}\xspace} +\providecommand{\tildez}{\ensuremath{\widetilde{z}}\xspace} +\providecommand{\tildeA}{\ensuremath{\widetilde{A}}\xspace} +\providecommand{\tildeB}{\ensuremath{\widetilde{B}}\xspace} +\providecommand{\tildeC}{\ensuremath{\widetilde{C}}\xspace} +\providecommand{\tildeD}{\ensuremath{\widetilde{D}}\xspace} +\providecommand{\tildeE}{\ensuremath{\widetilde{E}}\xspace} +\providecommand{\tildeF}{\ensuremath{\widetilde{F}}\xspace} +\providecommand{\tildeG}{\ensuremath{\widetilde{G}}\xspace} +\providecommand{\tildeH}{\ensuremath{\widetilde{H}}\xspace} +\providecommand{\tildeI}{\ensuremath{\widetilde{I}}\xspace} +\providecommand{\tildeJ}{\ensuremath{\widetilde{J}}\xspace} +\providecommand{\tildeK}{\ensuremath{\widetilde{K}}\xspace} +\providecommand{\tildeL}{\ensuremath{\widetilde{L}}\xspace} +\providecommand{\tildeM}{\ensuremath{\widetilde{M}}\xspace} +\providecommand{\tildeN}{\ensuremath{\widetilde{N}}\xspace} +\providecommand{\tildeO}{\ensuremath{\widetilde{O}}\xspace} +\providecommand{\tildeP}{\ensuremath{\widetilde{P}}\xspace} +\providecommand{\tildeQ}{\ensuremath{\widetilde{Q}}\xspace} +\providecommand{\tildeR}{\ensuremath{\widetilde{R}}\xspace} +\providecommand{\tildeS}{\ensuremath{\widetilde{S}}\xspace} +\providecommand{\tildeT}{\ensuremath{\widetilde{T}}\xspace} +\providecommand{\tildeU}{\ensuremath{\widetilde{U}}\xspace} +\providecommand{\tildeV}{\ensuremath{\widetilde{V}}\xspace} +\providecommand{\tildeW}{\ensuremath{\widetilde{W}}\xspace} +\providecommand{\tildeX}{\ensuremath{\widetilde{X}}\xspace} +\providecommand{\tildeY}{\ensuremath{\widetilde{Y}}\xspace} +\providecommand{\tildeZ}{\ensuremath{\widetilde{Z}}\xspace} + +\providecommand{\hata}{\ensuremath{\widehat{a}}\xspace} +\providecommand{\hatb}{\ensuremath{\widehat{b}}\xspace} +\providecommand{\hatc}{\ensuremath{\widehat{c}}\xspace} +\providecommand{\hatd}{\ensuremath{\widehat{d}}\xspace} +\providecommand{\hate}{\ensuremath{\widehat{e}}\xspace} +\providecommand{\hatf}{\ensuremath{\widehat{f}}\xspace} +\providecommand{\hatg}{\ensuremath{\widehat{g}}\xspace} +\providecommand{\hath}{\ensuremath{\widehat{h}}\xspace} +\providecommand{\hati}{\ensuremath{\widehat{i}}\xspace} +\providecommand{\hatj}{\ensuremath{\widehat{j}}\xspace} +\providecommand{\hatk}{\ensuremath{\widehat{k}}\xspace} +\providecommand{\hatl}{\ensuremath{\widehat{l}}\xspace} +\providecommand{\hatm}{\ensuremath{\widehat{m}}\xspace} +\providecommand{\hatn}{\ensuremath{\widehat{n}}\xspace} +\providecommand{\hato}{\ensuremath{\widehat{o}}\xspace} +\providecommand{\hatp}{\ensuremath{\widehat{p}}\xspace} +\providecommand{\hatq}{\ensuremath{\widehat{q}}\xspace} +\providecommand{\hatr}{\ensuremath{\widehat{r}}\xspace} +\providecommand{\hats}{\ensuremath{\widehat{s}}\xspace} +\providecommand{\hatt}{\ensuremath{\widehat{t}}\xspace} +\providecommand{\hatu}{\ensuremath{\widehat{u}}\xspace} +\providecommand{\hatv}{\ensuremath{\widehat{v}}\xspace} +\providecommand{\hatw}{\ensuremath{\widehat{w}}\xspace} +\providecommand{\hatx}{\ensuremath{\widehat{x}}\xspace} +\providecommand{\haty}{\ensuremath{\widehat{y}}\xspace} +\providecommand{\hatz}{\ensuremath{\widehat{z}}\xspace} +\providecommand{\hatA}{\ensuremath{\widehat{A}}\xspace} +\providecommand{\hatB}{\ensuremath{\widehat{B}}\xspace} +\providecommand{\hatC}{\ensuremath{\widehat{C}}\xspace} +\providecommand{\hatD}{\ensuremath{\widehat{D}}\xspace} +\providecommand{\hatE}{\ensuremath{\widehat{E}}\xspace} +\providecommand{\hatF}{\ensuremath{\widehat{F}}\xspace} +\providecommand{\hatG}{\ensuremath{\widehat{G}}\xspace} +\providecommand{\hatH}{\ensuremath{\widehat{H}}\xspace} +\providecommand{\hatI}{\ensuremath{\widehat{I}}\xspace} +\providecommand{\hatJ}{\ensuremath{\widehat{J}}\xspace} +\providecommand{\hatK}{\ensuremath{\widehat{K}}\xspace} +\providecommand{\hatL}{\ensuremath{\widehat{L}}\xspace} +\providecommand{\hatM}{\ensuremath{\widehat{M}}\xspace} +\providecommand{\hatN}{\ensuremath{\widehat{N}}\xspace} +\providecommand{\hatO}{\ensuremath{\widehat{O}}\xspace} +\providecommand{\hatP}{\ensuremath{\widehat{P}}\xspace} +\providecommand{\hatQ}{\ensuremath{\widehat{Q}}\xspace} +\providecommand{\hatR}{\ensuremath{\widehat{R}}\xspace} +\providecommand{\hatS}{\ensuremath{\widehat{S}}\xspace} +\providecommand{\hatT}{\ensuremath{\widehat{T}}\xspace} +\providecommand{\hatU}{\ensuremath{\widehat{U}}\xspace} +\providecommand{\hatV}{\ensuremath{\widehat{V}}\xspace} +\providecommand{\hatW}{\ensuremath{\widehat{W}}\xspace} +\providecommand{\hatX}{\ensuremath{\widehat{X}}\xspace} +\providecommand{\hatY}{\ensuremath{\widehat{Y}}\xspace} +\providecommand{\hatZ}{\ensuremath{\widehat{Z}}\xspace} + %---- calligraphic letters \providecommand{\cA}{\ensuremath{\mathcal{A}}\xspace} @@ -151,6 +532,222 @@ \providecommand{\ccY}{\ensuremath{\mathscr{Y}}\xspace} \providecommand{\ccZ}{\ensuremath{\mathscr{Z}}\xspace} +\providecommand{\dcA}{\ensuremath{\dot{\mathcal{A}}}\xspace} +\providecommand{\dcB}{\ensuremath{\dot{\mathcal{B}}}\xspace} +\providecommand{\dcC}{\ensuremath{\dot{\mathcal{C}}}\xspace} +\providecommand{\dcD}{\ensuremath{\dot{\mathcal{D}}}\xspace} +\providecommand{\dcE}{\ensuremath{\dot{\mathcal{E}}}\xspace} +\providecommand{\dcF}{\ensuremath{\dot{\mathcal{F}}}\xspace} +\providecommand{\dcG}{\ensuremath{\dot{\mathcal{G}}}\xspace} +\providecommand{\dcH}{\ensuremath{\dot{\mathcal{H}}}\xspace} +\providecommand{\dcI}{\ensuremath{\dot{\mathcal{I}}}\xspace} +\providecommand{\dcJ}{\ensuremath{\dot{\mathcal{J}}}\xspace} +\providecommand{\dcK}{\ensuremath{\dot{\mathcal{K}}}\xspace} +\providecommand{\dcL}{\ensuremath{\dot{\mathcal{L}}}\xspace} +\providecommand{\dcM}{\ensuremath{\dot{\mathcal{M}}}\xspace} +\providecommand{\dcN}{\ensuremath{\dot{\mathcal{N}}}\xspace} +\providecommand{\dcO}{\ensuremath{\dot{\mathcal{O}}}\xspace} +\providecommand{\dcP}{\ensuremath{\dot{\mathcal{P}}}\xspace} +\providecommand{\dcQ}{\ensuremath{\dot{\mathcal{Q}}}\xspace} +\providecommand{\dcR}{\ensuremath{\dot{\mathcal{R}}}\xspace} +\providecommand{\dcS}{\ensuremath{\dot{\mathcal{S}}}\xspace} +\providecommand{\dcT}{\ensuremath{\dot{\mathcal{T}}}\xspace} +\providecommand{\dcU}{\ensuremath{\dot{\mathcal{U}}}\xspace} +\providecommand{\dcV}{\ensuremath{\dot{\mathcal{V}}}\xspace} +\providecommand{\dcW}{\ensuremath{\dot{\mathcal{W}}}\xspace} +\providecommand{\dcX}{\ensuremath{\dot{\mathcal{X}}}\xspace} +\providecommand{\dcY}{\ensuremath{\dot{\mathcal{Y}}}\xspace} +\providecommand{\dcZ}{\ensuremath{\dot{\mathcal{Z}}}\xspace} + +\providecommand{\dccA}{\ensuremath{\dot{\mathscr{A}}}\xspace} +\providecommand{\dccB}{\ensuremath{\dot{\mathscr{B}}}\xspace} +\providecommand{\dccC}{\ensuremath{\dot{\mathscr{C}}}\xspace} +\providecommand{\dccD}{\ensuremath{\dot{\mathscr{D}}}\xspace} +\providecommand{\dccE}{\ensuremath{\dot{\mathscr{E}}}\xspace} +\providecommand{\dccF}{\ensuremath{\dot{\mathscr{F}}}\xspace} +\providecommand{\dccG}{\ensuremath{\dot{\mathscr{G}}}\xspace} +\providecommand{\dccH}{\ensuremath{\dot{\mathscr{H}}}\xspace} +\providecommand{\dccI}{\ensuremath{\dot{\mathscr{I}}}\xspace} +\providecommand{\dccJ}{\ensuremath{\dot{\mathscr{J}}}\xspace} +\providecommand{\dccK}{\ensuremath{\dot{\mathscr{K}}}\xspace} +\providecommand{\dccL}{\ensuremath{\dot{\mathscr{L}}}\xspace} +\providecommand{\dccM}{\ensuremath{\dot{\mathscr{M}}}\xspace} +\providecommand{\dccN}{\ensuremath{\dot{\mathscr{N}}}\xspace} +\providecommand{\dccO}{\ensuremath{\dot{\mathscr{O}}}\xspace} +\providecommand{\dccP}{\ensuremath{\dot{\mathscr{P}}}\xspace} +\providecommand{\dccQ}{\ensuremath{\dot{\mathscr{Q}}}\xspace} +\providecommand{\dccR}{\ensuremath{\dot{\mathscr{R}}}\xspace} +\providecommand{\dccS}{\ensuremath{\dot{\mathscr{S}}}\xspace} +\providecommand{\dccT}{\ensuremath{\dot{\mathscr{T}}}\xspace} +\providecommand{\dccU}{\ensuremath{\dot{\mathscr{U}}}\xspace} +\providecommand{\dccV}{\ensuremath{\dot{\mathscr{V}}}\xspace} +\providecommand{\dccW}{\ensuremath{\dot{\mathscr{W}}}\xspace} +\providecommand{\dccX}{\ensuremath{\dot{\mathscr{X}}}\xspace} +\providecommand{\dccY}{\ensuremath{\dot{\mathscr{Y}}}\xspace} +\providecommand{\dccZ}{\ensuremath{\dot{\mathscr{Z}}}\xspace} + +\providecommand{\tcA}{\ensuremath{\widetilde{\mathcal{A}}}\xspace} +\providecommand{\tcB}{\ensuremath{\widetilde{\mathcal{B}}}\xspace} +\providecommand{\tcC}{\ensuremath{\widetilde{\mathcal{C}}}\xspace} +\providecommand{\tcD}{\ensuremath{\widetilde{\mathcal{D}}}\xspace} +\providecommand{\tcE}{\ensuremath{\widetilde{\mathcal{E}}}\xspace} +\providecommand{\tcF}{\ensuremath{\widetilde{\mathcal{F}}}\xspace} +\providecommand{\tcG}{\ensuremath{\widetilde{\mathcal{G}}}\xspace} +\providecommand{\tcH}{\ensuremath{\widetilde{\mathcal{H}}}\xspace} +\providecommand{\tcI}{\ensuremath{\widetilde{\mathcal{I}}}\xspace} +\providecommand{\tcJ}{\ensuremath{\widetilde{\mathcal{J}}}\xspace} +\providecommand{\tcK}{\ensuremath{\widetilde{\mathcal{K}}}\xspace} +\providecommand{\tcL}{\ensuremath{\widetilde{\mathcal{L}}}\xspace} +\providecommand{\tcM}{\ensuremath{\widetilde{\mathcal{M}}}\xspace} +\providecommand{\tcN}{\ensuremath{\widetilde{\mathcal{N}}}\xspace} +\providecommand{\tcO}{\ensuremath{\widetilde{\mathcal{O}}}\xspace} +\providecommand{\tcP}{\ensuremath{\widetilde{\mathcal{P}}}\xspace} +\providecommand{\tcQ}{\ensuremath{\widetilde{\mathcal{Q}}}\xspace} +\providecommand{\tcR}{\ensuremath{\widetilde{\mathcal{R}}}\xspace} +\providecommand{\tcS}{\ensuremath{\widetilde{\mathcal{S}}}\xspace} +\providecommand{\tcT}{\ensuremath{\widetilde{\mathcal{T}}}\xspace} +\providecommand{\tcU}{\ensuremath{\widetilde{\mathcal{U}}}\xspace} +\providecommand{\tcV}{\ensuremath{\widetilde{\mathcal{V}}}\xspace} +\providecommand{\tcW}{\ensuremath{\widetilde{\mathcal{W}}}\xspace} +\providecommand{\tcX}{\ensuremath{\widetilde{\mathcal{X}}}\xspace} +\providecommand{\tcY}{\ensuremath{\widetilde{\mathcal{Y}}}\xspace} +\providecommand{\tcZ}{\ensuremath{\widetilde{\mathcal{Z}}}\xspace} + +\providecommand{\tccA}{\ensuremath{\widetilde{\mathscr{A}}}\xspace} +\providecommand{\tccB}{\ensuremath{\widetilde{\mathscr{B}}}\xspace} +\providecommand{\tccC}{\ensuremath{\widetilde{\mathscr{C}}}\xspace} +\providecommand{\tccD}{\ensuremath{\widetilde{\mathscr{D}}}\xspace} +\providecommand{\tccE}{\ensuremath{\widetilde{\mathscr{E}}}\xspace} +\providecommand{\tccF}{\ensuremath{\widetilde{\mathscr{F}}}\xspace} +\providecommand{\tccG}{\ensuremath{\widetilde{\mathscr{G}}}\xspace} +\providecommand{\tccH}{\ensuremath{\widetilde{\mathscr{H}}}\xspace} +\providecommand{\tccI}{\ensuremath{\widetilde{\mathscr{I}}}\xspace} +\providecommand{\tccJ}{\ensuremath{\widetilde{\mathscr{J}}}\xspace} +\providecommand{\tccK}{\ensuremath{\widetilde{\mathscr{K}}}\xspace} +\providecommand{\tccL}{\ensuremath{\widetilde{\mathscr{L}}}\xspace} +\providecommand{\tccM}{\ensuremath{\widetilde{\mathscr{M}}}\xspace} +\providecommand{\tccN}{\ensuremath{\widetilde{\mathscr{N}}}\xspace} +\providecommand{\tccO}{\ensuremath{\widetilde{\mathscr{O}}}\xspace} +\providecommand{\tccP}{\ensuremath{\widetilde{\mathscr{P}}}\xspace} +\providecommand{\tccQ}{\ensuremath{\widetilde{\mathscr{Q}}}\xspace} +\providecommand{\tccR}{\ensuremath{\widetilde{\mathscr{R}}}\xspace} +\providecommand{\tccS}{\ensuremath{\widetilde{\mathscr{S}}}\xspace} +\providecommand{\tccT}{\ensuremath{\widetilde{\mathscr{T}}}\xspace} +\providecommand{\tccU}{\ensuremath{\widetilde{\mathscr{U}}}\xspace} +\providecommand{\tccV}{\ensuremath{\widetilde{\mathscr{V}}}\xspace} +\providecommand{\tccW}{\ensuremath{\widetilde{\mathscr{W}}}\xspace} +\providecommand{\tccX}{\ensuremath{\widetilde{\mathscr{X}}}\xspace} +\providecommand{\tccY}{\ensuremath{\widetilde{\mathscr{Y}}}\xspace} +\providecommand{\tccZ}{\ensuremath{\widetilde{\mathscr{Z}}}\xspace} + +\providecommand{\hcA}{\ensuremath{\widehat{\mathcal{A}}}\xspace} +\providecommand{\hcB}{\ensuremath{\widehat{\mathcal{B}}}\xspace} +\providecommand{\hcC}{\ensuremath{\widehat{\mathcal{C}}}\xspace} +\providecommand{\hcD}{\ensuremath{\widehat{\mathcal{D}}}\xspace} +\providecommand{\hcE}{\ensuremath{\widehat{\mathcal{E}}}\xspace} +\providecommand{\hcF}{\ensuremath{\widehat{\mathcal{F}}}\xspace} +\providecommand{\hcG}{\ensuremath{\widehat{\mathcal{G}}}\xspace} +\providecommand{\hcH}{\ensuremath{\widehat{\mathcal{H}}}\xspace} +\providecommand{\hcI}{\ensuremath{\widehat{\mathcal{I}}}\xspace} +\providecommand{\hcJ}{\ensuremath{\widehat{\mathcal{J}}}\xspace} +\providecommand{\hcK}{\ensuremath{\widehat{\mathcal{K}}}\xspace} +\providecommand{\hcL}{\ensuremath{\widehat{\mathcal{L}}}\xspace} +\providecommand{\hcM}{\ensuremath{\widehat{\mathcal{M}}}\xspace} +\providecommand{\hcN}{\ensuremath{\widehat{\mathcal{N}}}\xspace} +\providecommand{\hcO}{\ensuremath{\widehat{\mathcal{O}}}\xspace} +\providecommand{\hcP}{\ensuremath{\widehat{\mathcal{P}}}\xspace} +\providecommand{\hcQ}{\ensuremath{\widehat{\mathcal{Q}}}\xspace} +\providecommand{\hcR}{\ensuremath{\widehat{\mathcal{R}}}\xspace} +\providecommand{\hcS}{\ensuremath{\widehat{\mathcal{S}}}\xspace} +\providecommand{\hcT}{\ensuremath{\widehat{\mathcal{T}}}\xspace} +\providecommand{\hcU}{\ensuremath{\widehat{\mathcal{U}}}\xspace} +\providecommand{\hcV}{\ensuremath{\widehat{\mathcal{V}}}\xspace} +\providecommand{\hcW}{\ensuremath{\widehat{\mathcal{W}}}\xspace} +\providecommand{\hcX}{\ensuremath{\widehat{\mathcal{X}}}\xspace} +\providecommand{\hcY}{\ensuremath{\widehat{\mathcal{Y}}}\xspace} +\providecommand{\hcZ}{\ensuremath{\widehat{\mathcal{Z}}}\xspace} + +\providecommand{\hccA}{\ensuremath{\widehat{\mathscr{A}}}\xspace} +\providecommand{\hccB}{\ensuremath{\widehat{\mathscr{B}}}\xspace} +\providecommand{\hccC}{\ensuremath{\widehat{\mathscr{C}}}\xspace} +\providecommand{\hccD}{\ensuremath{\widehat{\mathscr{D}}}\xspace} +\providecommand{\hccE}{\ensuremath{\widehat{\mathscr{E}}}\xspace} +\providecommand{\hccF}{\ensuremath{\widehat{\mathscr{F}}}\xspace} +\providecommand{\hccG}{\ensuremath{\widehat{\mathscr{G}}}\xspace} +\providecommand{\hccH}{\ensuremath{\widehat{\mathscr{H}}}\xspace} +\providecommand{\hccI}{\ensuremath{\widehat{\mathscr{I}}}\xspace} +\providecommand{\hccJ}{\ensuremath{\widehat{\mathscr{J}}}\xspace} +\providecommand{\hccK}{\ensuremath{\widehat{\mathscr{K}}}\xspace} +\providecommand{\hccL}{\ensuremath{\widehat{\mathscr{L}}}\xspace} +\providecommand{\hccM}{\ensuremath{\widehat{\mathscr{M}}}\xspace} +\providecommand{\hccN}{\ensuremath{\widehat{\mathscr{N}}}\xspace} +\providecommand{\hccO}{\ensuremath{\widehat{\mathscr{O}}}\xspace} +\providecommand{\hccP}{\ensuremath{\widehat{\mathscr{P}}}\xspace} +\providecommand{\hccQ}{\ensuremath{\widehat{\mathscr{Q}}}\xspace} +\providecommand{\hccR}{\ensuremath{\widehat{\mathscr{R}}}\xspace} +\providecommand{\hccS}{\ensuremath{\widehat{\mathscr{S}}}\xspace} +\providecommand{\hccT}{\ensuremath{\widehat{\mathscr{T}}}\xspace} +\providecommand{\hccU}{\ensuremath{\widehat{\mathscr{U}}}\xspace} +\providecommand{\hccV}{\ensuremath{\widehat{\mathscr{V}}}\xspace} +\providecommand{\hccW}{\ensuremath{\widehat{\mathscr{W}}}\xspace} +\providecommand{\hccX}{\ensuremath{\widehat{\mathscr{X}}}\xspace} +\providecommand{\hccY}{\ensuremath{\widehat{\mathscr{Y}}}\xspace} +\providecommand{\hccZ}{\ensuremath{\widehat{\mathscr{Z}}}\xspace} + +\providecommand{\bcA}{\ensuremath{\overline{\mathcal{A}}}\xspace} +\providecommand{\bcB}{\ensuremath{\overline{\mathcal{B}}}\xspace} +\providecommand{\bcC}{\ensuremath{\overline{\mathcal{C}}}\xspace} +\providecommand{\bcD}{\ensuremath{\overline{\mathcal{D}}}\xspace} +\providecommand{\bcE}{\ensuremath{\overline{\mathcal{E}}}\xspace} +\providecommand{\bcF}{\ensuremath{\overline{\mathcal{F}}}\xspace} +\providecommand{\bcG}{\ensuremath{\overline{\mathcal{G}}}\xspace} +\providecommand{\bcH}{\ensuremath{\overline{\mathcal{H}}}\xspace} +\providecommand{\bcI}{\ensuremath{\overline{\mathcal{I}}}\xspace} +\providecommand{\bcJ}{\ensuremath{\overline{\mathcal{J}}}\xspace} +\providecommand{\bcK}{\ensuremath{\overline{\mathcal{K}}}\xspace} +\providecommand{\bcL}{\ensuremath{\overline{\mathcal{L}}}\xspace} +\providecommand{\bcM}{\ensuremath{\overline{\mathcal{M}}}\xspace} +\providecommand{\bcN}{\ensuremath{\overline{\mathcal{N}}}\xspace} +\providecommand{\bcO}{\ensuremath{\overline{\mathcal{O}}}\xspace} +\providecommand{\bcP}{\ensuremath{\overline{\mathcal{P}}}\xspace} +\providecommand{\bcQ}{\ensuremath{\overline{\mathcal{Q}}}\xspace} +\providecommand{\bcR}{\ensuremath{\overline{\mathcal{R}}}\xspace} +\providecommand{\bcS}{\ensuremath{\overline{\mathcal{S}}}\xspace} +\providecommand{\bcT}{\ensuremath{\overline{\mathcal{T}}}\xspace} +\providecommand{\bcU}{\ensuremath{\overline{\mathcal{U}}}\xspace} +\providecommand{\bcV}{\ensuremath{\overline{\mathcal{V}}}\xspace} +\providecommand{\bcW}{\ensuremath{\overline{\mathcal{W}}}\xspace} +\providecommand{\bcX}{\ensuremath{\overline{\mathcal{X}}}\xspace} +\providecommand{\bcY}{\ensuremath{\overline{\mathcal{Y}}}\xspace} +\providecommand{\bcZ}{\ensuremath{\overline{\mathcal{Z}}}\xspace} + +\providecommand{\bccA}{\ensuremath{\overline{\mathscr{A}}}\xspace} +\providecommand{\bccB}{\ensuremath{\overline{\mathscr{B}}}\xspace} +\providecommand{\bccC}{\ensuremath{\overline{\mathscr{C}}}\xspace} +\providecommand{\bccD}{\ensuremath{\overline{\mathscr{D}}}\xspace} +\providecommand{\bccE}{\ensuremath{\overline{\mathscr{E}}}\xspace} +\providecommand{\bccF}{\ensuremath{\overline{\mathscr{F}}}\xspace} +\providecommand{\bccG}{\ensuremath{\overline{\mathscr{G}}}\xspace} +\providecommand{\bccH}{\ensuremath{\overline{\mathscr{H}}}\xspace} +\providecommand{\bccI}{\ensuremath{\overline{\mathscr{I}}}\xspace} +\providecommand{\bccJ}{\ensuremath{\overline{\mathscr{J}}}\xspace} +\providecommand{\bccK}{\ensuremath{\overline{\mathscr{K}}}\xspace} +\providecommand{\bccL}{\ensuremath{\overline{\mathscr{L}}}\xspace} +\providecommand{\bccM}{\ensuremath{\overline{\mathscr{M}}}\xspace} +\providecommand{\bccN}{\ensuremath{\overline{\mathscr{N}}}\xspace} +\providecommand{\bccO}{\ensuremath{\overline{\mathscr{O}}}\xspace} +\providecommand{\bccP}{\ensuremath{\overline{\mathscr{P}}}\xspace} +\providecommand{\bccQ}{\ensuremath{\overline{\mathscr{Q}}}\xspace} +\providecommand{\bccR}{\ensuremath{\overline{\mathscr{R}}}\xspace} +\providecommand{\bccS}{\ensuremath{\overline{\mathscr{S}}}\xspace} +\providecommand{\bccT}{\ensuremath{\overline{\mathscr{T}}}\xspace} +\providecommand{\bccU}{\ensuremath{\overline{\mathscr{U}}}\xspace} +\providecommand{\bccV}{\ensuremath{\overline{\mathscr{V}}}\xspace} +\providecommand{\bccW}{\ensuremath{\overline{\mathscr{W}}}\xspace} +\providecommand{\bccX}{\ensuremath{\overline{\mathscr{X}}}\xspace} +\providecommand{\bccY}{\ensuremath{\overline{\mathscr{Y}}}\xspace} +\providecommand{\bccZ}{\ensuremath{\overline{\mathscr{Z}}}\xspace} + %---- roman letters \providecommand{\rA}{\ensuremath{\mathrm{A}}\xspace} @@ -180,6 +777,114 @@ \providecommand{\rY}{\ensuremath{\mathrm{Y}}\xspace} \providecommand{\rZ}{\ensuremath{\mathrm{Z}}\xspace} +\providecommand{\drA}{\ensuremath{\dot{\mathrm{A}}}\xspace} +\providecommand{\drB}{\ensuremath{\dot{\mathrm{B}}}\xspace} +\providecommand{\drC}{\ensuremath{\dot{\mathrm{C}}}\xspace} +\providecommand{\drD}{\ensuremath{\dot{\mathrm{D}}}\xspace} +\providecommand{\drE}{\ensuremath{\dot{\mathrm{E}}}\xspace} +\providecommand{\drF}{\ensuremath{\dot{\mathrm{F}}}\xspace} +\providecommand{\drG}{\ensuremath{\dot{\mathrm{G}}}\xspace} +\providecommand{\drH}{\ensuremath{\dot{\mathrm{H}}}\xspace} +\providecommand{\drI}{\ensuremath{\dot{\mathrm{I}}}\xspace} +\providecommand{\drJ}{\ensuremath{\dot{\mathrm{J}}}\xspace} +\providecommand{\drK}{\ensuremath{\dot{\mathrm{K}}}\xspace} +\providecommand{\drL}{\ensuremath{\dot{\mathrm{L}}}\xspace} +\providecommand{\drM}{\ensuremath{\dot{\mathrm{M}}}\xspace} +\providecommand{\drN}{\ensuremath{\dot{\mathrm{N}}}\xspace} +\providecommand{\drO}{\ensuremath{\dot{\mathrm{O}}}\xspace} +\providecommand{\drP}{\ensuremath{\dot{\mathrm{P}}}\xspace} +\providecommand{\drQ}{\ensuremath{\dot{\mathrm{Q}}}\xspace} +\providecommand{\drR}{\ensuremath{\dot{\mathrm{R}}}\xspace} +\providecommand{\drS}{\ensuremath{\dot{\mathrm{S}}}\xspace} +\providecommand{\drT}{\ensuremath{\dot{\mathrm{T}}}\xspace} +\providecommand{\drU}{\ensuremath{\dot{\mathrm{U}}}\xspace} +\providecommand{\drV}{\ensuremath{\dot{\mathrm{V}}}\xspace} +\providecommand{\drW}{\ensuremath{\dot{\mathrm{W}}}\xspace} +\providecommand{\drX}{\ensuremath{\dot{\mathrm{X}}}\xspace} +\providecommand{\drY}{\ensuremath{\dot{\mathrm{Y}}}\xspace} +\providecommand{\drZ}{\ensuremath{\dot{\mathrm{Z}}}\xspace} + +\providecommand{\trA}{\ensuremath{\widetilde{\mathrm{A}}}\xspace} +\providecommand{\trB}{\ensuremath{\widetilde{\mathrm{B}}}\xspace} +\providecommand{\trC}{\ensuremath{\widetilde{\mathrm{C}}}\xspace} +\providecommand{\trD}{\ensuremath{\widetilde{\mathrm{D}}}\xspace} +\providecommand{\trE}{\ensuremath{\widetilde{\mathrm{E}}}\xspace} +\providecommand{\trF}{\ensuremath{\widetilde{\mathrm{F}}}\xspace} +\providecommand{\trG}{\ensuremath{\widetilde{\mathrm{G}}}\xspace} +\providecommand{\trH}{\ensuremath{\widetilde{\mathrm{H}}}\xspace} +\providecommand{\trI}{\ensuremath{\widetilde{\mathrm{I}}}\xspace} +\providecommand{\trJ}{\ensuremath{\widetilde{\mathrm{J}}}\xspace} +\providecommand{\trK}{\ensuremath{\widetilde{\mathrm{K}}}\xspace} +\providecommand{\trL}{\ensuremath{\widetilde{\mathrm{L}}}\xspace} +\providecommand{\trM}{\ensuremath{\widetilde{\mathrm{M}}}\xspace} +\providecommand{\trN}{\ensuremath{\widetilde{\mathrm{N}}}\xspace} +\providecommand{\trO}{\ensuremath{\widetilde{\mathrm{O}}}\xspace} +\providecommand{\trP}{\ensuremath{\widetilde{\mathrm{P}}}\xspace} +\providecommand{\trQ}{\ensuremath{\widetilde{\mathrm{Q}}}\xspace} +\providecommand{\trR}{\ensuremath{\widetilde{\mathrm{R}}}\xspace} +\providecommand{\trS}{\ensuremath{\widetilde{\mathrm{S}}}\xspace} +\providecommand{\trT}{\ensuremath{\widetilde{\mathrm{T}}}\xspace} +\providecommand{\trU}{\ensuremath{\widetilde{\mathrm{U}}}\xspace} +\providecommand{\trV}{\ensuremath{\widetilde{\mathrm{V}}}\xspace} +\providecommand{\trW}{\ensuremath{\widetilde{\mathrm{W}}}\xspace} +\providecommand{\trX}{\ensuremath{\widetilde{\mathrm{X}}}\xspace} +\providecommand{\trY}{\ensuremath{\widetilde{\mathrm{Y}}}\xspace} +\providecommand{\trZ}{\ensuremath{\widetilde{\mathrm{Z}}}\xspace} + +\providecommand{\hrA}{\ensuremath{\widehat{\mathrm{A}}}\xspace} +\providecommand{\hrB}{\ensuremath{\widehat{\mathrm{B}}}\xspace} +\providecommand{\hrC}{\ensuremath{\widehat{\mathrm{C}}}\xspace} +\providecommand{\hrD}{\ensuremath{\widehat{\mathrm{D}}}\xspace} +\providecommand{\hrE}{\ensuremath{\widehat{\mathrm{E}}}\xspace} +\providecommand{\hrF}{\ensuremath{\widehat{\mathrm{F}}}\xspace} +\providecommand{\hrG}{\ensuremath{\widehat{\mathrm{G}}}\xspace} +\providecommand{\hrH}{\ensuremath{\widehat{\mathrm{H}}}\xspace} +\providecommand{\hrI}{\ensuremath{\widehat{\mathrm{I}}}\xspace} +\providecommand{\hrJ}{\ensuremath{\widehat{\mathrm{J}}}\xspace} +\providecommand{\hrK}{\ensuremath{\widehat{\mathrm{K}}}\xspace} +\providecommand{\hrL}{\ensuremath{\widehat{\mathrm{L}}}\xspace} +\providecommand{\hrM}{\ensuremath{\widehat{\mathrm{M}}}\xspace} +\providecommand{\hrN}{\ensuremath{\widehat{\mathrm{N}}}\xspace} +\providecommand{\hrO}{\ensuremath{\widehat{\mathrm{O}}}\xspace} +\providecommand{\hrP}{\ensuremath{\widehat{\mathrm{P}}}\xspace} +\providecommand{\hrQ}{\ensuremath{\widehat{\mathrm{Q}}}\xspace} +\providecommand{\hrR}{\ensuremath{\widehat{\mathrm{R}}}\xspace} +\providecommand{\hrS}{\ensuremath{\widehat{\mathrm{S}}}\xspace} +\providecommand{\hrT}{\ensuremath{\widehat{\mathrm{T}}}\xspace} +\providecommand{\hrU}{\ensuremath{\widehat{\mathrm{U}}}\xspace} +\providecommand{\hrV}{\ensuremath{\widehat{\mathrm{V}}}\xspace} +\providecommand{\hrW}{\ensuremath{\widehat{\mathrm{W}}}\xspace} +\providecommand{\hrX}{\ensuremath{\widehat{\mathrm{X}}}\xspace} +\providecommand{\hrY}{\ensuremath{\widehat{\mathrm{Y}}}\xspace} +\providecommand{\hrZ}{\ensuremath{\widehat{\mathrm{Z}}}\xspace} + +\providecommand{\brA}{\ensuremath{\overline{\mathrm{A}}}\xspace} +\providecommand{\brB}{\ensuremath{\overline{\mathrm{B}}}\xspace} +\providecommand{\brC}{\ensuremath{\overline{\mathrm{C}}}\xspace} +\providecommand{\brD}{\ensuremath{\overline{\mathrm{D}}}\xspace} +\providecommand{\brE}{\ensuremath{\overline{\mathrm{E}}}\xspace} +\providecommand{\brF}{\ensuremath{\overline{\mathrm{F}}}\xspace} +\providecommand{\brG}{\ensuremath{\overline{\mathrm{G}}}\xspace} +\providecommand{\brH}{\ensuremath{\overline{\mathrm{H}}}\xspace} +\providecommand{\brI}{\ensuremath{\overline{\mathrm{I}}}\xspace} +\providecommand{\brJ}{\ensuremath{\overline{\mathrm{J}}}\xspace} +\providecommand{\brK}{\ensuremath{\overline{\mathrm{K}}}\xspace} +\providecommand{\brL}{\ensuremath{\overline{\mathrm{L}}}\xspace} +\providecommand{\brM}{\ensuremath{\overline{\mathrm{M}}}\xspace} +\providecommand{\brN}{\ensuremath{\overline{\mathrm{N}}}\xspace} +\providecommand{\brO}{\ensuremath{\overline{\mathrm{O}}}\xspace} +\providecommand{\brP}{\ensuremath{\overline{\mathrm{P}}}\xspace} +\providecommand{\brQ}{\ensuremath{\overline{\mathrm{Q}}}\xspace} +\providecommand{\brR}{\ensuremath{\overline{\mathrm{R}}}\xspace} +\providecommand{\brS}{\ensuremath{\overline{\mathrm{S}}}\xspace} +\providecommand{\brT}{\ensuremath{\overline{\mathrm{T}}}\xspace} +\providecommand{\brU}{\ensuremath{\overline{\mathrm{U}}}\xspace} +\providecommand{\brV}{\ensuremath{\overline{\mathrm{V}}}\xspace} +\providecommand{\brW}{\ensuremath{\overline{\mathrm{W}}}\xspace} +\providecommand{\brX}{\ensuremath{\overline{\mathrm{X}}}\xspace} +\providecommand{\brY}{\ensuremath{\overline{\mathrm{Y}}}\xspace} +\providecommand{\brZ}{\ensuremath{\overline{\mathrm{Z}}}\xspace} + %---- frak letters \providecommand{\ffa}{\ensuremath{\mathfrak{a}}\xspace} @@ -235,6 +940,218 @@ \providecommand{\ffY}{\ensuremath{\mathfrak{Y}}\xspace} \providecommand{\ffZ}{\ensuremath{\mathfrak{Z}}\xspace} +\providecommand{\dffa}{\ensuremath{\dot{\mathfrak{a}}}\xspace} +\providecommand{\dffb}{\ensuremath{\dot{\mathfrak{b}}}\xspace} +\providecommand{\dffc}{\ensuremath{\dot{\mathfrak{c}}}\xspace} +\providecommand{\dffd}{\ensuremath{\dot{\mathfrak{d}}}\xspace} +\providecommand{\dffe}{\ensuremath{\dot{\mathfrak{e}}}\xspace} +\providecommand{\dfff}{\ensuremath{\dot{\mathfrak{f}}}\xspace} +\providecommand{\dffg}{\ensuremath{\dot{\mathfrak{g}}}\xspace} +\providecommand{\dffh}{\ensuremath{\dot{\mathfrak{h}}}\xspace} +\providecommand{\dffi}{\ensuremath{\dot{\mathfrak{i}}}\xspace} +\providecommand{\dffj}{\ensuremath{\dot{\mathfrak{j}}}\xspace} +\providecommand{\dffk}{\ensuremath{\dot{\mathfrak{k}}}\xspace} +\providecommand{\dffl}{\ensuremath{\dot{\mathfrak{l}}}\xspace} +\providecommand{\dffm}{\ensuremath{\dot{\mathfrak{m}}}\xspace} +\providecommand{\dffn}{\ensuremath{\dot{\mathfrak{n}}}\xspace} +\providecommand{\dffo}{\ensuremath{\dot{\mathfrak{o}}}\xspace} +\providecommand{\dffp}{\ensuremath{\dot{\mathfrak{p}}}\xspace} +\providecommand{\dffq}{\ensuremath{\dot{\mathfrak{q}}}\xspace} +\providecommand{\dffr}{\ensuremath{\dot{\mathfrak{r}}}\xspace} +\providecommand{\dffs}{\ensuremath{\dot{\mathfrak{s}}}\xspace} +\providecommand{\dfft}{\ensuremath{\dot{\mathfrak{t}}}\xspace} +\providecommand{\dffu}{\ensuremath{\dot{\mathfrak{u}}}\xspace} +\providecommand{\dffv}{\ensuremath{\dot{\mathfrak{v}}}\xspace} +\providecommand{\dffw}{\ensuremath{\dot{\mathfrak{w}}}\xspace} +\providecommand{\dffx}{\ensuremath{\dot{\mathfrak{x}}}\xspace} +\providecommand{\dffy}{\ensuremath{\dot{\mathfrak{y}}}\xspace} +\providecommand{\dffz}{\ensuremath{\dot{\mathfrak{z}}}\xspace} +\providecommand{\dffA}{\ensuremath{\dot{\mathfrak{A}}}\xspace} +\providecommand{\dffB}{\ensuremath{\dot{\mathfrak{B}}}\xspace} +\providecommand{\dffC}{\ensuremath{\dot{\mathfrak{C}}}\xspace} +\providecommand{\dffD}{\ensuremath{\dot{\mathfrak{D}}}\xspace} +\providecommand{\dffE}{\ensuremath{\dot{\mathfrak{E}}}\xspace} +\providecommand{\dffF}{\ensuremath{\dot{\mathfrak{F}}}\xspace} +\providecommand{\dffG}{\ensuremath{\dot{\mathfrak{G}}}\xspace} +\providecommand{\dffH}{\ensuremath{\dot{\mathfrak{H}}}\xspace} +\providecommand{\dffI}{\ensuremath{\dot{\mathfrak{I}}}\xspace} +\providecommand{\dffJ}{\ensuremath{\dot{\mathfrak{J}}}\xspace} +\providecommand{\dffK}{\ensuremath{\dot{\mathfrak{K}}}\xspace} +\providecommand{\dffL}{\ensuremath{\dot{\mathfrak{L}}}\xspace} +\providecommand{\dffM}{\ensuremath{\dot{\mathfrak{M}}}\xspace} +\providecommand{\dffN}{\ensuremath{\dot{\mathfrak{N}}}\xspace} +\providecommand{\dffO}{\ensuremath{\dot{\mathfrak{O}}}\xspace} +\providecommand{\dffP}{\ensuremath{\dot{\mathfrak{P}}}\xspace} +\providecommand{\dffQ}{\ensuremath{\dot{\mathfrak{Q}}}\xspace} +\providecommand{\dffR}{\ensuremath{\dot{\mathfrak{R}}}\xspace} +\providecommand{\dffS}{\ensuremath{\dot{\mathfrak{S}}}\xspace} +\providecommand{\dffT}{\ensuremath{\dot{\mathfrak{T}}}\xspace} +\providecommand{\dffU}{\ensuremath{\dot{\mathfrak{U}}}\xspace} +\providecommand{\dffV}{\ensuremath{\dot{\mathfrak{V}}}\xspace} +\providecommand{\dffW}{\ensuremath{\dot{\mathfrak{W}}}\xspace} +\providecommand{\dffX}{\ensuremath{\dot{\mathfrak{X}}}\xspace} +\providecommand{\dffY}{\ensuremath{\dot{\mathfrak{Y}}}\xspace} +\providecommand{\dffZ}{\ensuremath{\dot{\mathfrak{Z}}}\xspace} + +\providecommand{\tffa}{\ensuremath{\widetilde{\mathfrak{a}}}\xspace} +\providecommand{\tffb}{\ensuremath{\widetilde{\mathfrak{b}}}\xspace} +\providecommand{\tffc}{\ensuremath{\widetilde{\mathfrak{c}}}\xspace} +\providecommand{\tffd}{\ensuremath{\widetilde{\mathfrak{d}}}\xspace} +\providecommand{\tffe}{\ensuremath{\widetilde{\mathfrak{e}}}\xspace} +\providecommand{\tfff}{\ensuremath{\widetilde{\mathfrak{f}}}\xspace} +\providecommand{\tffg}{\ensuremath{\widetilde{\mathfrak{g}}}\xspace} +\providecommand{\tffh}{\ensuremath{\widetilde{\mathfrak{h}}}\xspace} +\providecommand{\tffi}{\ensuremath{\widetilde{\mathfrak{i}}}\xspace} +\providecommand{\tffj}{\ensuremath{\widetilde{\mathfrak{j}}}\xspace} +\providecommand{\tffk}{\ensuremath{\widetilde{\mathfrak{k}}}\xspace} +\providecommand{\tffl}{\ensuremath{\widetilde{\mathfrak{l}}}\xspace} +\providecommand{\tffm}{\ensuremath{\widetilde{\mathfrak{m}}}\xspace} +\providecommand{\tffn}{\ensuremath{\widetilde{\mathfrak{n}}}\xspace} +\providecommand{\tffo}{\ensuremath{\widetilde{\mathfrak{o}}}\xspace} +\providecommand{\tffp}{\ensuremath{\widetilde{\mathfrak{p}}}\xspace} +\providecommand{\tffq}{\ensuremath{\widetilde{\mathfrak{q}}}\xspace} +\providecommand{\tffr}{\ensuremath{\widetilde{\mathfrak{r}}}\xspace} +\providecommand{\tffs}{\ensuremath{\widetilde{\mathfrak{s}}}\xspace} +\providecommand{\tfft}{\ensuremath{\widetilde{\mathfrak{t}}}\xspace} +\providecommand{\tffu}{\ensuremath{\widetilde{\mathfrak{u}}}\xspace} +\providecommand{\tffv}{\ensuremath{\widetilde{\mathfrak{v}}}\xspace} +\providecommand{\tffw}{\ensuremath{\widetilde{\mathfrak{w}}}\xspace} +\providecommand{\tffx}{\ensuremath{\widetilde{\mathfrak{x}}}\xspace} +\providecommand{\tffy}{\ensuremath{\widetilde{\mathfrak{y}}}\xspace} +\providecommand{\tffz}{\ensuremath{\widetilde{\mathfrak{z}}}\xspace} +\providecommand{\tffA}{\ensuremath{\widetilde{\mathfrak{A}}}\xspace} +\providecommand{\tffB}{\ensuremath{\widetilde{\mathfrak{B}}}\xspace} +\providecommand{\tffC}{\ensuremath{\widetilde{\mathfrak{C}}}\xspace} +\providecommand{\tffD}{\ensuremath{\widetilde{\mathfrak{D}}}\xspace} +\providecommand{\tffE}{\ensuremath{\widetilde{\mathfrak{E}}}\xspace} +\providecommand{\tffF}{\ensuremath{\widetilde{\mathfrak{F}}}\xspace} +\providecommand{\tffG}{\ensuremath{\widetilde{\mathfrak{G}}}\xspace} +\providecommand{\tffH}{\ensuremath{\widetilde{\mathfrak{H}}}\xspace} +\providecommand{\tffI}{\ensuremath{\widetilde{\mathfrak{I}}}\xspace} +\providecommand{\tffJ}{\ensuremath{\widetilde{\mathfrak{J}}}\xspace} +\providecommand{\tffK}{\ensuremath{\widetilde{\mathfrak{K}}}\xspace} +\providecommand{\tffL}{\ensuremath{\widetilde{\mathfrak{L}}}\xspace} +\providecommand{\tffM}{\ensuremath{\widetilde{\mathfrak{M}}}\xspace} +\providecommand{\tffN}{\ensuremath{\widetilde{\mathfrak{N}}}\xspace} +\providecommand{\tffO}{\ensuremath{\widetilde{\mathfrak{O}}}\xspace} +\providecommand{\tffP}{\ensuremath{\widetilde{\mathfrak{P}}}\xspace} +\providecommand{\tffQ}{\ensuremath{\widetilde{\mathfrak{Q}}}\xspace} +\providecommand{\tffR}{\ensuremath{\widetilde{\mathfrak{R}}}\xspace} +\providecommand{\tffS}{\ensuremath{\widetilde{\mathfrak{S}}}\xspace} +\providecommand{\tffT}{\ensuremath{\widetilde{\mathfrak{T}}}\xspace} +\providecommand{\tffU}{\ensuremath{\widetilde{\mathfrak{U}}}\xspace} +\providecommand{\tffV}{\ensuremath{\widetilde{\mathfrak{V}}}\xspace} +\providecommand{\tffW}{\ensuremath{\widetilde{\mathfrak{W}}}\xspace} +\providecommand{\tffX}{\ensuremath{\widetilde{\mathfrak{X}}}\xspace} +\providecommand{\tffY}{\ensuremath{\widetilde{\mathfrak{Y}}}\xspace} +\providecommand{\tffZ}{\ensuremath{\widetilde{\mathfrak{Z}}}\xspace} + +\providecommand{\hffa}{\ensuremath{\widehat{\mathfrak{a}}}\xspace} +\providecommand{\hffb}{\ensuremath{\widehat{\mathfrak{b}}}\xspace} +\providecommand{\hffc}{\ensuremath{\widehat{\mathfrak{c}}}\xspace} +\providecommand{\hffd}{\ensuremath{\widehat{\mathfrak{d}}}\xspace} +\providecommand{\hffe}{\ensuremath{\widehat{\mathfrak{e}}}\xspace} +\providecommand{\hfff}{\ensuremath{\widehat{\mathfrak{f}}}\xspace} +\providecommand{\hffg}{\ensuremath{\widehat{\mathfrak{g}}}\xspace} +\providecommand{\hffh}{\ensuremath{\widehat{\mathfrak{h}}}\xspace} +\providecommand{\hffi}{\ensuremath{\widehat{\mathfrak{i}}}\xspace} +\providecommand{\hffj}{\ensuremath{\widehat{\mathfrak{j}}}\xspace} +\providecommand{\hffk}{\ensuremath{\widehat{\mathfrak{k}}}\xspace} +\providecommand{\hffl}{\ensuremath{\widehat{\mathfrak{l}}}\xspace} +\providecommand{\hffm}{\ensuremath{\widehat{\mathfrak{m}}}\xspace} +\providecommand{\hffn}{\ensuremath{\widehat{\mathfrak{n}}}\xspace} +\providecommand{\hffo}{\ensuremath{\widehat{\mathfrak{o}}}\xspace} +\providecommand{\hffp}{\ensuremath{\widehat{\mathfrak{p}}}\xspace} +\providecommand{\hffq}{\ensuremath{\widehat{\mathfrak{q}}}\xspace} +\providecommand{\hffr}{\ensuremath{\widehat{\mathfrak{r}}}\xspace} +\providecommand{\hffs}{\ensuremath{\widehat{\mathfrak{s}}}\xspace} +\providecommand{\hfft}{\ensuremath{\widehat{\mathfrak{t}}}\xspace} +\providecommand{\hffu}{\ensuremath{\widehat{\mathfrak{u}}}\xspace} +\providecommand{\hffv}{\ensuremath{\widehat{\mathfrak{v}}}\xspace} +\providecommand{\hffw}{\ensuremath{\widehat{\mathfrak{w}}}\xspace} +\providecommand{\hffx}{\ensuremath{\widehat{\mathfrak{x}}}\xspace} +\providecommand{\hffy}{\ensuremath{\widehat{\mathfrak{y}}}\xspace} +\providecommand{\hffz}{\ensuremath{\widehat{\mathfrak{z}}}\xspace} +\providecommand{\hffA}{\ensuremath{\widehat{\mathfrak{A}}}\xspace} +\providecommand{\hffB}{\ensuremath{\widehat{\mathfrak{B}}}\xspace} +\providecommand{\hffC}{\ensuremath{\widehat{\mathfrak{C}}}\xspace} +\providecommand{\hffD}{\ensuremath{\widehat{\mathfrak{D}}}\xspace} +\providecommand{\hffE}{\ensuremath{\widehat{\mathfrak{E}}}\xspace} +\providecommand{\hffF}{\ensuremath{\widehat{\mathfrak{F}}}\xspace} +\providecommand{\hffG}{\ensuremath{\widehat{\mathfrak{G}}}\xspace} +\providecommand{\hffH}{\ensuremath{\widehat{\mathfrak{H}}}\xspace} +\providecommand{\hffI}{\ensuremath{\widehat{\mathfrak{I}}}\xspace} +\providecommand{\hffJ}{\ensuremath{\widehat{\mathfrak{J}}}\xspace} +\providecommand{\hffK}{\ensuremath{\widehat{\mathfrak{K}}}\xspace} +\providecommand{\hffL}{\ensuremath{\widehat{\mathfrak{L}}}\xspace} +\providecommand{\hffM}{\ensuremath{\widehat{\mathfrak{M}}}\xspace} +\providecommand{\hffN}{\ensuremath{\widehat{\mathfrak{N}}}\xspace} +\providecommand{\hffO}{\ensuremath{\widehat{\mathfrak{O}}}\xspace} +\providecommand{\hffP}{\ensuremath{\widehat{\mathfrak{P}}}\xspace} +\providecommand{\hffQ}{\ensuremath{\widehat{\mathfrak{Q}}}\xspace} +\providecommand{\hffR}{\ensuremath{\widehat{\mathfrak{R}}}\xspace} +\providecommand{\hffS}{\ensuremath{\widehat{\mathfrak{S}}}\xspace} +\providecommand{\hffT}{\ensuremath{\widehat{\mathfrak{T}}}\xspace} +\providecommand{\hffU}{\ensuremath{\widehat{\mathfrak{U}}}\xspace} +\providecommand{\hffV}{\ensuremath{\widehat{\mathfrak{V}}}\xspace} +\providecommand{\hffW}{\ensuremath{\widehat{\mathfrak{W}}}\xspace} +\providecommand{\hffX}{\ensuremath{\widehat{\mathfrak{X}}}\xspace} +\providecommand{\hffY}{\ensuremath{\widehat{\mathfrak{Y}}}\xspace} +\providecommand{\hffZ}{\ensuremath{\widehat{\mathfrak{Z}}}\xspace} + +\providecommand{\bffa}{\ensuremath{\overline{\mathfrak{a}}}\xspace} +\providecommand{\bffb}{\ensuremath{\overline{\mathfrak{b}}}\xspace} +\providecommand{\bffc}{\ensuremath{\overline{\mathfrak{c}}}\xspace} +\providecommand{\bffd}{\ensuremath{\overline{\mathfrak{d}}}\xspace} +\providecommand{\bffe}{\ensuremath{\overline{\mathfrak{e}}}\xspace} +\providecommand{\bfff}{\ensuremath{\overline{\mathfrak{f}}}\xspace} +\providecommand{\bffg}{\ensuremath{\overline{\mathfrak{g}}}\xspace} +\providecommand{\bffh}{\ensuremath{\overline{\mathfrak{h}}}\xspace} +\providecommand{\bffi}{\ensuremath{\overline{\mathfrak{i}}}\xspace} +\providecommand{\bffj}{\ensuremath{\overline{\mathfrak{j}}}\xspace} +\providecommand{\bffk}{\ensuremath{\overline{\mathfrak{k}}}\xspace} +\providecommand{\bffl}{\ensuremath{\overline{\mathfrak{l}}}\xspace} +\providecommand{\bffm}{\ensuremath{\overline{\mathfrak{m}}}\xspace} +\providecommand{\bffn}{\ensuremath{\overline{\mathfrak{n}}}\xspace} +\providecommand{\bffo}{\ensuremath{\overline{\mathfrak{o}}}\xspace} +\providecommand{\bffp}{\ensuremath{\overline{\mathfrak{p}}}\xspace} +\providecommand{\bffq}{\ensuremath{\overline{\mathfrak{q}}}\xspace} +\providecommand{\bffr}{\ensuremath{\overline{\mathfrak{r}}}\xspace} +\providecommand{\bffs}{\ensuremath{\overline{\mathfrak{s}}}\xspace} +\providecommand{\bfft}{\ensuremath{\overline{\mathfrak{t}}}\xspace} +\providecommand{\bffu}{\ensuremath{\overline{\mathfrak{u}}}\xspace} +\providecommand{\bffv}{\ensuremath{\overline{\mathfrak{v}}}\xspace} +\providecommand{\bffw}{\ensuremath{\overline{\mathfrak{w}}}\xspace} +\providecommand{\bffx}{\ensuremath{\overline{\mathfrak{x}}}\xspace} +\providecommand{\bffy}{\ensuremath{\overline{\mathfrak{y}}}\xspace} +\providecommand{\bffz}{\ensuremath{\overline{\mathfrak{z}}}\xspace} +\providecommand{\bffA}{\ensuremath{\overline{\mathfrak{A}}}\xspace} +\providecommand{\bffB}{\ensuremath{\overline{\mathfrak{B}}}\xspace} +\providecommand{\bffC}{\ensuremath{\overline{\mathfrak{C}}}\xspace} +\providecommand{\bffD}{\ensuremath{\overline{\mathfrak{D}}}\xspace} +\providecommand{\bffE}{\ensuremath{\overline{\mathfrak{E}}}\xspace} +\providecommand{\bffF}{\ensuremath{\overline{\mathfrak{F}}}\xspace} +\providecommand{\bffG}{\ensuremath{\overline{\mathfrak{G}}}\xspace} +\providecommand{\bffH}{\ensuremath{\overline{\mathfrak{H}}}\xspace} +\providecommand{\bffI}{\ensuremath{\overline{\mathfrak{I}}}\xspace} +\providecommand{\bffJ}{\ensuremath{\overline{\mathfrak{J}}}\xspace} +\providecommand{\bffK}{\ensuremath{\overline{\mathfrak{K}}}\xspace} +\providecommand{\bffL}{\ensuremath{\overline{\mathfrak{L}}}\xspace} +\providecommand{\bffM}{\ensuremath{\overline{\mathfrak{M}}}\xspace} +\providecommand{\bffN}{\ensuremath{\overline{\mathfrak{N}}}\xspace} +\providecommand{\bffO}{\ensuremath{\overline{\mathfrak{O}}}\xspace} +\providecommand{\bffP}{\ensuremath{\overline{\mathfrak{P}}}\xspace} +\providecommand{\bffQ}{\ensuremath{\overline{\mathfrak{Q}}}\xspace} +\providecommand{\bffR}{\ensuremath{\overline{\mathfrak{R}}}\xspace} +\providecommand{\bffS}{\ensuremath{\overline{\mathfrak{S}}}\xspace} +\providecommand{\bffT}{\ensuremath{\overline{\mathfrak{T}}}\xspace} +\providecommand{\bffU}{\ensuremath{\overline{\mathfrak{U}}}\xspace} +\providecommand{\bffV}{\ensuremath{\overline{\mathfrak{V}}}\xspace} +\providecommand{\bffW}{\ensuremath{\overline{\mathfrak{W}}}\xspace} +\providecommand{\bffX}{\ensuremath{\overline{\mathfrak{X}}}\xspace} +\providecommand{\bffY}{\ensuremath{\overline{\mathfrak{Y}}}\xspace} +\providecommand{\bffZ}{\ensuremath{\overline{\mathfrak{Z}}}\xspace} + %---- groups \providecommand{\OO}[1]{\ensuremath{\mathrm{O}(#1)}\xspace} @@ -246,10 +1163,25 @@ %---- algebras \providecommand{\liebraket}[2]{\ensuremath{\left[ #1,\, #2 \right]}} +\providecommand{\no}[1]{\ensuremath{\colon #1 \colon}\xspace} %---- inline partial derivatives (use $\ipd{s}$ for $\partial_s$) \providecommand{\ipd}[1]{\partial_{#1}} +\providecommand{\pd}{\ensuremath{\partial}\xspace} +\providecommand{\bpd}{\ensuremath{\overline{\partial}}\xspace} + +\providecommand{\lrpd}[2]{\frac{\overset{\leftrightarrow}{\partial} #1}{\partial #2}} +\providecommand{\lpd}[2]{\frac{\overset{\leftarrow}{\partial} #1}{\partial #2}} +\providecommand{\rpd}[2]{\frac{\overset{\rightarrow}{\partial} #1}{\partial #2}} + +\providecommand{\lripd}[1]{\overset{\leftrightarrow}{\partial_{#1}}} +\providecommand{\lipd}[1]{\overset{\leftarrow}{\partial_{#1}}} +\providecommand{\ripd}[1]{\overset{\rightarrow}{\partial_{#1}}} + +\providecommand{\lrfdv}[2]{\frac{\overset{\leftrightarrow}{\delta} #1}{\delta #2}} +\providecommand{\lfdv}[2]{\frac{\overset{\leftarrow}{\delta} #1}{\delta #2}} +\providecommand{\rfdv}[2]{\frac{\overset{\rightarrow}{\delta} #1}{\delta #2}} %---- metric (use $\dss[n]{s}$ for $\mathrm{d}s^n$, [n] is optional) diff --git a/sec/app/parameters.tex b/sec/app/parameters.tex index 23cafea..fdd3f2f 100644 --- a/sec/app/parameters.tex +++ b/sec/app/parameters.tex @@ -76,7 +76,7 @@ This would then imply We can finally show in details the computation of the parameters of the basis of hypergeometric functions used in the main text. The relation between these and the \SU{2} matrices can be computed requiring that the monodromies induced by the choice of the parameters equal the monodromies produced by the rotations of the D-branes. -The monodromy in $\omega_{\bt-1} = 0$ is simpler to compute given that we choose $\cL(\vb{n}_{\vb{0}})$ and $\cR(\widetilde{\vb{m}}_{\vb{0}})$ to be diagonal. +The monodromy in $\omega_{\bart-1} = 0$ is simpler to compute given that we choose $\cL(\vb{n}_{\vb{0}})$ and $\cR(\widetilde{\vb{m}}_{\vb{0}})$ to be diagonal. We impose: \begin{eqnarray} \mqty( \dmat{1, e^{-2\pi i c^{(L)}}} ) @@ -163,7 +163,7 @@ We therefore have: k_{a b}\in \Z. \label{eq:aL-bL} \end{equation} -The allowed values for $k_{\delta^{(L)}_{\vb{\infty}}}$ follow a construction similar to the monodromy around $\omega_{\bt-1} = 0$. +The allowed values for $k_{\delta^{(L)}_{\vb{\infty}}}$ follow a construction similar to the monodromy around $\omega_{\bart-1} = 0$. The main difference is given by the fact that $\frac{1}{2}(a^{(L)} + b^{(L)})$ may a priori take values in an interval of width $1$. As in the previous case we have $\alpha \le \delta_{\vb{\infty}}^{(L)} \le \alpha + \frac{1}{2}$ with $\alpha$ technically arbitrary. We cannot thus choose a vanishing $k_{\delta^{(L)}_{\vb{\infty}}}$ but we have to consider $k_{\delta^{(L)}_{\infty}} = 0,\, 1$. @@ -205,7 +205,7 @@ where \frac{n_{\vb{\infty}}^3}{n_{\vb{\infty}}}. \label{eq:cos_n1} \end{equation} -This expression is connected with rotation parameter in the third interaction point $\omega_{\bt+1} = 1$. +This expression is connected with rotation parameter in the third interaction point $\omega_{\bart+1} = 1$. In fact $\cos(2\pi \cA^{(L)}) = \cos(2\pi {n}_{\vb{1}})$. We then write \begin{equation} diff --git a/sec/app/reflection.tex b/sec/app/reflection.tex new file mode 100644 index 0000000..6d52c5d --- /dev/null +++ b/sec/app/reflection.tex @@ -0,0 +1,95 @@ +We provide details on how~\eqref{eq:reflection condition_out_field_generic_vacuum} can be computed. +First we introduce the projector of positive frequency and negative frequency modes for the NS fermion as +\begin{eqnarray} + P^{(+,\, 0)}(z,w) + & = & + \frac{+1}{z-w}, + \qquad + \abs{z} > \abs{w} + \\ + P^{(-,\, 0)}(z,w) + & = & + \frac{-1}{z-w}, + \qquad + \abs{z} < \abs{w}, +\end{eqnarray} +such that +\begin{equation} + \oint\limits_{\abs{z} > \abs{w}} \ddw + P^{(+,\, 0)}(z,w)\, + \Psi^{(0)}( 0 ) + = + \Psi^{(0,\, +)}( z ), +\end{equation} +and similarly for the negative frequency modes. +Likewise we introduce the projectors for the field with defects as +\begin{eqnarray} + P^{(+)}(z,\, w) + & = & + \frac{P\qty(z;\, \qty{x_{(t)},\, \rE_{(t)}} )\, + P\qty(w;\, \qty{x_{(t)},\, -\rE_{(t)}} ) + }{z-w}, + \qquad + \abs{z} > \abs{w} + \\ + P^{(-)}(z,\, w) + & = & + - + \frac{P\qty(z;\, \qty{x_{(t)},\, \rE_{(t)}} )\, + P\qty(w;\, \qty{x_{(t)}, -\rE_{(t)}} ) + }{z-w}, + \qquad + \abs{z} < \abs{w}, +\end{eqnarray} +with $P\qty(z;\, \qty{x_{(t)},\, \rE_{(t)}} ) = \finiteprod{t}{1}{N} \qty( 1- \frac{z}{x_{(t)}} )^{\rE_{(t)}}$ as in the main text. + +We then compute +\begin{equation} + \begin{split} + \qty(P^{(+)}\, P^{(+,\,0)})(z,\, w) + & = + \oint\limits_{\abs{z} > \abs{\zeta} > \abs{w}} \ddz + P^{(+)}(z,\, \zeta)\, + P^{(+,\, 0)}(\zeta,\, w) + = + P^{(+,\, 0)}(z,\, w) + \\ + \qty(P^{(+)}\, P^{(-,\, 0)})(z,\, w) + & = + \frac{P\qty(z;\, \qty{x_{(t)},\, \rE_{(t)}} )\, + P\qty(w;\, \qty{x_{(t)},\, -\rE_{(t)}} ) -1 + }{z-w}. + \end{split} +\end{equation} +The last equation is valid when $\rM=\finitesum{t}{1}{N} \rE_{(t)} \le 0$ and for $\abs{z}$ and $\abs{w}$ arbitrary. +Specializing the previous expressions to $\Psi^{(out)}( z )$, we need to constrain $\abs{z} > x_{(1)}$ and $\abs{w} > x_{(1)}$. + +Finally the vacuum in presence of defects can be described by +\begin{equation} + \begin{split} + \Psi^{(+)}( z ) \Gexcvacket + & = + \qty(P^{(+)}\, \Psi)( z ) \Gexcvacket + \\ + & = + \qty(P^{(+)}\, \Psi^{(out)})( z ) \Gexcvacket + \\ + & = + \left\lbrace + \qty(P^{(+)}\, P^{(+,\, 0)}\, \Psi^{(out)})( z ) + \right. + \\ + & + + \left. + \qty(P^{(+)}\, P^{(-,\, 0)}\, \Psi^{(out)})( z ) + \right\rbrace + \Gexcvacket + \\ + & = + 0, + \end{split} +\end{equation} +where we assumed $\abs{z} > x_{(1)}$. +The expression finally becomes~\eqref{eq:reflection condition_out_field_generic_vacuum}. + +% vim: ft=tex diff --git a/sec/part1/dbranes.tex b/sec/part1/dbranes.tex index 34f4662..2e958d3 100644 --- a/sec/part1/dbranes.tex +++ b/sec/part1/dbranes.tex @@ -152,16 +152,16 @@ We define the usual upper plane coordinates: & \in & \ccH \cup \qty{ z \in \C \mid \Im z = 0 }, \\ - \bu + \baru = x - i y = e^{\tau_E - i \sigma} & \in & - \overline{\ccH} \cup \qty{ z \in \C \mid \Im z = 0 }, + \bccH \cup \qty{ z \in \C \mid \Im z = 0 }, \end{eqnarray} -where $\ccH = \qty{ z \in \C \mid \Im z > 0 }$ is the upper complex plane and $\overline{\ccH} = \qty{ z \in \C \mid \Im z < 0 }$ is the lower complex plane. -In conformal coordinates $u$ and $\bu$, D-branes at $\sigma = 0$ and $\sigma = \pi$ are mapped onto the real axis $\Im z = 0$. +where $\ccH = \qty{ z \in \C \mid \Im z > 0 }$ is the upper complex plane and $\bccH = \qty{ z \in \C \mid \Im z < 0 }$ is the lower complex plane. +In conformal coordinates $u$ and $\baru$, D-branes at $\sigma = 0$ and $\sigma = \pi$ are mapped onto the real axis $\Im z = 0$. We use the symbol $D_{(t)}$ to label both the brane and the interval representing it on the real axis of the upper half plane: \begin{equation} D_{(t)} = \qty[ x_{(t)}, x_{(t-1)} ], @@ -185,8 +185,8 @@ In the global coordinates system associated to the subspace $\R^4 \subset \ccM^{ & = \frac{1}{2 \pi \ap} \iint\limits_{\ccH} - \dd{u} \dd{\bu}\, - \ipd{u} X^I\, \ipd{\bu} X^J\, + \dd{u} \dd{\baru}\, + \ipd{u} X^I\, \ipd{\baru} X^J\, \eta_{IJ} \\ & = @@ -202,10 +202,10 @@ In the global coordinates system associated to the subspace $\R^4 \subset \ccM^{ \end{split} \label{eq:string_action} \end{equation} -where $2\, \ipd{u} = \ipd{x} - i\, \ipd{y}$ and $2\, \ipd{\bu} = \ipd{x} + i\, \ipd{y}$. +where $2\, \ipd{u} = \ipd{x} - i\, \ipd{y}$ and $2\, \ipd{\baru} = \ipd{x} + i\, \ipd{y}$. The \eom in these coordinates are: \begin{equation} - \ipd{u} \ipd{\bu} X^I( u, \bu ) + \ipd{u} \ipd{\baru} X^I( u, \baru ) = \frac{1}{4} \qty( \ipd{x}^2 + \ipd{y}^2 ) X^I( x+iy, x-iy ) @@ -213,13 +213,13 @@ The \eom in these coordinates are: 0. \label{eq:string_equation_of_motion} \end{equation} -Their solution factorises as usual in holomorphic and anti-holomorphic components $X^I( u, \bu ) = X^I( u ) + \bX^I( \bu )$. +Their solution factorises as usual in holomorphic and anti-holomorphic components $X^I( u, \baru ) = X^I( u ) + \barX^I( \baru )$. In the well adapted frame~\eqref{eq:well-adapt-embed} we describe an open string with one of the endpoints on $D_{(t)}$ through the relations: \begin{eqnarray} \eval{\ipd{\sigma} X^i_{(t)}( \tau, \sigma )}_{\sigma = 0} = - \eval{\ipd{y} X^i_{(t)}( u, \bu )}_{y = 0} + \eval{\ipd{y} X^i_{(t)}( u, \baru )}_{y = 0} & = & 0, \qquad @@ -246,7 +246,7 @@ The simpler boundary conditions we consider in the global coordinates are: & = & i\, \tensor{\qty( R_{(t)} )}{^i_J} \qty( - \ipd{u} X^J( x + i\, 0^+ ) - \ipd{\bu} \bX^J( x - i\, 0^+ ) + \ipd{u} X^J( x + i\, 0^+ ) - \ipd{\baru} \barX^J( x - i\, 0^+ ) ) = 0, @@ -256,7 +256,7 @@ The simpler boundary conditions we consider in the global coordinates are: & = & i\, \tensor{\qty( R_{(t)} )}{^m_J} \qty( - \ipd{u} X^J( x + i\, 0^+ ) + \ipd{\bu} \bX^J( x - i\, 0^+ ) + \ipd{u} X^J( x + i\, 0^+ ) + \ipd{\baru} \barX^J( x - i\, 0^+ ) ) = 0, @@ -269,7 +269,7 @@ With the introduction of the target space embedding of the worldsheet interactio \ipd{u} X^I( x + i\, 0^+ ) & = \tensor{\qty( U_{(t)} )}{^I_J} - \ipd{\bu} \bX^J( x - i\, 0^+ ), + \ipd{\baru} \barX^J( x - i\, 0^+ ), \qquad x \in D_{(t)} \\ @@ -333,22 +333,22 @@ Information on $g_{(t)}$ is thus recovered through the global boundary condition \subsubsection{Doubling Trick and Branch Cut Structure} In conformal coordinates we thus introduced the discontinuities~\eqref{eq:discontinuity_bc} across each D-brane which define a non trivial cut structure on the plane. -One way to deal with them is to introduce the \emph{doubling trick} by gluing the relations along an arbitrary but fixed D-brane $D_{(\bt)}$: +One way to deal with them is to introduce the \emph{doubling trick} by gluing the relations along an arbitrary but fixed D-brane $D_{(\bart)}$: \begin{equation} \ipd{z} \cX(z) = \begin{cases} \ipd{u} X(u) & \qif - z = u \qand \Im z > 0 \qor z \in D_{(\bt)} + z = u \qand \Im z > 0 \qor z \in D_{(\bart)} \\ - U_{(\bt)}\, - \ipd{\bu} \bX(\bu) - & \qif z = \bu \qand \Im z < 0 \qor z \in D_{(\bt)} + U_{(\bart)}\, + \ipd{\baru} \barX(\baru) + & \qif z = \baru \qand \Im z < 0 \qor z \in D_{(\bart)} \end{cases}. \label{eq:real_doubling_trick} \end{equation} -Let then $\cU_{(t,\, t+1)} = U_{(t+1)}\, U_{(t)}$ and $\widetilde{\cU}_{(t,\, t+1)} = U_{(\bt)}\, U_{(t)}\, U_{(t+1)}\, U_{(\bt)}$. +Let then $\cU_{(t,\, t+1)} = U_{(t+1)}\, U_{(t)}$ and $\tcU_{(t,\, t+1)} = U_{(\bart)}\, U_{(t)}\, U_{(t+1)}\, U_{(\bart)}$. The boundary conditions in terms of the doubling field are: \begin{eqnarray} \ipd{z} \cX( x_t + e^{2 \pi i}( \eta + i\, 0^+ ) ) @@ -359,27 +359,27 @@ The boundary conditions in terms of the doubling field are: \\ \partial \cX( x_t + e^{2 \pi i}( \eta - i\, 0^+ ) ) & = & - \widetilde{\cU}_{(t,\, t+1)} + \tcU_{(t,\, t+1)} \ipd{z} \cX( x_t + \eta - i\, 0^+ ), \label{eq:bottom_monodromy} \end{eqnarray} for $0 < \eta < \min\qty( \abs{x_{(t-1)} - x_{(t)}}, \abs{x_{(t)} - x_{(t+1)}} )$ in order to consider only the two adjacent D-branes $D_{(t)}$ and $D_{(t+1)}$. -Matrices $\cU_{(t,\, t+1)}$ and $\widetilde{\cU}_{(t,\, t+1)}$ are the non trivial monodromies arising from the rotation of the D-branes. +Matrices $\cU_{(t,\, t+1)}$ and $\tcU_{(t,\, t+1)}$ are the non trivial monodromies arising from the rotation of the D-branes. -Since the relative rotations between consecutive D-branes are non Abelian, for each interaction point there are two monodromies \cU and $\widetilde{\cU}$ depending on the location of the base point of the closed loop: one for paths starting in the upper plane \ccH and one for paths starting in $\overline{\ccH}$. +Since the relative rotations between consecutive D-branes are non Abelian, for each interaction point there are two monodromies \cU and $\tcU$ depending on the location of the base point of the closed loop: one for paths starting in the upper plane \ccH and one for paths starting in $\bccH$. As a consequence of the geometry of the rotations of the D-branes, a path on the complex plane enclosing all of them does not present a monodromy: \begin{equation} \finiteprod{t}{1}{N_B}\, - \cU_{(\bt - t, \bt + 1 - t)} + \cU_{(\bart - t, \bart + 1 - t)} = \finiteprod{t}{1}{N_B}\, - \widetilde{\cU}_{(\bt + t, \bt + 1 + t)} + \tcU_{(\bart + t, \bart + 1 + t)} = \1_4. \label{eq:homotopy_rep} \end{equation} The complex plane has therefore branch cuts running between the D-branes at finite as shown in \Cref{fig:finite_cuts}. -We thus translated the rotations of the D-branes encoded in the matrices $R_{(t)}$ in terms of $\cU_{(t,\, t+1)}$ and $\widetilde{\cU}_{(t,\, t+1)}$ which are matrix representations of the homotopy group of the complex plane with the described branch cut structure. +We thus translated the rotations of the D-branes encoded in the matrices $R_{(t)}$ in terms of $\cU_{(t,\, t+1)}$ and $\tcU_{(t,\, t+1)}$ which are matrix representations of the homotopy group of the complex plane with the described branch cut structure. \begin{figure}[tbp] \centering @@ -408,10 +408,10 @@ In fact we can show that = \frac{1}{4 \pi \ap} \iint\limits_{\C} - \dd{z} \dd{\bz}\, + \dd{z} \dd{\barz}\, \ipd{z} \cX^T(z)\, - U_{(\bt)}\, - \ipd{\bz} \cX(\bz). + U_{(\bart)}\, + \ipd{\barz} \cX(\barz). \end{equation} As a matter of fact the action does not depend on the branch structure of the complex plane. @@ -449,7 +449,7 @@ The task is then to find the parameters of the hypergeometric functions producin We recall some of the properties of the isomorphism~\eqref{eq:su2isomorphism} in~\Cref{sec:isomorphism}. We define the spinor representation of $X$ as: \begin{equation} - X_{(s)}( u, \bu ) = X^I( u, \bu )\, \tau_I, + X_{(s)}( u, \baru ) = X^I( u, \baru )\, \tau_I, \end{equation} where $\tau = \qty( i\, \1_2,\, \vb{\sigma} )$ and $\vb{\sigma}$ is the vector of the Pauli matrices. Consider then: @@ -459,17 +459,17 @@ Consider then: \begin{cases} \ipd{u} X_{(s)}(u) & \qif - z \in \ccH \qor z \in D_{(\bt)} + z \in \ccH \qor z \in D_{(\bart)} \\ - U_{L}(\vb{n}_{(\bt)})\, - \ipd{\bu} X_{(s)}(\bu)\, - U_{R}^{\dagger}(\vb{m}_{(\bt)}) - & \qif z \in \overline{\ccH} \qor z \in D_{(\bt)} + U_{L}(\vb{n}_{(\bart)})\, + \ipd{\baru} X_{(s)}(\baru)\, + U_{R}^{\dagger}(\vb{m}_{(\bart)}) + & \qif z \in \bccH \qor z \in D_{(\bart)} \end{cases}. \label{eq:spinor_doubling_trick} \end{equation} -As in the real representation the discontinuities on the D-branes can be cast into monodromy factors with respect to the D-brane $D_{(\bt)}$. Branch cut structure and considerations on the homotopy group are left unchanged as long as we consider both left and right sectors of $\SU{2}_L \times \SU{2}_R$ at the same time. +As in the real representation the discontinuities on the D-branes can be cast into monodromy factors with respect to the D-brane $D_{(\bart)}$. Branch cut structure and considerations on the homotopy group are left unchanged as long as we consider both left and right sectors of $\SU{2}_L \times \SU{2}_R$ at the same time. Let $0 < \eta < \min\qty( \abs{x_{(t)} - x_{(t-1)}}, \abs{x_{(t+1)} - x_{(t)}} )$. We find: \begin{eqnarray} @@ -482,9 +482,9 @@ We find: \\ \ipd{z} \cX_{(s)}( x_t + e^{2 \pi i}( \eta - i\, 0^+) ) & = & - \widetilde{\cL}_{(t,\, t+1)}\, + \tcL_{(t,\, t+1)}\, \ipd{z} \cX_{(s)}( x_t + \eta - i\, 0^+ )\, - \widetilde{\cR}_{(t,\, t+1)}^{\dagger}, + \tcR_{(t,\, t+1)}^{\dagger}, \label{eq:bottom_spinor_monodromy} \end{eqnarray} where: @@ -494,24 +494,24 @@ We find: U_{L}(\vb{n}_{(t+1)})\, U_{L}^{\dagger}(\vb{n}_{(t)}), \\ - \widetilde{\cL}_{(t,\, t+1)} + \tcL_{(t,\, t+1)} & = & - U_{L}(\vb{n}_{(\bt)})\, + U_{L}(\vb{n}_{(\bart)})\, U_{L}^{\dagger}(\vb{n}_{(t)})\, U_{L}(\vb{n}_{(t+1)})\, - U_{L}^{\dagger}(\vb{n}_{(\bt)}), + U_{L}^{\dagger}(\vb{n}_{(\bart)}), \\ \cR_{(t,\, t+1)} & = & U_{R}(\vb{m}_{(t+1)})\, U_{R}^{\dagger}(\vb{m}_{(t)}), \\ - \widetilde{\cR}_{(t,\, t+1)} + \tcR_{(t,\, t+1)} & = & - U_{R}(\vb{m}_{(\bt)})\, + U_{R}(\vb{m}_{(\bart)})\, U_{R}^{\dagger}(\vb{m}_{(t)})\, U_{R}(\vb{m}_{(t+1)})\, - U_{R}^{\dagger}(\vb{m}_{(\bt)}). + U_{R}^{\dagger}(\vb{m}_{(\bart)}). \end{eqnarray} In spinor representation the action~\eqref{eq:string_action} becomes @@ -521,18 +521,18 @@ In spinor representation the action~\eqref{eq:string_action} becomes & = \frac{1}{4 \pi \ap} \iint\limits_{\ccH} - \dd{u} \dd{\bu}\, - \tr(\ipd{u} X_{(s)}(u, \bu) \cdot \ipd{\bu} X^{\dagger}_{(s)}(u, \bu)) + \dd{u} \dd{\baru}\, + \tr(\ipd{u} X_{(s)}(u, \baru) \cdot \ipd{\baru} X^{\dagger}_{(s)}(u, \baru)) \\ & = \frac{1}{8 \pi \ap} \iint\limits_{\C} - \dd{z} \dd{\bz}\, + \dd{z} \dd{\barz}\, \tr( - U_{L}(\vb{n}_{(\bt)})\, - \ipd{z} \cX_{(s)}(z, \bz)\, - U_{R}^{\dagger}(\vb{m}_{(\bt)})\, - \ipd{\bz} \cX_{(s)}^{\dagger}(z, \bz) + U_{L}(\vb{n}_{(\bart)})\, + \ipd{z} \cX_{(s)}(z, \barz)\, + U_{R}^{\dagger}(\vb{m}_{(\bart)})\, + \ipd{\barz} \cX_{(s)}^{\dagger}(z, \barz) ). \end{split} \label{eq:action_doubling_fields_spinor_representation} @@ -586,25 +586,25 @@ In what follows we start the investigation of the relation between the hypergeom We build the spinorial representation with \SU{2} matrices and solutions of Fuchsian equations with $N_B$ regular singular points. We are specifically interested in a solution with $N_B = 3$. -We fix the usual \SL{2}{\R} invariance by mapping the three intersection points $x_{(\bt-1)}$, $x_{(\bt+1)}$ and $x_{(\bt)}$ to $\omega_{\bt-1} = \omega_{x_{(\bt-1)}} = 0$, $\omega_{\bt+1} = \omega_{x_{(\bt+1)}} = 1$ and $\omega_{\bt} = \omega_{x_{(\bt)}} = \infty$ respectively through: +We fix the usual \SL{2}{\R} invariance by mapping the three intersection points $x_{(\bart-1)}$, $x_{(\bart+1)}$ and $x_{(\bart)}$ to $\omega_{\bart-1} = \omega_{x_{(\bart-1)}} = 0$, $\omega_{\bart+1} = \omega_{x_{(\bart+1)}} = 1$ and $\omega_{\bart} = \omega_{x_{(\bart)}} = \infty$ respectively through: \begin{equation} \omega_{u} = - \frac{u - x_{(\bt-1)}}{u - x_{(\bt)}} + \frac{u - x_{(\bart-1)}}{u - x_{(\bart)}} \cdot - \frac{x_{(\bt+1)} - x_{(\bt-1)}}{x_{(\bt+1)} - x_{(\bt)}} + \frac{x_{(\bart+1)} - x_{(\bart-1)}}{x_{(\bart+1)} - x_{(\bart)}} \label{eq:def_omega} \end{equation} The cut structure for this choice is presented in~\Cref{fig:hypergeometric_cuts}. -The map also defines $\arg(\omega_t - \omega_z) \in \left[ 0,\, 2\pi \right)$ for $t = \bt-1,\, \bt+1$. -We choose $\bt = 1$ in what follows. +The map also defines $\arg(\omega_t - \omega_z) \in \left[ 0,\, 2\pi \right)$ for $t = \bart-1,\, \bart+1$. +We choose $\bart = 1$ in what follows. \begin{figure}[tbp] \centering \def\svgwidth{0.35\linewidth} \import{img}{threebranes_plane.pdf_tex} \caption{% - Fixing the \SL{2}{\R} invariance for $N_B = 3$ and $\bt = 1$ leads to a cut structure with all the cuts defined on the real axis towards $\omega_{\bt} = \infty$.} + Fixing the \SL{2}{\R} invariance for $N_B = 3$ and $\bart = 1$ leads to a cut structure with all the cuts defined on the real axis towards $\omega_{\bart} = \infty$.} \label{fig:hypergeometric_cuts} \end{figure} @@ -642,7 +642,7 @@ The choice of the branch cuts follows from the cut on $\left[ 1, +\infty \right) As argued in~\eqref{eq:homotopy_rep}, the homotopy group of the complex plane with the branch cut structure of~\Cref{fig:hypergeometric_cuts} is such that a closed loop around all the singularities is homotopically trivial. The corresponding product of the monodromy matrices~\eqref{eq:homotopy_rep} is the unit matrix. -Let for instance $\cM_{\omega_z}^{\pm}$ be the monodromy matrix which represents a closed loop around $\omega_z$ (the $+$ sign represents a path starting in $\ccH$, while $-$ is a path with base point in $\overline{\ccH}$). +Let for instance $\cM_{\omega_z}^{\pm}$ be the monodromy matrix which represents a closed loop around $\omega_z$ (the $+$ sign represents a path starting in $\ccH$, while $-$ is a path with base point in $\bccH$). The triviality property is realised through: \begin{equation} \cM_{\vb{0}}^+\, @@ -656,7 +656,7 @@ The triviality property is realised through: \1_2 \label{eq:monodromy_relations} \end{equation} -The monodromy matrix $\omega_{\bt+1} = 1$ can thus be recovered as a product of monodromies around $0$ and $\infty$ given the properties +The monodromy matrix $\omega_{\bart+1} = 1$ can thus be recovered as a product of monodromies around $0$ and $\infty$ given the properties \begin{equation} \begin{split} \cM_{\vb{0}}^+ @@ -680,7 +680,7 @@ Using the basis in $z = 0$~\eqref{eq:basis_0} it is straightforward to find the \rM_{\vb{0}}( c ) = \mqty( \dmat{1, e^{-2\pi i c}} ). \label{eq:monodromy_zero} \end{equation} -The computation of the monodromy matrix $\rM_{\vb{\infty}}$ representing the monodromy in $\omega_z =\infty$ in the basis \eqref{eq:basis_0} requires to first compute the monodromy representation $\widetilde{\rM}_{\vb{\infty}}$ of the abstract monodromy $\cM_{\vb{\infty}}$ in the basis of hypergeometric functions around $z = \infty$: +The computation of the monodromy matrix $\rM_{\vb{\infty}}$ representing the monodromy in $\omega_z =\infty$ in the basis \eqref{eq:basis_0} requires to first compute the monodromy representation $\trM_{\vb{\infty}}$ of the abstract monodromy $\cM_{\vb{\infty}}$ in the basis of hypergeometric functions around $z = \infty$: \begin{equation} B_{\vb{\infty}}(z) = @@ -709,7 +709,7 @@ This basis is connected to~\eqref{eq:basis_0} through the transition matrix as $B_{\vb{0}}(z) = \cC(a,\, b,\, c)~B_{\vb{\infty}}(z)$. Through the loop $z \mapsto z e^{-2\pi i}$ we find: \begin{equation} - \widetilde{\rM}_{\vb{\infty}}( a,\, b ) + \trM_{\vb{\infty}}( a,\, b ) = \mqty( \dmat{e^{2\pi i a}, e^{2\pi i b}} ). \end{equation} @@ -718,7 +718,7 @@ Finally we can build the desired monodromy: \rM_{\vb{\infty}} = \cC(a,\, b,\, c)\, - \widetilde{\rM}_{\vb{\infty}}(a,\, b)\, + \trM_{\vb{\infty}}(a,\, b)\, \cC^{-1}(a,\, b,\, c). \label{eq:monodromy_infty} \end{equation} @@ -841,31 +841,31 @@ where we defined \begin{eqnarray} \cL(\vb{n}_{\vb{0}}) & = & - \cL_{(\bt-1,\,\bt)} + \cL_{(\bart-1,\,\bart)} = - U_L(\vb{n}_{(\bt)})\, - U_L^{\dagger}(\vb{n}_{(\bt-1)}), + U_L(\vb{n}_{(\bart)})\, + U_L^{\dagger}(\vb{n}_{(\bart-1)}), \\ \cL(\vb{n}_{\vb{\infty}}) & = & - \cL_{(\bt,\, \bt+1)} + \cL_{(\bart,\, \bart+1)} = - U_L(\vb{n}_{(\bt+1)}) - U_L^{\dagger}(\vb{n}_{(\bt)}), + U_L(\vb{n}_{(\bart+1)}) + U_L^{\dagger}(\vb{n}_{(\bart)}), \\ \cR(\vb{m}_{\vb{0}}) & = & - \cR_{(\bt-1,\, \bt)} + \cR_{(\bart-1,\, \bart)} = - U_R(\vb{n}_{(\bt)}) - U_R^{\dagger}(\vb{n}_{(\bt-1)}), + U_R(\vb{n}_{(\bart)}) + U_R^{\dagger}(\vb{n}_{(\bart-1)}), \\ \cR(\vb{m}_{\vb{\infty}}) & = & - \cR_{(\bt,\, \bt+1)} + \cR_{(\bart,\, \bart+1)} = - U_R(\vb{n}_{(\bt+1)}) - U_R^{\dagger}(\vb{n}_{(\bt)}). + U_R(\vb{n}_{(\bart+1)}) + U_R^{\dagger}(\vb{n}_{(\bart)}). \end{eqnarray} The range of $\delta_{\vb{0}}^{(L)}$ is \begin{equation} @@ -877,7 +877,7 @@ We then choose $\alpha = 0$ for simplicity. The same considerations hold true for all the other additional parameters $\delta_{\vb{0}}^{(R)}$ and $\delta_{\vb{\infty}}^{(L,\,R)}$. Since we are interested in relative rotations of the D-branes, we choose the -rotation in $\omega_{\bt-1} = 0$ in the maximal torus of $\SU{2}_L \times \SU{2}_R$ without loss of generality: as we have two independent groups, we can in fact fix the orientation of both vectors $\vb{n}_{\vb{0}}$ and $\vb{m}_{\vb{0}}$. +rotation in $\omega_{\bart-1} = 0$ in the maximal torus of $\SU{2}_L \times \SU{2}_R$ without loss of generality: as we have two independent groups, we can in fact fix the orientation of both vectors $\vb{n}_{\vb{0}}$ and $\vb{m}_{\vb{0}}$. In particular we set: \begin{eqnarray} \vb{n}_{\vb{0}} @@ -895,7 +895,7 @@ In particular we set: \label{eq:maximal_torus_right} \end{eqnarray} where $n_{\vb{0}}^3 = 0$ is excluded to avoid considering a trivial rotation. -We then define the parameters of the rotation in $\omega_{\bt} = \infty$ to be the most general +We then define the parameters of the rotation in $\omega_{\bart} = \infty$ to be the most general \begin{equation} \begin{split} \vb{n}_{\vb{\infty}} @@ -966,7 +966,7 @@ We find: \end{eqnarray} where $f^{(L)} \in \qty{ 0,\, 1 }$. For the sake of brevity we defined two auxiliary functions, namely $\cG(a,\, b,\, c) = \gfun{1-a}\, \gfun{1-b}\, \gfun{a+1-c}\, \gfun{b+1-c}$ and $\cF(a,\, b,\, c) = \sin(\pi c)\, \sin(\pi(a-b))$. -We also introduced the norm $n_{\vb{1}} = \norm{\vb{n}_{\vb{1}}}$ of the rotation vector around $\omega_{\bt+1} = 1$. +We also introduced the norm $n_{\vb{1}} = \norm{\vb{n}_{\vb{1}}}$ of the rotation vector around $\omega_{\bart+1} = 1$. Its dependence on the other parameters is encoded in~\eqref{eq:monodromy_relations}, where $\rM^+_{\vb{1}} = \rM^{-1}_{\vb{0}}\, \rM^{-1}_{\vb{\infty}}$, and the composition rule~\eqref{eq:product_in_SU2}: \begin{equation} \cos(2\pi n_{\vb{1}}) @@ -1266,7 +1266,7 @@ We could however use the symbolic solution~\eqref{eq:symbolic_solutions_using_P} As a matter of fact, finding the possible solutions with finite action can be recast to finding conditions such that the field $\ipd{z} \cX(z)$ is finite by itself. Linearity of this condition ensures a simpler approach with respect to the quadratic action of the string. From~\eqref{eq:action_doubling_fields_spinor_representation} it is clear that the action can be expressed as the sum of the product of any possible couple of elements of the expansion~\eqref{eq:formal_solution}. -We thus need to take into examination all possible pairs of contributions $\ipd{z} \cX_{l_1 r_1}(z)~ \ipd{\bz} \cX_{l_2 r_2}(\bz)$. +We thus need to take into examination all possible pairs of contributions $\ipd{z} \cX_{l_1 r_1}(z)~ \ipd{\barz} \cX_{l_2 r_2}(\barz)$. Near its singular points, the behavior of any element of solution~\eqref{eq:formal_solution} can be easily read from its symbolic representation~\eqref{eq:symbolic_solutions_using_P}: \begin{equation} \ipd{z} \cX(z) @@ -1277,22 +1277,22 @@ Near its singular points, the behavior of any element of solution~\eqref{eq:form \end{equation} It can be verified that the convergence of the action both at finite and infinite intersection points is ensured by the same constraints found when imposing the convergence at any point of the classical solution \begin{equation} - X_{(s)}(u,\, \bu) + X_{(s)}(u,\, \baru) = - f_{(s)\, (\bt-1)} + f_{(s)\, (\bart-1)} + - \finiteint{u'}{x_{(\bt-1)}}{u} + \finiteint{u'}{x_{(\bart-1)}}{u} \ipd{u'} \cX_{(s)}(u') + - U_L^{\dagger}(\vb{n}_{{\bt}}) + U_L^{\dagger}(\vb{n}_{{\bart}}) \qty[ - \finiteint{\bu'}{x_{(\bt-1)}}{\bu} - \ipd{\bu'} \cX_{(s)}(\bu') + \finiteint{\baru'}{x_{(\bart-1)}}{\baru} + \ipd{\baru'} \cX_{(s)}(\baru') ] - U_R(\vb{m}_{{\bt}}), + U_R(\vb{m}_{{\bart}}), \label{eq:classical_solution} \end{equation} -which follows in spinor representation from~\eqref{eq:spinor_doubling_trick} and where $f_{(s),\, (\bt-1)} = f^I_{(\bt-1)}\, \tau_I$. +which follows in spinor representation from~\eqref{eq:spinor_doubling_trick} and where $f_{(s),\, (\bart-1)} = f^I_{(\bart-1)}\, \tau_I$. We specifically find: \begin{equation} \begin{split} @@ -2034,19 +2034,19 @@ The general solution for $\ipd{\omega} \cX$ is therefore: The final solution depends only on two complex constants, $C_1$ and $C_2$, which we can fix imposing the global conditions in \eqref{eq:discontinuity_bc}, that is the second equation in the solution \eqref{eq:classical_solution}. As the three intersection points in target space always define a triangle on a 2-dimensional plane, we impose the boundary conditions knowing two angles formed by the sides of the triangle (i.e.\ the branes between two intersections) and the length of one of them. Since we already fixed the parameters associated to the rotations, we need to compute the length of one of the sides. -Consider for instance the length of $X(x_{\bt+1},\, x_{\bt+1}) - X(x_{\bt-1},\, x_{\bt-1})$. +Consider for instance the length of $X(x_{\bart+1},\, x_{\bart+1}) - X(x_{\bart-1},\, x_{\bart-1})$. Explicitly we impose the four real equations in spinorial formalism \begin{equation} \finiteint{\omega}{0}{1} \ipd{\omega} \cX(\omega) + - U_L^{\dagger}(\vb{n}_{{\bt}}) + U_L^{\dagger}(\vb{n}_{{\bart}}) \qty[ \finiteint{\bomega}{0}{1} \ipd{\bomega} \cX(\bomega) ] - U_R(\vb{m}_{{\bt}}) + U_R(\vb{m}_{{\bart}}) = - f_{{\bt+1}\, (s)} - f_{{\bt-1}\, (s)}, + f_{{\bart+1}\, (s)} - f_{{\bart-1}\, (s)}, \end{equation} where we used the mapping~\eqref{eq:def_omega} to write the integrals in the $\omega$ variables. This equation has enough degrees of freedom to fix completely the two complex parameters $C_1$ and $C_2$. @@ -2393,29 +2393,29 @@ such that $\sum\limits_{t} \varepsilon_{\vb{t}} = 1$, and \label{eq:Abelian_rotation_second} \end{equation} where $\sum\limits_{t} \varphi_{\vb{t}} = 2$, in order to approach the usual notation in the literature. -As usual we have $\ipd{\omega_z} \cZ^1( \omega_z ) \neq \qty[ \ipd{\omega_z} \overline{\cZ}^1( \omega_z ) ]^*$. +As usual we have $\ipd{\omega_z} \cZ^1( \omega_z ) \neq \qty[ \ipd{\omega_z} \bcZ^1( \omega_z ) ]^*$. We can now build the Abelian solution to show the analytical structure of the limit. We have \begin{equation} - \mqty( i \overline{Z}^1( u,\, \bu ) & Z^2( u,\, \bu ) + \mqty( i \barZ^1( u,\, \baru ) & Z^2( u,\, \baru ) \\ - \overline{Z}^2( u,\, \bu ) & i Z^1( u,\, \bu ) + \barZ^2( u,\, \baru ) & i Z^1( u,\, \baru ) ) = - \mqty( i \overline{f}^1_{(\bt - 1)} + i \finiteint{\omega}{0}{\bomega_{\bu}}\, \ipd{\omega} \cZ^1 + \mqty( i \barf^1_{(\bart - 1)} + i \finiteint{\omega}{0}{\bomega_{\baru}}\, \ipd{\omega} \cZ^1 & - f^2_{(\bt -1)} + \finiteint{\omega}{0}{\omega_u}\, \ipd{\omega} \cZ^2 + f^2_{(\bart -1)} + \finiteint{\omega}{0}{\omega_u}\, \ipd{\omega} \cZ^2 \\ - \overline{f}^2_{(\bt - 1)} + \finiteint{\omega}{0}{\bomega_{\bu}}\, \ipd{\omega} \cZ^2 + \barf^2_{(\bart - 1)} + \finiteint{\omega}{0}{\bomega_{\baru}}\, \ipd{\omega} \cZ^2 & - i f^1_{(\bt-1)} + i \finiteint{\omega}{0}{\omega_u}\, \ipd{\omega_z} \cZ^1 + i f^1_{(\bart-1)} + i \finiteint{\omega}{0}{\omega_u}\, \ipd{\omega_z} \cZ^1 ) \end{equation} -where we chose $R_{(\bt)} = \1_4$ so that $U_{(\bt)}$ in~\eqref{eq:Umatrices} is mapped to $(i \sigma_1, i \sigma_1) \in \SU{2} \times \SU{2}$. +where we chose $R_{(\bart)} = \1_4$ so that $U_{(\bart)}$ in~\eqref{eq:Umatrices} is mapped to $(i \sigma_1, i \sigma_1) \in \SU{2} \times \SU{2}$. Notice however that $\vb{n}_{\vb{t}} = n_{\vb{t}}^3\, \vb{k}$ implies that $v^3_{(t)} = 0$ in~\eqref{eq:special_UL_brane_t}. Hence $U_L$ and $U_R$ are always off diagonal and their action on~\eqref{eq:Abelian_sol_example} is to fill the first column. -From the previous relations we can also recover the usual holomorphicity $\overline{Z}^1(\bu) = \qty[ Z^1(u) ]^*$ of the sector with $\sum\limits_t \varepsilon_{\vb{t}} = 1$ and $\overline{Z}^2(\bu) = \qty[ Z^2(u) ]^*$ of the sector with $\sum\limits_t \varphi_{\vb{t}} = 2$. +From the previous relations we can also recover the usual holomorphicity $\barZ^1(\baru) = \qty[ Z^1(u) ]^*$ of the sector with $\sum\limits_t \varepsilon_{\vb{t}} = 1$ and $\barZ^2(\baru) = \qty[ Z^2(u) ]^*$ of the sector with $\sum\limits_t \varphi_{\vb{t}} = 2$. \subsubsection{Abelian Limits} @@ -2472,13 +2472,13 @@ For $x_{(t)} < x < x_{(t-1)}$ we have: X_R(x-iy) + Y, \end{equation} where $Y \in \R$ is a constant factor which cannot depend on the particular D-brane $D_{(t)}$. -In fact the continuity of $X_L(u)$ and $X_R(\bu)$ on the worldsheet intersection point ensures that +In fact the continuity of $X_L(u)$ and $X_R(\baru)$ on the worldsheet intersection point ensures that \begin{equation} \lim\limits_{x \to x_{(t)}^+} X(x, x) = \lim\limits_{x \to x_{(t)}^-} X(x, x), \end{equation} -which does not allow $Y$ to depend on the specific D-brane while the reality of $X(u,\bu)$ implies that $\Im Y = 0$. +which does not allow $Y$ to depend on the specific D-brane while the reality of $X(u,\baru)$ implies that $\Im Y = 0$. Now~\eqref{eq:area_tmp} becomes: \begin{equation} \begin{split} @@ -2512,9 +2512,9 @@ where $g^{(\perp)}_{(t)\, I} = \sum_{m \in \{3, 4\}} \tensor{\qty( R_{(t)}^{-1} In this case there are global complex coordinates for which the string solution is holomorphic: \begin{equation} - Z^i(u, \bu) = Z^i_L(u), + Z^i(u, \baru) = Z^i_L(u), \qquad - \overline{Z}^i(u, \bu) = \bar{Z}^i(\bu) = \qty( Z^i_L(u) )^*, + \barZ^i(u, \baru) = \bar{Z}^i(\baru) = \qty( Z^i_L(u) )^*, \end{equation} where $i = 1$ in the Abelian case and $i=1,\, 2$ in the \SU{2} case. We also have $f^i_{(t)} = Z^i_L(x_{(t)} + i\, 0^+)$. diff --git a/sec/part1/fermions.tex b/sec/part1/fermions.tex index 1ea999c..eb3b9f4 100644 --- a/sec/part1/fermions.tex +++ b/sec/part1/fermions.tex @@ -9,8 +9,8 @@ Non Abelian cases have been considered~\cite{Inoue:1987:NonAbelianOrbifolds,Inou Despite the existence of an efficient method based on bosonization~\cite{Friedan:1986:ConformalInvarianceSupersymmetry} and old methods based on the Reggeon vertex~\cite{Sciuto:1969:GeneralVertexFunction,DellaSelva:1970:SimpleExpressionSciuto,Schwarz:1973:EvaluationDualFermion,DiVecchia:1990:VertexIncludingEmission,Nilsson:1990:GeneralNSRString,DiBartolomeo:1990:GeneralPropertiesVertices,Engberg:1993:AlgorithmComputingFourRamond,Petersen:1989:CovariantSuperreggeonCalculus}, we take into examination the computation of spin field correlators and propose a new method to compute them. We hope to be able to extend this approach to correlators involving twist fields and non Abelian spin and twist fields. We would also like to investigate the reason of the non existence of an approach equivalent to bosonization for twist fields. -At the same time we are interested to explore what happens to a CFT in presence of defects. -It turns out that, despite the defects, it is still possible to define a radial time dependent stress-energy tensor which satisfies the canonical OPE. +At the same time we are interested to explore what happens to a \cft in presence of defects. +It turns out that, despite the defects, it is still possible to define a radial time dependent stress-energy tensor which satisfies the canonical \ope. Moreover the boundary changing defects in the construction can be associated with excited spin fields enabling the computation of correlators involving excited spin fields without resorting to bosonization. @@ -28,7 +28,7 @@ Their two-dimensional Minkowski action defined on the strip $\Sigma$ is: \finiteint{\sigma}{0}{\pi} \qty( \frac{1}{2}\, \bpsi_i( \tau,\, \sigma )\, - \qty( -i \gamma^{\alpha} \lrpartial{\alpha} )\, + \qty( -i \gamma^{\alpha} \lripd{\alpha} )\, \psi^i( \tau,\, \sigma ) ), \label{eq:cft-action_full} @@ -68,7 +68,7 @@ In components the action reads: \infinfint{\xi_+} \infinfint{\xi_-} \qty( - \psi^*_{-,\, i} \lrpartial{+} \psi^i_- + \psi^*_{+,\, i} \lrpartial{-} \psi^i_+ + \psi^*_{-,\, i} \lripd{+} \psi^i_- + \psi^*_{+,\, i} \lripd{-} \psi^i_+ ), \label{eq:cft-action} \end{equation} @@ -348,11 +348,11 @@ Using the usual Nöther's procedure we get the on-shell non vanishing components \begin{split} \cT_{++}( \xi_+ ) & = - -i \frac{T}{4} \psi_{+,\, i}^*( \xi_+ ) \lrpartial{+} \psi_+^i( \xi_+ ), + -i \frac{T}{4} \psi_{+,\, i}^*( \xi_+ ) \lripd{+} \psi_+^i( \xi_+ ), \\ \cT_{--}( \xi_- ) & = - -i \frac{T}{4} \psi_{-,\, i}^*( \xi_- )\lrpartial{-} \psi_-^i( \xi_- ). + -i \frac{T}{4} \psi_{-,\, i}^*( \xi_- )\lripd{-} \psi_-^i( \xi_- ). \end{split} \label{eq:stress-energy-tensor-lightcone} \end{equation} @@ -395,9 +395,9 @@ reads:\footnotemark{} - i \frac{T}{4} \int \Delta\tau \qty( - 2 \eval{\psi_{+,\, i}^*\, \lrpartial{\tau} \psi_+^i}^{\sigma = \pi}_{\sigma = 0} + 2 \eval{\psi_{+,\, i}^*\, \lripd{\tau} \psi_+^i}^{\sigma = \pi}_{\sigma = 0} - - \eval{\psi_{+,\, i}^* \tensor{\qty( R_{(t)}^{\dagger} \lrpartial{\tau} R_{(t)} )}{^i_j} \psi_+^j}_{\sigma = 0} + \eval{\psi_{+,\, i}^* \tensor{\qty( R_{(t)}^{\dagger} \lripd{\tau} R_{(t)} )}{^i_j} \psi_+^j}_{\sigma = 0} ) \neq 0. @@ -412,7 +412,7 @@ The corresponding condition for the Hamiltonian $\rH$ is: & = - i \frac{T}{4} \int \Delta\tau - \qty( \eval{\psi_{+,\, i}^* \tensor{\qty( R_{(t)}^{\dagger} \lrpartial{\tau} R_{(t)} )}{^i_j} \psi_+^j}_{\sigma = 0} ) + \qty( \eval{\psi_{+,\, i}^* \tensor{\qty( R_{(t)}^{\dagger} \lripd{\tau} R_{(t)} )}{^i_j} \psi_+^j}_{\sigma = 0} ) = 0 \quad \Leftrightarrow @@ -457,13 +457,13 @@ The fields $\psi^i$ (and the fields $\Psi^i$) are then a superposition of such m \begin{equation} \psi^i_{\pm}( \xi_{\pm} ) = - \sum\limits_{n \in \Z} b_n\, \psi^i_{n,\, \pm}( \xi_{\pm} ) + \infinfsum{n} b_n\, \psi^i_{n,\, \pm}( \xi_{\pm} ) \qquad \Rightarrow \qquad \Psi^i( \xi ) = - \sum\limits_{n \in \Z} b_n\, \Psi^i_n( \xi ). + \infinfsum{n} b_n\, \Psi^i_n( \xi ). \label{eq:usual-mode-expansion} \end{equation} @@ -522,7 +522,7 @@ This task is however easier to address in a Euclidean formulation. In the Euclidean reformulation the solution to the \eom might be easier to study than its Lorentzian worldsheet form. This is specifically the case when $R_{(t)} \in \U{1}^{N_f} \subset \U{N_f}$. -The presence of a time dependent Hamiltonian is not standard and we can neither blindly apply the usual Wick rotation nor the usual CFT techniques. +The presence of a time dependent Hamiltonian is not standard and we can neither blindly apply the usual Wick rotation nor the usual \cft techniques. \subsubsection{Fields on the Strip} @@ -535,9 +535,9 @@ Performing the Wick rotation as $\tau_E = i \tau$ such that $e^{i S} = e^{-S_E}$ \iint \dd{\xi} \dd{\bxi}\, \frac{1}{2}\, \qty( - \hpsi_{E,\, +,\, i}^*\, \lrpartial{\bxi} \hpsi_{E,\, +}^i + \hpsi_{E,\, +,\, i}^*\, \lripd{\bxi} \hpsi_{E,\, +}^i + - \hpsi_{E,\, -,\, i}^*\, \lrpartial{\xi} \hpsi_{E,\, -}^i + \hpsi_{E,\, -,\, i}^*\, \lripd{\xi} \hpsi_{E,\, -}^i ), \label{eq:S_Eu_strip} \end{equation} @@ -620,23 +620,23 @@ where $\halpha^*_E$ and $\hbeta_E$ are the Euclidean counterparts of the generic In the Euclidean context we have to explicitly write $\halpha^*_E$ because it is no longer the ``complex conjugate'' of $\halpha_E$ in the traditional sense. The product is conserved only when it couples two solutions which have different boundary conditions as in~\eqref{eq:bc_eu_strip}. -The definition of the stress-energy tensor in~\eqref{eq:stress-energy-tensor-lightcone} requires a change in the numerical pre-factor to use the usual CFT normalization. +The definition of the stress-energy tensor in~\eqref{eq:stress-energy-tensor-lightcone} requires a change in the numerical pre-factor to use the usual \cft normalization. Introducing a spacetime variable central charge as well the components of the stress-energy tensor become:\footnotemark{} \footnotetext{% - The canonical coefficient in front of the CFT stress-energy tensor is such that the Euclidean Hamiltonian $\rL_{0}$ is normalized such that - \begin{equation*} + The canonical coefficient in front of the \cft stress-energy tensor is such that the Euclidean Hamiltonian $\rL_{0}$ is normalized such that + \begin{equation} \cT_{\zeta\zeta}( \zeta ) = \sum_n \rL_{n} e^{-n \zeta} - \end{equation*} + \end{equation} (we are anticipating the double strip notation defined in the next subsection for simplicity). We thus get: - \begin{equation*} + \begin{equation} \rH_E = \rL_{0} = \int\limits_{0}^{2\pi} \frac{\dd{\phi}}{2 \pi} \cT_{\zeta \zeta}( \tau_E + i\, \phi ) - \end{equation*} + \end{equation} therefore $\cT_{\zeta\zeta}( \zeta ) = 2 \pi\, \cT^{(can)}_{\zeta\zeta}( \zeta )$. } \begin{equation} @@ -644,20 +644,20 @@ Introducing a spacetime variable central charge as well the components of the st \cT_{\xi \xi}( \xi ) & = - \frac{\pi T}{2}\, - \hpsi_{E,\, +,\, i}^*( \xi )\, \lrpartial{\xi} \hpsi^i_{E,\, +}( \xi ) + \hpsi_{E,\, +,\, i}^*( \xi )\, \lripd{\xi} \hpsi^i_{E,\, +}( \xi ) + \widehat{\cC}( \xi ), \\ \cT_{\bxi \bxi}( \bxi ) & = - \frac{\pi T}{2}\, - \hpsi_{E,\, -,\, i}^*( \bxi ) \lrpartial{\bxi} + \hpsi_{E,\, -,\, i}^*( \bxi ) \lripd{\bxi} \hpsi^i_{E,\, -}( \bxi ) + - \widehat{\overline{\cC}}( \bxi ), + \widehat{\bcC}( \bxi ), \end{split} \end{equation} -where $\widehat{\cC}$ and $\widehat{\overline{\cC}}$ are the leftover terms after the regularization of the singularities due to the normal ordering. +where $\widehat{\cC}$ and $\widehat{\bcC}$ are the leftover terms after the regularization of the singularities due to the normal ordering. The canonical anti-commutation relations are then \begin{equation} \eval{ @@ -685,11 +685,11 @@ We can then expand the fields as \begin{cases} \hpsi^i_{E,\, +}(\xi) & = - \sum\limits_{n \in \Z} b_n\, \hpsi^i_{E,\, +,\, n}(\xi) + \infinfsum{n} b_n\, \hpsi^i_{E,\, +,\, n}(\xi) \\ \hpsi^i_{E,\, -}(\bxi) & = - \sum\limits_{n \in \Z} b_n\, \hpsi^i_{E,\, -,\, n}(\bxi) + \infinfsum{n} b_n\, \hpsi^i_{E,\, -,\, n}(\bxi) \end{cases} \end{equation} and @@ -697,11 +697,11 @@ and \begin{cases} \hpsi^*_{E,\, +,\, i}(\xi) & = - \sum\limits_{n \in \Z} b^*_n\, \hpsi^*_{E,\, +,\, n,\, i}(\xi) + \infinfsum{n} b^*_n\, \hpsi^*_{E,\, +,\, n,\, i}(\xi) \\ \hpsi^*_{E,\, -,\, i}(\bxi) & = - \sum\limits_{n \in \Z} b^*_n\, \hpsi^*_{E,\, -,\, n,\, i}(\bxi) + \infinfsum{n} b^*_n\, \hpsi^*_{E,\, -,\, n,\, i}(\bxi) \end{cases} \end{equation} in order to extract the operators through the conserved product @@ -729,7 +729,7 @@ It is natural to use the doubling trick on the strip to simplify the previous ex Define the coordinate $\zeta = \tau_E + i\, \phi$ with $0 \le \phi \le 2\pi$. We then have \begin{equation} - \Hpsi( \zeta ) + \hPsi( \zeta ) = \begin{cases} \hpsi_{E,\, +}(\zeta) @@ -743,23 +743,23 @@ We then have \phi = 2\pi - \sigma \in \qty[ \pi, 2 \pi ] \end{cases} \end{equation} -on-shell (and similarly for $\Hpsi^*( \zeta )$ with the substitution $\hpsi_{E,\, \pm} \to \hpsi_{E,\, \pm}^*$). +on-shell (and similarly for $\hPsi^*( \zeta )$ with the substitution $\hpsi_{E,\, \pm} \to \hpsi_{E,\, \pm}^*$). The ``complex conjugation'' $\star$ acts on the off-shell double fields as \begin{equation} - \qty[ \Hpsi^i(\zeta,\, \bzeta) ]^\star = \Hpsi_i^*(-\bzeta,\, -\zeta), + \qty[ \hPsi^i(\zeta,\, \bzeta) ]^\star = \hPsi_i^*(-\bzeta,\, -\zeta), \end{equation} while the boundary conditions are translated into \begin{equation} \begin{cases} - \Hpsi^i( \tau_E + 2 \pi i^- ) + \hPsi^i( \tau_E + 2 \pi i^- ) = -\tensor{\qty( R_{(t)} )}{^i_j}\, - \Hpsi^j( \tau_E + i\, 0^+ ) + \hPsi^j( \tau_E + i\, 0^+ ) \\ - \Hpsi^{* i}( \tau_E + 2 \pi i^- ) + \hPsi^{* i}( \tau_E + 2 \pi i^- ) = -\tensor{\qty( R_{(t)}^* )}{^i_j}\, - \Hpsi^{* j}( \tau_E + i\, 0^+ ) + \hPsi^{* j}( \tau_E + i\, 0^+ ) \end{cases} \end{equation} for $\tau_E \in \qty( \htau_{E,\, (t)}, \htau_{E,\, (t-1)} )$. @@ -778,7 +778,7 @@ The holomorphic stress-energy tensor is then \cT_{\zeta \zeta}( \zeta ) = - \frac{\pi T}{2}\, - \Hpsi_{i}^*( \zeta )\, \lrpartial{\zeta} \Hpsi^i( \zeta ) + \hPsi_{i}^*( \zeta )\, \lripd{\zeta} \hPsi^i( \zeta ) + \widehat{\cC}(\zeta) \end{equation} @@ -786,9 +786,9 @@ and the canonical anti-commutation relations are now \begin{equation} \eval{ \qty[ - \Hpsi^i( \zeta_1 ) + \hPsi^i( \zeta_1 ) , - \Hpsi_{j}^*( \zeta_2 ) + \hPsi_{j}^*( \zeta_2 ) ]_+ }_{\Re\zeta_1 = \Re\zeta_2} = @@ -797,40 +797,40 @@ and the canonical anti-commutation relations are now The double field formulation shows that we need only one coefficient $b_n$ (or $b_n^*$) for both $\psi_{E,\, +}$ and $\psi_{E,\, -}$ (or for both $\psi^*_{E,\, +}$ and $\psi^*_{E,\, -}$). -In fact, given the Euclidean modes $\Hpsi^i_{n}$ and $\Hpsi^*_{n,\, i}$ where $n \in \Z$, we define the dual modes $\dual{\Hpsi}^i_{n}$ and $\dual{\Hpsi}^*_{n,\, i}$ such that: +In fact, given the Euclidean modes $\hPsi^i_{n}$ and $\hPsi^*_{n,\, i}$ where $n \in \Z$, we define the dual modes $\dual{\hPsi}^i_{n}$ and $\dual{\hPsi}^*_{n,\, i}$ such that: \begin{equation} - \lconsprod{\dual{\Hpsi}^*_{n}}{\Hpsi_{m}} + \lconsprod{\dual{\hPsi}^*_{n}}{\hPsi_{m}} = - \lconsprod{\dual{\Hpsi}_{n}}{\Hpsi^*_{m}} + \lconsprod{\dual{\hPsi}_{n}}{\hPsi^*_{m}} = \delta_{n,m}. \end{equation} We expand the double fields as \begin{equation} - \Hpsi^i(\zeta) + \hPsi^i(\zeta) = - \sum\limits_{n \in \Z} b_n \Hpsi^i_{n}(\zeta), + \infinfsum{n} b_n \hPsi^i_{n}(\zeta), \qquad - \Hpsi^*_{i}(\zeta) + \hPsi^*_{i}(\zeta) = - \sum\limits_{n \in \Z} b^*_n \Hpsi^*_{n}(\zeta) + \infinfsum{n} b^*_n \hPsi^*_{n}(\zeta) \end{equation} Operators are then extracted as \begin{equation} b_n = - \lconsprod{\dual{\Hpsi}^*_{n}}{\Hpsi}, + \lconsprod{\dual{\hPsi}^*_{n}}{\hPsi}, \qquad b^*_n = - \lconsprod{\dual{\Hpsi}_{n}}{\Hpsi^*}. + \lconsprod{\dual{\hPsi}_{n}}{\hPsi^*}. \label{eq:upper-half-extraction} \end{equation} Finally we get the anti-commutation relations as \begin{equation} \eval{ \qty[ b_n, b^*_m ]_+ }_{\tau_E = \tau_{E,\, 0}} = - \frac{2 \cN}{T} \lconsprod{\dual{\Hpsi}^*_{n}}{\dual{\Hpsi}_m}. + \frac{2 \cN}{T} \lconsprod{\dual{\hPsi}^*_{n}}{\dual{\hPsi}_m}. \end{equation} @@ -840,7 +840,7 @@ Finally we get the anti-commutation relations as \centering \includegraphics[width=0.5\linewidth]{img/complex-plane} \caption{% - Fields are glued on the $x < 0$ semi-axis with non trivial discontinuities for $x_t < x < x_{t-1}$ for $t = 1,\, 2,\, \dots,\, N$ and where $x_t = \exp( \htau_{E,\, (t)} )$. + Fields are glued on the $x < 0$ semi-axis with non trivial discontinuities for $x_{(t)} < x < x_{(t-1)}$ for $t = 1,\, 2,\, \dots,\, N$ and where $x_{(t)} = \exp( \htau_{E,\, (t)} )$. } \label{fig:complex-plane} \end{figure} @@ -861,55 +861,55 @@ Under this change of coordinates the Euclidean action~\eqref{eq:S_Eu_strip} beco S_E & = \frac{T}{2} - \iint \dd{u}\dd{\bu}\, + \iint \dd{u}\dd{\baru}\, \frac{1}{2}\, \qty( - \frac{1}{u}\, \hpsi_{E,\, +,\, i}^* \lrpartial{\bu} \hpsi_{E,\, +}^i + \frac{1}{u}\, \hpsi_{E,\, +,\, i}^* \lripd{\baru} \hpsi_{E,\, +}^i + - \frac{1}{\bu}\, \hpsi_{E,\, -,\, i}^* \lrpartial{u} \hpsi_{E,\, -}^i + \frac{1}{\baru}\, \hpsi_{E,\, -,\, i}^* \lripd{u} \hpsi_{E,\, -}^i ) \\ & = \frac{T}{2} - \iint \dd{u}\dd{\bu}\, + \iint \dd{u}\dd{\baru}\, \frac{1}{2}\, \qty( - \psi_{E,\, +,\, i}^* \lrpartial{\bu} \psi_{E,\, +}^i + \psi_{E,\, +,\, i}^* \lripd{\baru} \psi_{E,\, +}^i + - \psi_{E,\, -,\, i}^* \lrpartial{u} \psi_{E,\, -}^i + \psi_{E,\, -,\, i}^* \lripd{u} \psi_{E,\, -}^i ), \end{split} \end{equation} where we introduce the off-shell field redefinitions: \begin{equation} - \psi_{E,\, +}^i(u,\, \bu) + \psi_{E,\, +}^i(u,\, \baru) = \frac{1}{\sqrt{u}}\, \hpsi_{E,\, +}^i( \xi,\, \bxi ), \qquad - \psi_{E,\, -}^i(u,\, \bu) + \psi_{E,\, -}^i(u,\, \baru) = - \frac{1}{\sqrt{\bu}}\, \hpsi_{E,\, -}^i( \xi,\, \bxi ). + \frac{1}{\sqrt{\baru}}\, \hpsi_{E,\, -}^i( \xi,\, \bxi ). \label{eq:euclidean-off-shell-redefinitions} \end{equation} Fields with the hat sign on top thus represent strip and double strip definitions, while fields without the hat sign are defined on $\ccH$ or $\C$.\footnotemark{} \footnotetext{% - We could have anticipated these redefinitions from a CFT argument where - \begin{equation*} + We could have anticipated these redefinitions from a \cft argument where + \begin{equation} \psi( u ) = \eval{\qty( \dv{u}{\xi} )^{-\frac{1}{2}} {\hpsi}(\xi)}_{\xi = \ln( u )}, - \end{equation*} - but we cannot and do not rely on CFT properties since we have not shown that the theory is a CFT yet. + \end{equation} + but we cannot and do not rely on \cft properties since we have not shown that the theory is a \cft yet. } Notice that this is the result one would expect from the engineering dimension: in this case it works since the theory is essentially free. Using the redefinitions~\eqref{eq:euclidean-off-shell-redefinitions}, the off-shell ``complex conjugation'' $\star$ then becomes \begin{equation} - \qty[ \psi_{E,\, +,\, i}( u,\, \bu ) ]^\star + \qty[ \psi_{E,\, +,\, i}( u,\, \baru ) ]^\star = - \frac{1}{\bu}\, \psi_{E,\, +,\, i}^*\qty( \frac{1}{\bu},\, \frac{1}{u} ), + \frac{1}{\baru}\, \psi_{E,\, +,\, i}^*\qty( \frac{1}{\baru},\, \frac{1}{u} ), \qquad - \qty[ \psi_{E,\, -,\, i}( u,\, \bu ) ]^\star + \qty[ \psi_{E,\, -,\, i}( u,\, \baru ) ]^\star = - \frac{1}{u}\, \psi_{E,\, -,\, i}^*\qty( \frac{1}{\bu},\, \frac{1}{u} ). + \frac{1}{u}\, \psi_{E,\, -,\, i}^*\qty( \frac{1}{\baru},\, \frac{1}{u} ). \end{equation} We choose to insert the cut of the square root on the real negative axis. @@ -945,8 +945,8 @@ The product~\eqref{eq:euclidean-conserved-product-strip} is then \int\limits_{\widehat{\Sigma}} \dd{u} \alpha^*_{+,\, i}(u) \beta_+^i(u) - - \int\limits_{\widehat{\overline{\Sigma}}} \dd{\bu} - \alpha^*_{-,\, i}(\bu) \beta_-^i(\bu) + \int\limits_{\widehat{\overline{\Sigma}}} \dd{\baru} + \alpha^*_{-,\, i}(\baru) \beta_-^i(\baru) ], \label{eq:prod_H} \end{equation} @@ -955,14 +955,14 @@ The stress-energy tensor becomes:\footnotemark{} \footnotetext{% While rewriting the operator part of the stress-energy tensor from the strip formulation into the coordinates in $\ccH$ we actually get - \begin{equation*} + \begin{equation} \cT_{\xi \xi}( \xi(u) ) = u^2\, \cT_{u u}( u ). - \end{equation*} + \end{equation} The reason of the presence of $u^2$ factor can be explained in two ways. Using GR we know that $\cT_{\xi \xi}( \xi ) \dss[2]{\xi} = \cT_{u u}( u ) \dss[2]{u}$. Another way is to notice that a translation in $\xi$ is a dilatation of $u$: the infinitesimal generator of $\xi$ translation must be the infinitesimal generator of $u$ dilatation, that is: - \begin{equation*} + \begin{equation} P_{\xi} \sim \int \dd{\sigma}\, \cT_{\xi \xi} @@ -970,23 +970,23 @@ The stress-energy tensor becomes:\footnotemark{} D_u \sim \int \dd{u}\, u\, \cT_{u u}. - \end{equation*} + \end{equation} } \begin{equation} \begin{split} \cT_{u u}( u ) & = - \frac{\pi T}{2}\, - \psi_{E,\, +,\, i}^*( u )\, \lrpartial{u} \psi^i_{E,\, +}( u ) + \psi_{E,\, +,\, i}^*( u )\, \lripd{u} \psi^i_{E,\, +}( u ) + \widehat{\cC}(u ), \\ - \cT_{\bu \bu}( \bu ) + \cT_{\baru \baru}( \baru ) & = - \frac{\pi T}{2}\, - \psi_{E,\, -,\, i}^*( \bu )\, \lrpartial{\bu} \psi^i_{E,\, -}( \bu ) + \psi_{E,\, -,\, i}^*( \baru )\, \lripd{\baru} \psi^i_{E,\, -}( \baru ) + - \widehat{\overline{\cC}}( \bu ). + \widehat{\bcC}( \baru ). \end{split} \end{equation} Finally the anti-commutation relations are @@ -994,9 +994,9 @@ Finally the anti-commutation relations are \begin{cases} \eval{ \qty[ - \psi_{E,\, +}^i( u_1,\, \bu_1 ) + \psi_{E,\, +}^i( u_1,\, \baru_1 ) , - \psi_{E,\, +,\, j}^*( u_2,\, \bu_2 ) + \psi_{E,\, +,\, j}^*( u_2,\, \baru_2 ) ]_+ }_{\abs{u_1} = \abs{u_2}} & = @@ -1004,13 +1004,13 @@ Finally the anti-commutation relations are \\ \eval{ \qty[ - \psi_{E,\, -}^i( u_1,\, \bu_1 ) + \psi_{E,\, -}^i( u_1,\, \baru_1 ) , - \psi_{E,\, -,\, j}^*( u_2,\, \bu_2 ) + \psi_{E,\, -,\, j}^*( u_2,\, \baru_2 ) ]_+ }_{\abs{u_1} = \abs{u_2}} & = - \frac{2}{T \bu_1}\, \tensor{\delta}{^i_j}\, \delta( \arg(u_1) - \arg(u_2) ), + \frac{2}{T \baru_1}\, \tensor{\delta}{^i_j}\, \delta( \arg(u_1) - \arg(u_2) ), \end{cases} \end{equation} which despite the strange look of the expression are perfectly compatible with the definition~\eqref{eq:upper-half-extraction} leading to: @@ -1029,11 +1029,11 @@ We expand the fields in modes as: \begin{cases} \psi^i_{E,\, +}(u) = - \sum\limits_{n \in \Z} b_n\, \psi^i_{E,\, +,\, n}(u) + \infinfsum{n} b_n\, \psi^i_{E,\, +,\, n}(u) \\ - \psi^i_{E,\, -}(\bu) + \psi^i_{E,\, -}(\baru) = - \sum\limits_{n \in \Z} b_n\, \psi^i_{E,\, -,\, n}(\bu) + \infinfsum{n} b_n\, \psi^i_{E,\, -,\, n}(\baru) \end{cases} \end{equation} and @@ -1041,11 +1041,11 @@ and \begin{cases} \psi^*_{E,\, +,\, i}(u) = - \sum\limits_{n \in \Z} b^*_n\, \psi^*_{E,\, +,\, n,\, i}(u) + \infinfsum{n} b^*_n\, \psi^*_{E,\, +,\, n,\, i}(u) \\ - \psi^*_{E,\, -,\, i}(\bu) + \psi^*_{E,\, -,\, i}(\baru) = - \sum\limits_{n \in \Z} b^*_n\, \psi^*_{E,\, -,\, n,\, i}(\bu) + \infinfsum{n} b^*_n\, \psi^*_{E,\, -,\, n,\, i}(\baru) \end{cases} \end{equation} and $\dual{\psi}_{E,\, n}$ and $\dual{\psi}^*_{E,\, n}$ are the corresponding dual modes on the upper half plane. @@ -1062,9 +1062,9 @@ We use again the doubling trick to define the fields on the subset $\C \setminus & \qfor z = u \in \ccH \setminus \qty[ x_{(N)}, x_{(1)} ] \\ - \psi_{E,\, -}(\bu) + \psi_{E,\, -}(\baru) & \qfor - z = \bu \in \overline{\ccH} \setminus \qty[ x_{(N)}, x_{(1)} ] + z = \baru \in \overline{\ccH} \setminus \qty[ x_{(N)}, x_{(1)} ] \end{cases} \end{equation} where $z = \exp( \tau_E + i\, \phi ) = x + i y$ and $\overline{\ccH} = \qty{ w \in \C \mid \Im w \le 0 }$. @@ -1072,9 +1072,9 @@ The same procedure applies also to $\Psi^*$ with the exchange $\psi_{E,\, \pm} \ In this case the ``complex conjugation'' $\star$ acts off-shell as \begin{equation} - \qty[ \Psi^i( z, \bz ) ]^\star + \qty[ \Psi^i( z, \barz ) ]^\star = - \frac{1}{\bz}\, \Psi_i^*\qty(\frac{1}{\bz}, \frac{1}{z}). + \frac{1}{\barz}\, \Psi_i^*\qty(\frac{1}{\barz}, \frac{1}{z}). \label{eq:complex-plane-conjugate} \end{equation} The boundary conditions then become: @@ -1117,7 +1117,7 @@ In the same way we can recast the stress-energy tensor components~\eqref{eq:stre \cT( z ) = - \frac{\pi T}{2}\, - \Psi^*_i( z )\, \lrpartial{z} \Psi^i( z ) + \Psi^*_i( z )\, \lripd{z} \Psi^i( z ) + \cC( z ), \end{equation} @@ -1138,11 +1138,11 @@ The fields expansion in modes thus reads \begin{equation} \Psi^i(z) = - \sum\limits_{n \in \Z} b_n\, \Psi^i_{n}(z), + \infinfsum{n} b_n\, \Psi^i_{n}(z), \qquad \Psi^*_{i}(z) = - \sum\limits_{n \in \Z} b^*_n\, \Psi^*_{n. i}(z). + \infinfsum{n} b^*_n\, \Psi^*_{n. i}(z). \label{eq:complex-plane-mode-expansion} \end{equation} The anti-commutation relations among the operators are @@ -1226,13 +1226,13 @@ Consider the NS expansion in modes of the double fields: \begin{eqnarray} \Psi^i( z ) & = & - \sum\limits_{n \in \Z}\, + \infinfsum{n}\, \sum\limits_{i_0}\, b_{(n,\, i_0)}\, \Psi^i_{( n,\, i_0 )}( z ), \\ \Psi^{*}_i( z ) & = & - \sum\limits_{n \in \Z}\, + \infinfsum{n}\, \sum\limits_{i_0}\, b^*_{(n,\, i_0)}\, \Psi^{*}_{( n,\, i_0 ),\, i}( z ), \end{eqnarray} @@ -1255,4 +1255,1190 @@ and \end{equation} +\subsubsection{Twisted Complex Fermions: Preliminaries} + +We can then generalise the discussion about $N_f = 1$ complex fermions in the presence of $N$ point-like defects which we will show to be primary boundary changing operators (i.e. plain and excited spin fields). +Let +\begin{equation} + \begin{cases} + R_{(t)} & = e^{i \pi \alpha_{( t )}} \in \U{ 1 } + \\ + R_{(t)}^* & = e^{-i \pi \alpha_{( t )}} \in \U{ 1 } + \end{cases} +\end{equation} +such that $0 < \alpha_{( t )} < 2$. The boundary conditions are: +\begin{equation} + \begin{cases} + \Psi( x - i\, 0^+ ) & = e^{i \pi \alpha_{( t )}} \Psi( x + i\, 0^+) + \\ + \Psi^*( x - i\, 0^+ ) & = e^{-i \pi \alpha_{( t )}} \Psi^*( x + i\, 0^+ ) + \end{cases}, +\end{equation} +for $x \in ( x_{(t)}, x_{(t-1)} )$, and +\begin{equation} + \begin{cases} + \Psi( x - i\, 0^+ ) & = \Psi( x + i\, 0^+) + \\ + \Psi^*( x - i\, 0^+ ) & = \Psi^*( x + i\, 0^+ ) + \end{cases}, +\end{equation} +for $x < 0$. +These boundary conditions can be recast in the form of monodromy factors. +With a loop around $x_{(t)}$ we find +\begin{equation} + \Psi\qty( x_{(t)} + \delta e^{i 0^+} ) + = + e^{i \pi \qty( \alpha_{( t )} - \alpha_{( t + 1 )} )}\, + \Psi( x_{(t)} + \delta e^{2 \pi i} ), +\end{equation} +where $\delta \in \R^+$ is small enough and the $\pm$ in the phase represents the position relative to the real axis ($+$ is in the upper half plane \ccH, while $-$ in the lower half plane \bccH).\footnotemark{} +\footnotetext{% + More precisely $0 < \delta < \min\qty( \abs{x_{(t-1)} - x_{(t)}}, \abs{x_{(t)} - x_{t+1}} )$. +} +We then define the convenient combination: +\begin{equation} + \epsilon_{(t)} + = + \alpha_{( t+1 )} - \alpha_{( t )} + + + \theta( \alpha_{( t )} -\alpha_{( t+1 )} - 1 ) + - + \theta( \alpha_{( t+1 )} - \alpha_{( t )} - 1 ) +\end{equation} +such that:\footnotemark{} +\footnotetext{% + \label{foot:other_range} + Notice that the choice of the range for $\epsilon_{(t)}$ is not unique. + We can choose $0 < \alpha_{( t )} < 2$ leading to $\epsilon_{(t)} = \alpha_{( t+1 )} - \alpha_{( t )} + 2 \theta( \alpha_{( t )} - \alpha_{( t+1 )} )$. + Then in this case $\epsilon_{(t)} = 2 - \bepsilon_{(t)}$ and $\epsilon_{(t)}, \, \bepsilon_{(t)} \in \qty( 0, 2 )$. + We will however stick to the first definition in the following sections since it allows us to consider the NS case as special. +} +\begin{equation} + -1 < \epsilon_{(t)} < 1, \qquad \forall t = 1, 2, \dots, N. +\end{equation} +The previous loop around $x_{(t)}$ induces a monodromy +\begin{equation} + \begin{cases} + \Psi( x_{(t)} + \delta e^{i 0^+} ) + & = + e^{-i \pi \epsilon_{(t)}} \Psi( x_{(t)} + \delta e^{2 i \pi^+} ) + \\ + \Psi^*( x_{(t)} + \delta e^{i 0^+} ) + & = + e^{-i \pi \bepsilon_{(t)}} \Psi^*( x_{(t)} + \delta e^{2 i \pi^+} ), + \end{cases} + \label{eq:monodromy-factors} +\end{equation} +where $\bepsilon_{(t)} = - \epsilon_{(t)} \Rightarrow -1 < \bepsilon_{(t)} < 1$, thus showing a symmetry under the exchange of: +\begin{equation} + \Psi \longleftrightarrow \Psi^* + \qquad \Rightarrow \qquad + \epsilon_{(t)} \longleftrightarrow \bepsilon_{(t)}. +\end{equation} + + +\subsubsection{Usual Twisted Fermions} +\label{sec:usual-twisted-fermion} + +As a reference for future discussion, we consider the case of one complex fermion in the presence of one twisted boundary condition with the defects located at zero and infinity. +We take $N = 2$ and $x_{(1)} = \infty$ and $x_{(2)} = 0$. +For simplicity we denote with $\epsilon$ the argument of the monodromy factor arising from the presence of the cut on the interval $( 0, +\infty)$. + +In order to fulfill the requests \eqref{eq:monodromy-factors} we write the +modes as: +\begin{equation} + \begin{split} + \Psi_n^{( \rE )} & = \cN_{\Psi}\, z^{-n + \rE}, + \\ + \Psi_n^{*\, ( \brE )} & = \cN_{\Psi}\, z^{-n + \brE}, + \end{split} + \label{eq:usual-twisted-modes} +\end{equation} +such that +\begin{equation} + \begin{split} + \rE = n_{\rE} + \frac{\epsilon}{2}, + & \quad + n_{\rE} \in \Z, + \\ + \brE = n_{\brE} + \frac{\bepsilon}{2}, + & \quad + n_{\brE} \in \Z. + \end{split} +\end{equation} +Together with the integer factor $n_{\rE}$ and $n_{\brE}$ we also define a third integer for later convenience:\footnotemark +\footnotetext{% + The choice discussed in \Cref{foot:other_range} implies $\rL = n_{\rE} + n_{\brE} +1$. + We can swap the definitions by exchangin $\bepsilon_{(t)} \leftrightarrow \bepsilon_{(t)} + 2$ and $n_{\brE} \leftrightarrow n_{\brE} - 1$. +} +\begin{equation} + \rL = \rE + \brE = n_{\rE} + n_{\brE} \in \Z. + \label{eq:usual_integer_factor} +\end{equation} +To extract creation and annihilation operators with the conserved product~\eqref{eq:conserved-product-complex-plane}, we define the dual basis as: +\begin{eqnarray} + \dual{\Psi}_n^{( \brE )}( z ) + & = & + \frac{1}{2 \pi \cN\, \cN_{\Psi}} z^{n - 1 - \brE}, + \\ + \dual{\Psi}_n^{*\, ( \rE )}( z ) + & = & + \frac{1}{2 \pi \cN\, \cN_{\Psi}} z^{n - 1 - \rE}. +\end{eqnarray} +This way we compute the usual anti-commutation relations as +\begin{equation} + \lconsprod{\dual{\Psi}_n^{*\, ( \rE )}}{\dual{\Psi}_m^{( \brE )}} + = + \frac{\delta_{n + m, 1 + \rL}}{2 \pi \cN\, \cN_{\Psi}^2} + \quad \Rightarrow \quad + \qty[ b_n,\, b_m^* ]_+ + = + \frac{1}{\pi T \cN_{\Psi}^2} \delta_{n + m, 1 + \rL}, + \label{eq:twisted-fermion-algebra} +\end{equation} +which are constant in time independently from $\rE$ or $\brE$ since the only possible singularities are located at $z = 0$ and $z = \infty$. +We can then expand the fields $\Psi(z)$ and $\Psi^*( z )$ using this or a more conventional basis: +\begin{eqnarray} + \Psi( z ) + & = & + \infinfsum{n} b^{(\rE)}_n \Psi_n^{( \rE )}( z ) + = + \infinfsum{n} b_{n + n_{\rE}} \Psi_n^{( \frac{\epsilon}{2} )}( z ), + \label{eq:usual-twisted-mode-expansion} + \\ + \Psi^*( z ) + & = & + \infinfsum{n} b^{*\, (\brE)}_n\, \Psi_n^{*\, (\brE)}(z) + = + \infinfsum{n} b^{*}_{n+n_{\brE}}\, \Psi_n^{*\, (-\frac{\epsilon}{2})}(z), + \label{eq:usual-twisted-mode-expansion-conjugate} +\end{eqnarray} +where we used the notation $b = b^{( \frac{\epsilon}{2} )}$ and $b^* = b^{*\,( \frac{\epsilon}{2} )}$. + + +\subsubsection{Generic Case With Defects} +\label{sec:generic-twisted} + +We consider one complex fermion in the presence of $N$ defects such that the modes satisfy: +\begin{equation} + \Psi_n( x_{(t)} + \delta e^{2 \pi i^+} ) + = + e^{i \pi \epsilon_{(t)}}\, \Psi_n( x_{(t)} + \delta e^{i 0^+} ) +\end{equation} +for $t = 1, 2, \dots, N$ and $\delta > 0$. +We define the basis of solutions as: +\begin{eqnarray} + \Psi_n( z;\, \qty{x_{(t)},\, \rE_{(t)}} ) + & = & + \cN_{\Psi}\, + z^{-n}\, + \finiteprod{t}{1}{N} \qty( 1 - \frac{z}{x_{(t)}} )^{\rE_{(t)}}, + \label{eq:generic-case-basis} + \\ + \Psi^*_n( z;\, \qty{x_{(t)},\, \brE_{(t)}} ) + & = & + \cN_{\Psi}\, + z^{-n}\, + \finiteprod{t}{1}{N} \qty( 1 - \frac{z}{x_{(t)}} )^{\brE_{(t)}}, + \label{eq:generic-case-basis-conjugate} +\end{eqnarray} +where we generalise the definition of +\begin{eqnarray} + \rE_{(t)} + & = & + n_{\rE_{(t)}} + \frac{\epsilon_{(t)}}{2}, + \quad + n_{\rE_{(t)}} \in \Z, + \\ + \brE_{(t)} + & = & + n_{\brE_{(t)}} + \frac{\bepsilon_{(t)}}{2} + \quad + n_{\brE_{(t)}} \in \Z +\end{eqnarray} +and we define $N$ integer factors analogous to~\eqref{eq:usual_integer_factor}: +\begin{equation} + \rL_{(t)} + = + \rE_{(t)} + \brE_{(t)} + = + n_{\rE_{(t)}} + n_{\brE_{(t)}} \in \Z, + \qquad + \forall t \in \qty{ 1,\, 2,\, \dots,\, N }. +\end{equation} +From the definition of the conserved product~\eqref{eq:conserved-product-complex-plane}, we compute the dual basis: +\begin{eqnarray} + \dual{\Psi}_n( z ) + & = & + \frac{1}{2 \pi \cN\, \cN_{\Psi}}\, + z^{n-1}\, + \finiteprod{t}{1}{N} \qty( 1 - \frac{z}{x_{(t)}} )^{-\brE_{(t)}}, + \\ + \dual{\Psi}^*_n( z ) + & = & + \frac{1}{2 \pi \cN\, \cN_{\Psi}}\, + z^{n-1}\, + \finiteprod{t}{1}{N} \qty( 1 - \frac{z}{x_{(t)}} )^{-\rE_{(t)}}, +\end{eqnarray} +and the conserved products between dual modes: +\begin{equation} + \lconsprod{\dual{\Psi}_n^*}{\dual{\Psi}_m} + = + \frac{1}{2 \pi \cN\, \cN_{\Psi}^2}\, + \oint \ddz\, + z^{n+m-2}\, + \finiteprod{t}{1}{N} \qty( 1 - \frac{z}{x_{(t)}} )^{-\rL_{(t)}}. +\end{equation} +Notice that the products are radially invariant only if +\begin{equation} + \rL_{(t)} \le 0, + \qquad + \forall t \in \qty{ 1,\, 2,\, \dots, N }, + \label{eq:generic-case-negativity-condition} +\end{equation} +since the integrand must not present time dependent singularities on the integration path, thus +\begin{equation} + \begin{split} + \lconsprod{\dual{\Psi}_n^*}{\dual{\Psi}_m} + & = + \frac{1}{2 \pi \cN\, \cN_{\Psi}^2}\, + \oint \ddz\, + \finiteprod{t}{1}{N}\, + \finitesum{k_t}{0}{\abs{\rL_{(t)}}}\, + \binom{\abs{\rL_{(t)}}}{k_t} + \qty( - \frac{1}{x_{(t)}} )^{k_t}\, + z^{k_t + n + m - 2} + \\ + & = + \frac{1}{2 \pi \cN\, \cN_{\Psi}^2}\, p_{1 - n - m}, + \end{split} +\end{equation} +where we defined +\begin{equation} + p_k + = + \finiteprod{t}{1}{N}\, + \finitesum{k_t}{0}{\abs{\rL_{(t)}}}\, + \binom{\abs{\rL_{(t)}}}{k_t} + \qty( - \frac{1}{x_{(t)}} )^{k_t}\, + \delta_{\finitesum{t}{1}{N} k_t,\, k} + \label{eq:generic-conserved-product-factor} +\end{equation} +such that +\begin{eqnarray} + p_{0 \le k \le \finitesum{t}{1}{N} \abs{\rL_{(t)}}} + & \neq & + 0, + \\ + p_{k \le -1} + = + p_{k \ge \finitesum{t}{1}{N} \abs{\rL_{(t)}} + 1} + & = & + 0. +\end{eqnarray} +We can finally write +\begin{equation} + \qty[ b_n,\, b^*_m ]_+ + = + \frac{1}{\pi T \cN_{\Psi}^2}\, p_{1 - n - m}, + \qquad + 1 - \finitesum{t}{1}{N} \abs{\rL_{(t)}} \le n + m \le 1. + \label{eq:generic-case-anti-commutation} +\end{equation} + + +\subsection{Representation of the Algebra: Definition of the In-Vacuum} +\label{sec:invacuum} + +In the previous section we computed the algebra of the operators for different theories. +We now define in-vacua where they are represented to be able to compute the relevant amplitudes. +We show how to recover the NS vacuum and the usual twisted vacuum. +Finally we discuss the vacuum in the presence of a generic number of defects. + + +\subsubsection{NS Fermions} + +The in-vacuum can be correctly obtained either requiring $\Psi( z )$ and $\Psi^*( z )$ to be non singular as $z \to 0$ when applied on the vacuum. +The same request can also be made on $\hPsi( \xi )$ and $\hPsi^*( \xi )$. +In both cases we get the same vacuum which turns out to be \SL{2}{\R} invariant: +\begin{equation} + b_{( n,\, i_0 )} \regvacuum + = + b^*_{( n,\, i_0 )} \regvacuum + = + 0, + \qquad + n \ge 1. +\label{eq:NS_SL2_vacuum} +\end{equation} +The spectrum of the theory is constructed acting with operators $b_{( n,\, i_0 )}$ and $b^*_{( n,\, i_0 )}$ with $n \le 0$. + + +\subsubsection{Twisted Fermion} +\label{sec:twisted-fermion} + +Consider the case of the usual twisted fermion in \Cref{sec:usual-twisted-fermion}. +Define the excited vacuum $\excvacket$ as: +\begin{equation} + b^{(\rE)}_n \excvacket + = + b^{*\, ( \brE )}_n \excvacket + = + 0, + \qquad n \ge 1. + \label{eq:usual-excited-vacuum} +\end{equation} +The introduction of $\rE$ and $\brE$ is necessary to define the vacuum as in previous cases, that is with a range of $n$ independent from them and without singularities as $z \to 0$. +Explicitly we have: +\begin{equation} + \Psi(z) \excvacket + \sim + z^{\rE}\, (\dots), + \qquad + \Psi^*(z) \excvacket + \sim + z^{\brE}\, (\dots). + \label{eq:asymp_beha_Psi_on_exc_vac} +\end{equation} +Comparing with~\eqref{eq:generic-case-basis} and~\eqref{eq:generic-case-basis-conjugate}, the behavior suggests the existence of a hidden operator in $x_{(t)}$ creating $\excvacket$ with $\rE = \rE_{(t)}$ and $\brE = \brE_{(t)}$. + +These relations are subject to consistency conditions since +\begin{equation} + \excvacket + = + \pi T \cN_{\Psi}^2 \qty[b^{(E)}_n,\, b^{*\, ( \brE )}_{\rL+1-n}]_+ \excvacket, +\end{equation} +that is we cannot have two in-annihilators (namely both $b^{(\rE)}_n$ and $b^{*\, ( \brE )}_{\rL+1-n}$) with non vanishing anti-commutation relations. +Specifically we have: +\begin{equation} + 1 \le n \le \rL + \qquad \Rightarrow \qquad + b^{(\rE)}_n \excvacket = 0, + \quad + b^{*\, ( \brE )}_{\rL + 1 - n} \excvacket = 0, +\end{equation} +that is +\begin{equation} + \excvacket + = + \pi T \cN_{\Psi}^2 \qty[ b^{(\rE)}_n,\, b^{*\, ( \brE )}_{\rL + 1 - n} ]_+ \excvacket + = + 0, +\end{equation} +which is not consistent (see~\Cref{fig:inconsistent-theories} for a graphical reference): the theory does not exist. +We shall therefore consider only cases such that +\begin{equation} + \rL \le 0, +\end{equation} +analogously to~\eqref{eq:generic-case-negativity-condition}. + +Moreover notice that for $\rL \le -1$ both $b^{(\rE)}_{\rL \le n \le 0}$ and $b^{*\, ( \brE )}_{\rL \le n \le 0}$ are in- and out-creation operators. + +\begin{figure}[tbp] + \centering + \includegraphics[width=0.5\linewidth]{img/in-annihilators.pdf} + \caption{As a consistency condition, we have to exclude the values of + $\rL$ for which both $b^{( +E)}_n$ and $b^{*\, ( \brE )}_{\rL + 1 - n}$ are in-annihilators + with a non vanishing anti-commutation relation.} + \label{fig:inconsistent-theories} +\end{figure} + +The vacuum $\excvacket$ is not however associated to the lowest energy. +In fact the usual way to build the vacuum would be to require $\Psi( z )$ and $\Psi^*( z )$ to be non singular as $z \to 0$ for the in-vacuum so that $b^{(\rE)}_n \twsvacket = 0$ for $n > \rE$, and $b^{*\, ( \brE )}_n \twsvacket = 0$ for $n > \brE$. +However this procedure almost always fails to give a good definition of the vacuum. +In fact it works only for NS fermions. +For example when $\epsilon > 0$ we have: +\begin{equation} + 0 + = + \pi T \cN_{\Psi}^2\, + \qty[ b^{(\rE)}_{1 + n_{\rE} },\, b^{*\, ( \brE )}_{n_{\brE} } ]_+ + \twsvacket + = + \twsvacket, +\end{equation} +which is not consistent since both $b^{(\rE)}_{1 + n_{\rE}}$ and $b^{*\, ( \brE )}_{n_{\brE}}$ are annihilators as $1 + n_{\rE} > E$ and $n_{\brE} > \brE$. + +The minimum energy vacuum is instead defined in a proper way on the strip. +Requiring that the action of $\hPsi( \xi )$ and $\hPsi^*( \xi )$ for $\xi \to -\infty$ on the vacuum is well defined we get +\begin{eqnarray} + b^{(\rE)}_n \twsvacket + & = & + 0, + \qquad + n > \rE + \frac{1}{2}, + \\ + b^{*\, ( \brE )}_m \twsvacket + & = & + 0, + \qquad + m > \brE + \frac{1}{2}. +\end{eqnarray} +This is a good definition of the vacuum as $-\frac{1}{2} < \frac{\epsilon}{2} = -\frac{\bepsilon}{2} < \frac{1}{2}$ implies that $b^{(\rE)}_n$ and $b^{*\, ( \brE )}_m$ are annihilation operators for $n \ge n_{\rE}+1 > \rE + \frac{1}{2}$ and $m \ge n_{\brE}+1 > \brE + \frac{1}{2}$ so that +\begin{equation} + 0 + = + \pi T \cN_{\Psi}^2\, + \qty[ b^{(\rE)}_{n},\, b^{*\, ( \brE )}_{m} ]_+ + \twsvacket + = + \delta_{n + m,\, \rE+\brE+1 } + \twsvacket + = + 0. +\end{equation} +This way we get a consistent definition of the twisted vacuum. +This is however not generally \SL{2}{\R} invariant as we will show later when defining stress-energy tensor.\footnotemark{} +\footnotetext{% + Notice that the second choice of $\epsilon$ interval discussed in~\Cref{foot:other_range} needs to distinguish between two cases: $0 < \frac{\epsilon}{2} < \frac{1}{2}$ (and $\frac{1}{2} < \frac{\bepsilon}{2} < 1$) and $\frac{1}{2} < \frac{\epsilon}{2} < 1$ (and $0 < \frac{\bepsilon}{2} < \frac{1}{2}$). +} + +The vacua $\excvacket$ and $\twsvacket$ are related. +Consider for example the case $n_{\rE} \ge 1$ and the definition: +\begin{eqnarray} + b^{(\rE)}_n \excvacket & = & 0, \qquad n \ge 1, + \\ + b^{(\rE)}_n \twsvacket & = & 0, \qquad n \ge 1 + n_{\rE}. +\end{eqnarray} +Then for $1 \le n \le n_{\rE}$ the modes $b^{(\rE)}_n$ act as a annihilation operator on $\excvacket$ and as a creation operator on $\twsvacket$: +\begin{equation} + \excvacket + \propto + b^{(\rE)}_{n_{\rE}}\, b^{(\rE)}_{n_{\rE} - 1}\, \dots\, b^{(\rE)}_1 + \twsvacket. + \label{eq:usual-twisted-vacuum-relation} +\end{equation} +Moreover, since $\rL = n_{\rE} + n_{\brE} \le 0 \Rightarrow n_{\brE} \le -1$, we have: +\begin{eqnarray} + b^{*\, ( \brE )}_m \excvacket & = & 0, \qquad m \ge 1, + \\ + b^{*\, ( \brE )}_n \twsvacket & = & 0, \qquad m \ge 1 - \abs{n_{\brE}}, +\end{eqnarray} +which leads to: +\begin{equation} + \twsvacket + \propto + b^{*\, ( \brE )}_0\, b^{*\, ( \brE )}_1\, \dots\, b^{*\, ( \brE )}_{1 - \abs{n_{\brE}}} + \excvacket. + \label{eq:usual-twisted-fermion-conformal-twisted} +\end{equation} +The consistency of the definition can be checked requiring +\begin{equation} + \excvacket + = + \qty( \pi T \cN_{\Psi}^2 )^{n_{\rE}}\, + b^{(\rE)}_{n_{\rE}}\, + b^{(\rE)}_{n_{\rE}-1}\, + \dots\, + b^{(\rE)}_1\, + b^{*\, ( \brE )}_0\, + b^{*\, ( \brE )}_1\, + \dots\, + b^{*\, ( \brE )}_{1 - \abs{n_{\brE}}} + \excvacket, +\end{equation} +where the number of $b$ operators has to match the number of $b^*$ + operators: +\begin{equation} + n_{\rE} + n_{\brE} + = + \rE + \brE + = + \rL + = + 0. + \label{eq:twisted-fermion-consistency} +\end{equation} +The same procedure applies also in the case $n_{\rE} \le 0$, leading to the same result. +As a consequence of~\eqref{eq:twisted-fermion-consistency}, we express the twisted vacuum as: +\begin{eqnarray} + b^{(\rE)}_n \twsvacket & = & 0, \qquad n \ge 1 + n_{\rE}, + \\ + b^{*\, ( \brE )}_m \twsvacket & = & 0, \qquad m \ge 1 - n_{\rE}. +\end{eqnarray} + + +\subsubsection{Generic Case with Defects} + +Since the fields in presence of defects behave as NS fields in the limit $z \to 0$, we define the vacuum in the usual fashion by requiring a finite limit $\lim\limits_{z \to 0} \Psi( z ) \Gexcvacket$. +As in the NS we get the representation: +\begin{equation} + b_{n} \Gexcvacket + = + b^*_{n} \Gexcvacket + = + 0, + \qquad n \ge 1. + \label{eq:generic_vacuum} +\end{equation} + + +\subsection{Asymptotic Fields and Vacua} +\label{sect:asymp_fields} + +In this section we define the asymptotic in-field and out-field and discuss how their vacua are related to the theory with defects. +The relation is ``radial time dependent'' explicitly showing that an interaction is hidden in the defects. +In particular the vacuum for the theory with defects can be identified with \SL{2}{\R} in-field vacuum while it is connected by a Bogoliubov transformation to the \SL{2}{\R} out-field vacuum. + +In the following we use the expansion of +\begin{equation} + P\qty(z;\, \qty{ x_{(t)},\, \rE_{(t)} } ) + = + \finiteprod{t}{1}{N} + \qty( 1- \frac{z}{x_{(t)}} )^{\rE_{(t)}}, +\end{equation} +around the origin and infinity with coefficients +\begin{eqnarray} + \cmode{k}{0}{\rE_{(t)}}{x_{(t)}} + & = & + \sum_{ \qty{k_t} \in \N^N} + \finiteprod{t}{1}{N} + \qty[ \binom{\rE_{(t)}}{k_t} \qty( - \frac{1}{x_{(t)}} )^{k_t} ] + \delta_{\finitesum{t}{1}{N} k_t, k} + \\ + \cmode{k}{\infty}{\rE_{(t)}}{x_{(t)}} + & = & + \sum_{ \qty{k_t} \in \N^N} + \finiteprod{t}{1}{N} + \qty[ \binom{\rE_{(t)}}{k_t} \qty( - x_{(t)} )^{k_t - \rE_{(t)}} ] + \delta_{\finitesum{t}{1}{N} k_t, k}, +\end{eqnarray} +so that we can write +\begin{equation} + \begin{split} + P\qty(z;\, \qty{x_{(t)}, \rE_{(t)}}) + = + \begin{cases} + \zeroinfsum{k} + \cmode{k}{0}{\rE_{(t)}}{x_{(t)}}\, z^k, + & \qfor + \abs{z} < x_{(N)} + \\ + \zeroinfsum{k} + \cmode{k}{\infty}{\rE_{(t)}}{x_{(t)}}\, z^{-k+\finitesum{t}{1}{N} \rE_{(t)}}, + & \qfor + \abs{z} > x_{(1)} + \end{cases} + \end{split} +\end{equation} +We do not discuss intermediate fields, that is expansions for $x_{(t)} < \abs{z} < x_{(t-1)}$, as it is not possible to disentangle the effects of defects before and after this range. +The vacuum in the presence of defects is in fact related to the radial ordering of the operators associated with the defects as we argue later on. + + +\subsubsection{Asymptotic In-field and Its Vacuum} + +Consider the definitions of the basis of solutions~\eqref{eq:generic-case-basis} and expand around $z = 0$.\footnotemark{} +\footnotetext{% + Similarly we could have considered~\eqref{eq:generic-case-basis-conjugate} to begin with. + Analogous relations can in fact be written for $b_n^{*\, ( 0 )}$ with the substitutions of $\rE_{(t)} $ with $\brE_{(t)}$. +} +We get for $0 \le \abs{z} < x_{(N)}$: +\begin{equation} + \Psi_n( z ) + = + \zeroinfsum{k} + \cmode{k}{0}{\rE_{(t)}}{x_{(t)}}\, + \Psi^{( 0 )}_{n-k}( z ), + \label{eq:expansion-around-0} +\end{equation} +where $\Psi^{( 0 )}_n( z ) = \cN_{\Psi} z^{-n}$ as in~\eqref{eq:usual-twisted-modes} with $\rE = 0$ which are the modes of a untwisted fermion, i.e.\ a plain NS fermion. +The previous expansion connects the asymptotic behavior of the modes of the fermion with defects with the modes of a NS fermion close to the origin. +We can now connect the operators of the system with defects with those of the asymptotic in-field. +To this purpose we substitute the expansion~\eqref{eq:expansion-around-0} with the usual expression of the modes~\eqref{eq:complex-plane-mode-expansion}: +\begin{equation} + \Psi( z ) + = + \infinfsum{n} + b_n\, \Psi_n( z ) + \underset{\abs{z} < x_{(N)}}{=} + \Psi^{(\text{in})}( z ) + = + \infinfsum{n} + b_n^{( 0 )}\, \Psi_n^{( 0 )}( z ) +\end{equation} +thus leading to +\begin{equation} + b_n^{( 0 )} + = + \zeroinfsum{k} + b_{n + k}\, \cmode{k}{0}{\rE_{(t)}}{x_{(t)}}. +\end{equation} +These expressions can be inverted writing $\Psi^{( 0 )}_{n}( z ) = \Psi_{n}( z )\, P\qty(z;\, \qty{x_{(t)},\, -\rE_{(t)}} )$. +We then get: +\begin{equation} + \begin{split} + b_n + & = + \zeroinfsum{k} + \cmode{k}{0}{-\rE_{(t)}}{x_{(t)}}\, b^{(0)}_{n + k}, + \end{split} +\end{equation} +Annihilation operators of the asymptotic theory, i.e.\ operators with positive index, are therefore expressed only using annihilation operators of the theory with defects. +We can thus set +\begin{equation} + \Gexcvacket = \regvacuumin. +\end{equation} + + +\subsubsection{Asymptotic Vacua and Bogoliubov Transformations} + +Revisiting the previous section, we can also explicitly compute the expansion for $\abs{z} > x_{(1)}$. +Define for simplicity $\rM = \finitesum{t}{1}{N} \rE_{(t)}$). +We then get: +\begin{equation} + \Psi_n( z ) + = + \zeroinfsum{k} + \cmode{k}{\infty}{\rE_{(t)}}{x_{(t)}}\, + \Psi^{( 0 )}_{n+k-\rM}( z ). +\end{equation} +The formula connects the asymptotic behavior of the modes of the fermion with defects to modes of a NS fermion which can be seen close to the infinity. + +Out-operators can thus be connected to the theory with defect through:\footnotemark{} +\footnotetext{% + To avoid a redundant notation we do not introduce an object $\Psi^{(\infty)}_n( z )$. + Even though it would have been in principle correct, we would have also found that $\Psi^{(\infty)}_n( z ) = \Psi^{(0)}_n( z )$. +} +\begin{equation} + \Psi(z) + = + \infinfsum{n} + b_n\, \Psi_n( z ) + \underset{|z|>x_1}{=} + \Psi^{(\text{out})}( z ) + = + \infinfsum{n} + b_n^{( \infty )}\, \Psi_n^{( 0 )}( z ). +\end{equation} +We get: +\begin{equation} + b_n^{( \infty )} + = + \zeroinfsum{k} + b_{n + \rM - k}\, + \cmode{k}{\infty}{\rE_{(t)}}{x_{(t)}}. + \label{eq:b_inf-b} +\end{equation} +The inverse of the expression is: +\begin{equation} + \begin{split} + b_n + & = + \zeroinfsum{k} + \cmode{k}{\infty}{-\rE_{(t)}}{x_{(t)}}\, + b^{( \infty )}_{n + \rM - k}. + \end{split} +\end{equation} +As we will show later however $\rM = 0$. +Annihilation operators of the asymptotic theory, i.e.\ operators with positive index, are thus expressed using both annihilation and creation operators of the theory with defects while creators, i.e.\ operators with non negative index, are expressed by means of creation operators only. +It follows from the vacuum definition that: +\begin{equation} + \begin{split} + \qty(% + \tffC_0\, b_1^{(\infty)} + + + \text{creation op.} + ) \Gexcvacket + & = + 0, + \\ + \qty(% + \tffC_0\, b_2^{(\infty)} + + + \tffC_1\, b_1^{(\infty)} + + + \text{creation op.} + ) \Gexcvacket + & = + 0, + \\ + & \vdots + \end{split} +\end{equation} +where $\tffC_n = \cmode{n}{\infty}{-\rE_{(t)}}{x_{(t)}}$ for brevity. +This means that the vacuum for the asymptotic out-field is non trivially connected to the vacuum of the theory with defects. +More explicitly we have: +\begin{equation} + \begin{split} + \Gexcvacket + & = + \cN_{(\text{out})}(\qty{x_{(t)},\, \rE_{(t)},\, \brE_{(t)}}) + \\ + & \times + e^{% + \infzerosum{m} \infzerosum{n} + \cM_{m n}^{(\text{out})}\qty(\qty{x_{(t)},\, \rE_{(t)},\, \brE_{(t)}})\, + b^{*\, ( \infty )}_{m}\, b^{( \infty )}_{n} + } + \regvacuumout, + \end{split} +\end{equation} +so that the two \SL{2}{\R} vacua are connected by a Bogoliubov transformation. +More precisely we get (see appendix \ref{sec:details_reflection} for details) +\begin{equation} + \qty(% + \Psi^{(\text{out},\, +)}( z ) + + + \ccL_{(1)}\qty[\Psi^{(\text{out},\, -)}]( z ) + ) + \Gexcvacket + = + 0, + \qquad + \abs{z} > x_{(1)}, + \label{eq:reflection condition_out_field_generic_vacuum} +\end{equation} +where +\begin{equation} + \ccL_{(t)}\qty[\Psi]( z ) + = + \oint\limits_{\abs{w} > x_{(t)}} \ddw + \frac{P\qty( z;\, \qty{x_{(t)},\, \rE_{(t)}} )\, P\qty( w;\, \qty{x_{(t)},\, -\rE_{(t)}} ) - 1}{z - w}\, + \Psi( w ). +\end{equation} +The corresponding equation for $\Psi^{*\, (\text{out})}( z )$ can be found with the substitution $\rE \rightarrow \brE$. +Notice that the kernel of the integral is nothing else (up to a multiplicative constant) but the regularised propagator, that is the propagator in the presence of the defects to which the NS propagator has been subtracted. +The previous equation is solved explicitly by: +\begin{equation} + \begin{split} + \Gexcvacket + & = + \cN\qty(\qty{x_{(t)},\, \rE_{(t)},\, \brE_{(t)}})~ + \\ + & \times + e^{% + \oint_{\abs{z} > x_{(1)}} \ddz + \Psi^{*\, (\text{out},\, -)}( z )\, + \ccL\qty[ \Psi^{(\text{out},\, -)} ]( z ) + } + \regvacuumout. + \end{split} +\end{equation} +In the previous equation there is no need to specify the relation between $\abs{z}$ and $\abs{w}$ since $\Psi^{*\, (\text{out}, -)}( z )$ and $\Psi^{(\text{out}, -)}( w )$ anti-commute. +From the same expression for $\Psi^{*\, (\text{out}, -)}( z )$ we deduce that $\brE_{(t)} = -\rE_{(t)}$. + + +\subsection{Contractions and Stress-energy Tensor} +\label{sec:contraction_and_T} + +Given the definition of the algebra of the operators and its representation, we can finally define the normal ordering operation and proceed to compute the contractions and \ope of the operators. +The procedure ultimately leads to the definition of the stress-energy tensor. This is enough to show that the theory is a time dependent \cft since the stress-energy tensor satisfies the canonical \ope. + + +\subsubsection{NS Complex Fermion} + +First of all we deal with the simple case of NS fermions. +Using the algebra~\eqref{eq:ns-algebra} we compute the \ope of fermion fields as: +\begin{equation} + \Psi^i( z )\, \Psi^*_j( w ) + = + \no{\Psi^i( z )\, \Psi^*_j( z )} + + + \frac{1}{\pi T} \frac{\tensor{\delta}{^i_j}}{ z - w }, + \qquad + \abs{w} < \abs{z}, +\end{equation} +where the operation $\no{\cdot}$ is the normal ordering with respect to the \SL{2}{R} vacuum defined in~\eqref{eq:NS_SL2_vacuum}. +We then get the expression of the stress-energy tensor: +\begin{equation} + \begin{split} + \cT( z ) + & = + \lim\limits_{w \to z} + \qty[% + -\frac{ \pi T }{2} + \qty( \Psi^*_i( z )\, \partial_w \Psi^i( w ) - \partial_z \Psi^*_i( z )\, \Psi^i( w ) ) + + + \frac{N_f}{ ( z - w )^2 } + ] + \\ + & = + -\frac{\pi T}{2} \no{ \Psi^*_i( z ) \lripd{z} \Psi^i( z ) }. + \end{split} +\end{equation} +We are then able to derive the necessary minimal subtraction: +\begin{equation} + \ffh( z - w ) = \frac{N_f}{( z - w )^2}. +\end{equation} + + +\subsubsection{Twisted Fermion} + +We now focus on $N_f = 1$ theories. +First of all we consider the simplest case of the usual twisted fermion with the mode expansion~\eqref{eq:usual-twisted-mode-expansion} and~\eqref{eq:usual-twisted-mode-expansion-conjugate}. +Using the anti-commutation relations~\eqref{eq:twisted-fermion-algebra} we can compute the \ope: +\begin{equation} + \Psi( z )\, \Psi^*( w ) + = + \noE{ \Psi( z )\, \Psi^*( w ) } + + + \frac{1}{\pi T} \qty( \frac{z}{w} )^{\rE} \frac{1}{z - w}, + \qquad + \abs{w} < \abs{z}, +\end{equation} +and +\begin{equation} + \Psi^*( w )\, \Psi( z ) + = + \noE{ \Psi^*( w )\, \Psi( z ) } + + + \frac{1}{\pi T} \qty( \frac{w}{z} )^{\brE} \frac{1}{w - z}, + \qquad + \abs{w} > \abs{z}. +\end{equation} +If we require that the previous results can be assembled in a well defined continuous radial ordering $\rR\qty( \Psi( z ) \Psi^*( w ) )$ we need to set $\rE = -\brE$ so we can write +\begin{equation} + \rR\qty( \Psi( z ) \Psi^*( w ) ) + = + \noE{ \Psi( z )\, \Psi^*( w ) } + + + \frac{1}{\pi T} \qty( \frac{z}{w} )^{\rE} \frac{1}{z - w}. +\end{equation} + +The same result can be reached by computing the stress-energy tensor starting from the previous expressions. +We have two ways to construct it depending on the ordering of the classical expression: +\begin{equation} + \begin{split} + \cT( z ) + & = + \lim\limits_{\substack{w \to z \\ \abs{w} < \abs{z}}} + \qty[% + -\frac{\pi T}{2} + \qty(% + \Psi^*( z ) \partial_w \Psi( w ) + - + \partial_z \Psi^*( z ) \Psi( w ) + ) + + + \frac{1}{( z- w )^2} + ] + \\ + & = + -\frac{\pi T}{2} + \no{ \Psi^*( z ) \lripd{z} \Psi( z ) } + + + \frac{\rE^2}{2 z^2}, + \end{split} + \label{eq:T_excited_vacuum} +\end{equation} +or +\begin{equation} + \begin{split} + \cT( z ) + & = + \lim\limits_{\substack{w \to z \\ \abs{w} < \abs{z}}} + \qty[% + -\frac{\pi T}{2} + \qty(% + -\partial_z \Psi( z ) \Psi^*( w ) + + + \Psi( z ) \partial_w \Psi^*( w ) + ) + + + \frac{1}{( z - w )^2} + ] + \\ + & = + -\frac{\pi T}{2} + \no{ \Psi^*( z ) \lripd{z} \Psi( z ) } + + + \frac{\brE^2}{2 z^2}, + \end{split} +\end{equation} +which however must coincide for consistency. +Since +\begin{equation} + \no{ \Psi( z ) \lripd{z} \Psi^*( z ) } + = + \no{ \Psi^*( z ) \lripd{z} \Psi( z ) } +\end{equation} +then we must then require $\rE^2 = \brE^2$. +We can get a stronger constraint by computing the \ope $\cT( z )\, \cT( w )$. +In fact the cancellation of the cubic divergence requires $\rE + \brE = 0$. +From now on we will therefore use the notation \eexcvacket instead of \excvacket. + +From the usual definition of the stress-energy tensor in terms of the Virasoro generators $\cT( z ) = \infinfsum{k} L_k z^{-k-2}$, we extract the operators $L_k$ from any of the previous definitions: +\begin{equation} + \begin{split} + L_{(\rE) k} + & = + -\frac{\pi T}{2}\, \cN_{\Psi}^2\, + \infinfsum{k} + \noE{b^{*\, ( \brE )}_n\, b^{(\rE)}_{k + 1 - n}} + ( 2n - k + 2 \rE - 1 ) + + + \frac{\rE^2}{2} \delta_{k,0} + \\ + & = + \frac{\pi T}{2}\, \cN_{\Psi}^2\, + \finitesum{n}{1}{+\infty} + \left[ + ( 2n - k + 2 \rE - 1 ) + \noE{b^{(\rE)}_{k + 1 - n}\, b^{*\, ( \brE )}_n} + \right. + \\ + & + + \left. + ( 2n - k - 2 \rE - 1 ) + \noE{b^{*\, ( \brE )}_{k + 1 - n}\, b^{(\rE)}_n} + \right] + + + \frac{\rE^2}{2} \delta_{k,0}. + \end{split} +\end{equation} + +%%% TODO %%% + + Looking back at the analysis of the excited and twisted vacua, we already + hinted to the fact that they are not in general \SL{2}{R} invariant. + In particular we can see that that the excited vacua $\eexcvacket$ + is a primary field + \begin{equation} + L_{(\rE) k > 0} \eexcvacket =0, + \qquad + L_{(\rE) 0} \eexcvacket = \frac{\rE^2}{2} \eexcvacket + , + \end{equation} + with non trivial conformal dimensions + $\Delta\qty( \eexcvacket ) = \frac{\rE^2}{2}$. + This operator is an excited spin field $\rS_{\rE_{(t)}}\qty( x )$ + inserted at $x=0$ whose bosonized expression is given by + \begin{equation} + \rS_{\rE}\qty( x ) = e^{i \rE \phi( x )}, + \end{equation} + where $\phi$ is such that + \begin{equation} + \left\langle \phi( z ) \phi( w ) \right\rangle = -\frac{1}{( z - + w)^2} + . + \end{equation} + + In fact the minimal conformal dimension is achieved + for $n_{\rE}=n_{\brE}=0$, i.e. + $ + \Delta\qty( \twsvacket ) = \frac{\epsilon^2}{8} + $ + and we know this is the basic spin field. + We can further check this idea + by showing that the conformal dimensions are consistent. + Using \eqref{eq:usual-twisted-fermion-conformal-twisted} we get + \begin{equation} + \begin{split} + L_{(\rE) 0}\twsvacket + & = + L_0 + \qty( b^{*\, ( \brE )}_0 b^{*\, ( \brE )}_{-1} \dots b^{*\, ( \brE )}_{2-n_{\rE}} \eexcvacket) + \\ + & = + \left[ \sum\limits^{n_{\rE}}_{n = 1} ( n - \frac{\rE + 1}{2} ) + +\frac{\rE^2}{2} + \right] \twsvacket + = +\frac{1}{8} \epsilon^2\twsvacket . + \end{split} + \end{equation} + + + \subsection{Generic Case With Defects} + + We will now apply the same procedure to the generic case of one complex + fermion in the presence of an arbitrary number of spin fields with respect + to the vacuum we introduced in \eqref{eq:generic_vacuum}. We will consider the mode expansion + \eqref{eq:generic-case-basis} and \eqref{eq:generic-case-basis-conjugate} as + well as the anti-commutation relations + \eqref{eq:generic-case-anti-commutation}. + + + As in the usual twisted case, we will first consider the contraction of + the field $\Psi$ and $\Psi^*$ and then move to the stress-energy + tensor. Using the anti-commutation relations + and $\sum\limits_{k \in \mathds{Z}} p_k z^k = \finiteprod{t}{1}{N} \qty( 1 - \frac{z}{x_{(t)}} )^{-\rL_{(t)}}$ + where $p_k$ is defined in \eqref{eq:generic-conserved-product-factor}. + We have: + \begin{equation} + \Psi( z ) \Psi^*( w ) = \no{ \Psi( z ) \Psi^*( w ) } + + \frac{1}{\pi T} \frac{1}{z - w} \finiteprod{t}{1}{N} \qty( 1 - \frac{z}{x_{(t)}} + )^{\rE_{(t)}} \qty( 1 - \frac{w}{x_{(t)}} )^{-\rE_{(t)}}, + \end{equation} + as well as + \begin{equation} + \Psi^*( z ) \Psi( w ) = \no{ \Psi^*( z ) \Psi( w ) } + + \frac{1}{\pi T} \frac{1}{z - w} \finiteprod{t}{1}{N} \qty( 1 - \frac{z}{x_{(t)}} + )^{\brE_{(t)}} \qty( 1 - \frac{w}{x_{(t)}} )^{-\brE_{(t)}}, + \end{equation} + both for $\abs{w} < \abs{z}$. + If we require that the previous results can be assembled in a + well defined continuous radial ordering + $ R\qty[ \Psi( z ) \Psi^*( w ) ] + $ we need to set $\rE_{(t)}=-\brE_{(t)}$ so we can write + \begin{equation} + R\qty[ \Psi( z ) \Psi^*( w ) ] + = + \no{ \Psi( z ) \Psi^*( w ) } + + \frac{1}{\pi T} \frac{1}{z - w} + \finiteprod{t}{1}{N} \qty( 1 - \frac{z}{x_{(t)}})^{\rE_{(t)}} + \qty( 1 - \frac{w}{x_{(t)}} )^{-\rE_{(t)}} + . + \label{eq:gen_Radial_order} + \end{equation} + We can then expand the results around $z$: + \begin{eqnarray} + \begin{aligned} + R\qty[\Psi( z ) \Psi^*( w )] + & = + \no{ \qty(\Psi\Psi^*)( z ) } + + + \no{ \qty(\Psi\partial\Psi^*)( z ) }\, (w-z) + \\ + & + \frac{1}{\pi T} \left[ + \frac{-1}{w- z} + + \finitesum{t}{1}{N} \frac{\rE_{(t)}}{z - x_{(t)}} \right. + \\ + & \left. + - \frac{1}{2} \qty( + \finitesum{t}{1}{N} \sum\limits_{u \neq t} + \frac{\rE_{(t)} \rE_{(u)}}{( z - x_{(t)}) ( z - x_{(u)} )} + + \finitesum{t}{1}{N} \frac{\rE_{(t)} \qty( \rE_{(t)} - 1 )}{( z - x_{(t)} )^2} ) + ( w - z ) + \right] + \\ + & + \order{( w - z )^2} + , + \end{aligned} + \end{eqnarray} + and around $w$ + \begin{eqnarray} + \begin{aligned} + R\qty[\Psi( z ) \Psi^*( w )] + & = + \no{ \qty(\Psi\Psi^*)( w ) } + + + \no{ \qty(\partial\Psi\Psi^*)( w ) }\, (z-w) + \\ + & + \frac{1}{\pi T} \left[ + \frac{1}{z- w} + + \finitesum{t}{1}{N} \frac{\rE_{(t)}}{w - x_{(t)}} \right. + \\ + & \left. + + \frac{1}{2} \qty( + \finitesum{t}{1}{N} \sum\limits_{u \neq t} + \frac{\rE_{(t)} \rE_{(u)}}{( w - x_{(t)}) ( w - x_{(u)} )} + + \finitesum{t}{1}{N} \frac{\rE_{(t)} \qty( \rE_{(t)} - 1 )}{( w - x_{(t)} )^2} ) + ( z - w ) + \right] + \\ + & + \order{( z - w )^2} + , + \end{aligned} + \end{eqnarray} + so that the stress-energy tensor becomes: + \begin{align*} + \cT( z ) & = -\frac{\pi T}{2} \no{ \Psi( z ) \lripd{z} + \Psi^*( z ) } + + \frac{1}{2} \qty( \finitesum{t}{1}{N} \frac{\rE_{(t)}}{z - x_{(t)}})^2 + \nonumber\\ + &= + \frac{\pi T }{2} \cN_{\Psi}^2 + \sum_{n,m} : b_n b^*_m: + z^{-n-m} + \qty[ \frac{m-n}{z}+2\finitesum{t}{1}{N} \frac{\rE_{(t)}}{z-x_{(t)}} ] + + \frac{1}{2} \qty( \finitesum{t}{1}{N} \frac{\rE_{(t)}}{z - x_{(t)}})^2 + . + \end{align*} + The last expression shows that the energy momentum tensor $\cT( z )$ + is radial time dependent but it satisfies the usual \ope. + + Notice first of all that the vacuum $\Gexcvacket$ is actually + $\GGexcvacket$, i.e. it depends only on $x_{(t)}$ and $\rE_{(t)}$. + Then we can try to interpret the previous result in the light of the + usual \cft approach. + In particular we can refine the idea we discussed after + \eqref{eq:asymp_beha_Psi_on_exc_vac} that + the singularity in the modes \eqref{eq:generic-case-basis} and + \eqref{eq:generic-case-basis-conjugate} + at the point $x_{(t)}$ + is associated with a primary conformal + operator which creates $\eexcvacket$ with $\rE=\rE_{(t)}$. + In fact by comparison with the stress energy tensor of a + excited vacuum + \eqref{eq:T_excited_vacuum}, + we can read from the second order singularity + that at the points $x_{(t)}$ there is an operator + which creates the excited vacuum $\GGexcvacket$ from the + \SL{2}{R} vacuum \regvacuum. + Given the discussion in the previous section this is an excited + spin field $\rS_{\rE_{(t)}}\qty( x_{(t)}) = e^{i \rE_{(t)} \phi( x_{(t)} )}$. + The first order singularities in $x_{(u)}-x_{(t)}$ are then the result + of the interaction between two of the previous excited spin + fields. + We can try to be more precise. + Using the usual \cft operatorial approach + we can suppose that the following + identification holds + \begin{align} + \GGexcvacket + &= + \cN(\{x_{(t)}, \rE_{(t)}\})~ + \rS_{\rE_{(1)}}\qty( x_1 ) \dots \rS_{\rE_{(N)}}\qty( x_N ) \regvacuum + \nonumber\\ + &= + \cN(\{x_{(t)}, \rE_{(t)} \})~ + R\qty[ \finiteprod{t}{1}{N} \rS_{\rE_{(t)}}\qty( x_{(t) } ) ] \regvacuum + , + \label{eq:vacuum_R_prod_spin_fields} + \end{align} + then we get + \begin{align*} + \cT( z ) \GGexcvacket + &= + \cN(\{x_{(t)}, \rE_{(t)} \})~ + R\qty[ + \cT( z ) + \finiteprod{t}{1}{N} \rS_{\rE_{(t)}}\qty( x_{(t)} ) ] \regvacuum + . +% \label{eq:T_on_vacuum} + \end{align*} + The fact that $ \cT( z )$ enters the radial ordering may +seem strange but the left hand side is well defined for all $z$ and +the only well defined expression for the right hand side is the one +with the radial ordering. +In fact an operatorial expression like $\cT( z ) +R\qty[\partial\phi(x_1)\partial\phi(x_2)] \regvacuum$ +is only defined for $|z|> x_{1,2}$. + It then follows that + \begin{equation} + \cT( z ) \GGexcvacket + = + \finitesum{t}{1}{N} \qty( + \frac{\rE_{(t)}^2 /2 }{ (z-x_{(t)})^2} + + + \frac{\ \ipd{x_{(t)}} - \ipd{x_{(t)}} \log \cN }{ z-x_{(t)}} + ) \GGexcvacket + + \mbox{regular terms in $z$} + , +% \label{eq:generic-case-stress-energy-tensor} + \end{equation} +which allows to write +\begin{align*} + \cN(\{x_{(t)}, \rE_{(t)} \})~ + &R\qty[ + \ipd{x_{(t)}} \rS_{\rE_{(t)}}\qty( x_{(t)} ) + \prod_{u\ne t} \rS_{\rE_{(u)}}\qty( x_{(u)} ) ] \regvacuum + \nonumber\\ + &= + \rE_{(t)}\, + \qty[ + \pi T \cN_{\Psi}^2\, + \sum_{n,m=0}^\infty \frac{ b_n b^*_m}{x_{(t)}^{n+m}} + + + \sum_{u\ne t} \frac{\rE_{(u)}}{x_{(t)}-x_{(u)}} + ] + \GGexcvacket + . +\end{align*} +This result shows the way non primary operators are represented in +this formalism and is consistent with the computation of the excited +spin fields correlator performed in section \ref{sec:spin_correlators}. + + % vim: ft=tex diff --git a/sec/part1/introduction.tex b/sec/part1/introduction.tex index 18d4458..fa40fb8 100644 --- a/sec/part1/introduction.tex +++ b/sec/part1/introduction.tex @@ -188,13 +188,13 @@ while its conservation $\partial^{\alpha} T_{\alpha\beta} = 0$ becomes:\footnote Since we fix $\gamma_{\alpha\beta}(\tau, \sigma) \propto \eta_{\alpha\beta}$ we do not need to account for the components of the connection and we can replace the covariant derivative $\nabla^{\alpha}$ with a standard derivative $\partial^{\alpha}$. } \begin{equation} - \bpd T_{\xi\xi}( \xi,\, \bxi ) = \pd \bT_{\bxi\bxi}( \xi,\, \bxi ) = 0. + \bpd T_{\xi\xi}( \xi,\, \bxi ) = \pd \barT_{\bxi\bxi}( \xi,\, \bxi ) = 0. \end{equation} The last equation finally implies \begin{equation} T_{\xi\xi}( \xi,\, \bxi ) = T_{\xi\xi}( \xi ) = T( \xi ), \qquad - \bT_{\bxi\bxi}( \xi,\, \bxi ) = \bT_{\bxi\bxi}( \bxi ) = \bT( \bxi ), + \barT_{\bxi\bxi}( \xi,\, \bxi ) = \barT_{\bxi\bxi}( \bxi ) = \barT( \bxi ), \end{equation} which are respectively the holomorphic and the anti-holomorphic components of the bidimensional stress energy tensor. @@ -240,7 +240,7 @@ An additional conformal transformation \begin{equation} z = e^{\xi} = e^{\tau_e + i \sigma} \in \qty{ z \in \C | \Im z \ge 0 }, \qquad - \bz = e^{\bxi} = e^{\tau_e - i \sigma} \in \qty{ z \in \C | \Im z \le 0 } + \barz = e^{\bxi} = e^{\tau_e - i \sigma} \in \qty{ z \in \C | \Im z \le 0 } \end{equation} maps the worldsheet of the string to the complex plane. On this Riemann surface the usual time ordering becomes a \emph{radial ordering} as constant time surfaces are circles around the origin (see the contours $\ccC_{(0)}$ and $\ccC_{(1)}$ in \Cref{fig:conf:complex_plane}). @@ -254,7 +254,7 @@ In these coordinates the conserved charge associated to the transformation $z \m + \cint{0} \ddbz - \bepsilon(\bz)\, \bT(\bz), + \bepsilon(\barz)\, \barT(\barz), \end{equation} where $\ccC_0$ is an anti-clockwise constant radial time path around the origin. The transformation on a field $\phi_{\omega, \bomega}$ of weight $(\omega, \bomega)$ is thus given by the commutator with $Q_{\epsilon, \bepsilon}$: @@ -262,17 +262,17 @@ The transformation on a field $\phi_{\omega, \bomega}$ of weight $(\omega, \bome \begin{split} \delta_{\epsilon, \bepsilon} \phi_{\omega, \bomega} & = - \liebraket{Q_{\epsilon, \bepsilon}}{\phi_{\omega, \bomega}( w, \bw )} + \liebraket{Q_{\epsilon, \bepsilon}}{\phi_{\omega, \bomega}( w, \barw )} \\ & = - \cint{0} \ddz \epsilon(z) \qty[ T(z), \phi_{\omega, \bomega}( w, \bw ) ] + \cint{0} \ddz \epsilon(z) \qty[ T(z), \phi_{\omega, \bomega}( w, \barw ) ] + - \cint{0} \ddbz \bepsilon(\bz) \qty[ \bT(\bz), \phi_{\omega, \bomega}( w, \bw ) ] + \cint{0} \ddbz \bepsilon(\barz) \qty[ \barT(\barz), \phi_{\omega, \bomega}( w, \barw ) ] \\ & = - \cint{w} \ddz \epsilon(z)\, \rR\!\qty( T(z)\, \phi_{\omega, \bomega}( w, \bw ) ) + \cint{w} \ddz \epsilon(z)\, \rR\!\qty( T(z)\, \phi_{\omega, \bomega}( w, \barw ) ) + - \cint{\bw} \ddbz \bepsilon(\bz)\, \rR\!\qty( \bT(\bz)\, \phi_{\omega, \bomega}( w, \bw ) ), + \cint{\barw} \ddbz \bepsilon(\barz)\, \rR\!\qty( \barT(\barz)\, \phi_{\omega, \bomega}( w, \barw ) ), \end{split} \end{equation} where in the last passage we used the fact that the difference of ordered integrals becomes the contour integral of the radially ordered product computed surrounding $w$. @@ -281,58 +281,58 @@ Equating the result with the expected variation \begin{split} \delta_{\epsilon, \bepsilon} \phi_{\omega, \bomega} & = - \omega\, \ipd{w} \epsilon( w )\, \phi_{\omega, \bomega}( w, \bw ) + \omega\, \ipd{w} \epsilon( w )\, \phi_{\omega, \bomega}( w, \barw ) + - \epsilon( w )\, \ipd{w} \phi_{\omega, \bomega}( w, \bw ) + \epsilon( w )\, \ipd{w} \phi_{\omega, \bomega}( w, \barw ) \\ & + - \bomega\, \ipd{\bw} \bepsilon( \bw )\, \phi_{\omega, \bomega}( w, \bw ) + \bomega\, \ipd{\barw} \bepsilon( \barw )\, \phi_{\omega, \bomega}( w, \barw ) + - \epsilon( \bw )\, \ipd{\bw} \phi_{\omega, \bomega}( w, \bw ) + \epsilon( \barw )\, \ipd{\barw} \phi_{\omega, \bomega}( w, \barw ) \end{split} \end{equation} -we find the short distance singularities of the components of the stress-energy tensor with the field $\phi_{\omega, \bomega}( w, \bw )$: +we find the short distance singularities of the components of the stress-energy tensor with the field $\phi_{\omega, \bomega}( w, \barw )$: \begin{equation} \begin{split} - T( z )\, \phi_{\omega, \bomega}( w, \bw ) + T( z )\, \phi_{\omega, \bomega}( w, \barw ) & = - \frac{\omega}{(z - w)^2}\, \phi_{\omega, \bomega}( w, \bw ) + \frac{\omega}{(z - w)^2}\, \phi_{\omega, \bomega}( w, \barw ) + - \frac{1}{z - w}\, \ipd{w} \phi_{\omega, \bomega}( w, \bw ) + \frac{1}{z - w}\, \ipd{w} \phi_{\omega, \bomega}( w, \barw ) + \order{1}, \\ - \bT( \bz )\, \phi_{\omega, \bomega}( w, \bw ) + \barT( \barz )\, \phi_{\omega, \bomega}( w, \barw ) & = - \frac{\bomega}{(\bz - \bw)^2}\, \phi_{\omega, \bomega}( w, \bw ) + \frac{\bomega}{(\barz - \barw)^2}\, \phi_{\omega, \bomega}( w, \barw ) + - \frac{1}{\bz - \bw}\, \ipd{\bw} \phi_{\omega, \bomega}( w, \bw ) + \frac{1}{\barz - \barw}\, \ipd{\barw} \phi_{\omega, \bomega}( w, \barw ) + \order{1}, \end{split} \label{eq:conf:primary} \end{equation} where we drop the radial ordering symbol $\rR$ for simplicity. -Since the contour $\ccC_{w}$ is infinitely small around $w$, the conformal properties of $\phi_{\omega, \bomega}( w, \bw )$ are entirely defined by these relations. -In fact $\phi_{\omega, \bomega}( w, \bw )$ is a \emph{primary field} if its short distance behaviour with the stress-energy tensor is as such. +Since the contour $\ccC_{w}$ is infinitely small around $w$, the conformal properties of $\phi_{\omega, \bomega}( w, \barw )$ are entirely defined by these relations. +In fact $\phi_{\omega, \bomega}( w, \barw )$ is a \emph{primary field} if its short distance behaviour with the stress-energy tensor is as such. This is an example of an \emph{operator product expansion} (\ope) \begin{equation} - \phi^{(i)}_{\omega_i, \bomega_i}( z, \bz )\, \phi^{(j)}_{\omega_j, \bomega_j}( w, \bw ) + \phi^{(i)}_{\omega_i, \bomega_i}( z, \barz )\, \phi^{(j)}_{\omega_j, \bomega_j}( w, \barw ) = \sum\limits_{k} \cC_{ijk} (z - w)^{\omega_k - \omega_i - \omega_j}\, - (\bz - \bw)^{\bomega_k - \bomega_i - \bomega_j}\, - \phi^{(k)}_{\omega_k, \bomega_k}( w, \bw ) + (\barz - \barw)^{\bomega_k - \bomega_i - \bomega_j}\, + \phi^{(k)}_{\omega_k, \bomega_k}( w, \barw ) \label{eq:conf:ope} \end{equation} which is an asymptotic expansion containing the full information on the singularities.\footnotemark{} \footnotetext{% The expression \eqref{eq:conf:ope} is valid assuming the normalisation of the 2-points function \begin{equation*} - \left\langle \phi^{(i)}_{\omega_i, \bomega_i}( z, \bz )\, \phi^{(j)}_{\omega_j, \bomega_j}( w, \bw ) \right\rangle + \left\langle \phi^{(i)}_{\omega_i, \bomega_i}( z, \barz )\, \phi^{(j)}_{\omega_j, \bomega_j}( w, \barw ) \right\rangle = - \frac{\delta_{ij}}{(z - w)^{\omega_i + \omega_j} (\bz - \bw)^{\bomega_i + \bomega_j}}. + \frac{\delta_{ij}}{(z - w)^{\omega_i + \omega_j} (\barz - \barw)^{\bomega_i + \bomega_j}}. \end{equation*} } The constant coefficients $\cC_{ijk}$ are subject to restrictive constraints given by the properties of the conformal theories to the point that a \cft is completely specified by the spectrum of the weights $(\omega_i, \bomega_i)$ and the coefficients $\cC_{ijk}$ \cite{Friedan:1986:ConformalInvarianceSupersymmetry}. @@ -349,27 +349,27 @@ Focusing on the holomorphic component we find + \frac{1}{z - w}\, \ipd{w} T(w), \\ - \bT( \bz )\, \bT( \bw ) + \barT( \barz )\, \barT( \barw ) & = - \frac{\frac{\overline{c}}{2}}{(\bz - \bw)^4} + \frac{\frac{\barc}{2}}{(\barz - \barw)^4} + - \frac{2}{(\bz - \bw)^2}\, \bT(\bw) + \frac{2}{(\barz - \barw)^2}\, \barT(\barw) + - \frac{1}{\bz - \bw}\, \ipd{\bw} \bT(\bw). + \frac{1}{\barz - \barw}\, \ipd{\barw} \barT(\barw). \end{split} \label{eq:conf:TTexpansion} \end{equation} The components of the stress-energy tensor are therefore not primary fields and show an anomaly in the behaviour encoded by the constant $c \in \R$. -This is a reflection of the anomalous algebra of the operator modes $L_n$ and $\bL_n$ computed from the Laurent expansion +This is a reflection of the anomalous algebra of the operator modes $L_n$ and $\barL_n$ computed from the Laurent expansion \begin{equation} \begin{split} T( z ) = \infinfsum{n} L_n\, z^{-n -2} & \Rightarrow L_n = \cint{0} \ddz z^{n + 1} T(z), \\ - \bT( \bz ) = \infinfsum{n} \bL_n\, \bz^{-n -2} + \barT( \barz ) = \infinfsum{n} \barL_n\, \barz^{-n -2} & \Rightarrow - \bL_n = \cint{0} \ddbz \bz^{n + 1} \bT(\bz). + \barL_n = \cint{0} \ddbz \barz^{n + 1} \barT(\barz). \end{split} \label{eq:conf:Texpansion} \end{equation} @@ -380,39 +380,39 @@ This ultimately leads to the quantum algebra & = (n - m)\, L_{n + m} + \frac{c}{12}\, n\, (n^2 - 1)\, \delta_{n, -m}, \\ - \liebraket{\bL_n}{\bL_m} + \liebraket{\barL_n}{\barL_m} & = - (n - m)\, \bL_{n + m} + \frac{\overline{c}}{12}\, n\, (n^2 - 1)\, \delta_{n, -m}, + (n - m)\, \barL_{n + m} + \frac{\barc}{12}\, n\, (n^2 - 1)\, \delta_{n, -m}, \\ - \liebraket{L_n}{\bL_m} + \liebraket{L_n}{\barL_m} & = 0, \end{split} \label{eq:conf:virasoro} \end{equation} known as Virasoro algebra, unique central extension of the classical de Witt algebra, with central charge $c$. -Operators $L_n$ and $\bL_n$ are called Virasoro operators.\footnotemark{} +Operators $L_n$ and $\barL_n$ are called Virasoro operators.\footnotemark{} \footnotetext{% Notice that the subset of Virasoro operators $\qty{ L_{-1}, L_0, L_1 }$ forms a closed subalgebra generating the group $\SL{2}{\R}$. } -Notice that $L_0 + \bL_0$ is the generator of the dilations on the complex plane. -In terms of radial quantization this translates to time translations and $L_0 + \bL_0$ can be considered to be the Hamiltonian of the theory. +Notice that $L_0 + \barL_0$ is the generator of the dilations on the complex plane. +In terms of radial quantization this translates to time translations and $L_0 + \barL_0$ can be considered to be the Hamiltonian of the theory. In the same fashion as~\eqref{eq:conf:Texpansion}, fields can be expanded in modes: \begin{equation} - \phi_{\omega, \bomega}( w, \bw ) + \phi_{\omega, \bomega}( w, \barw ) = \sum\limits_{n,\, m = -\infty}^{+\infty} \phi_{\omega, \bomega}^{(n, m)}\, z^{-n -\omega}\, - \bz^{-m -\bomega}. + \barz^{-m -\bomega}. \label{eq:conf:expansion} \end{equation} From the previous relations we can finally define the ``asymptotic'' in-states as one-to-one correspondence with conformal operators: \begin{equation} \ket{\phi_{\omega, \bomega}} = - \lim\limits_{z,\, \bz \to 0} + \lim\limits_{z,\, \barz \to 0} \phi_{\omega, \bomega} \regvacuum. \end{equation} @@ -431,7 +431,7 @@ As a consequence also \begin{equation} L_n \regvacuum = - \bL_n \regvacuum + \barL_n \regvacuum = 0, \qquad @@ -442,16 +442,16 @@ Finally the definitions of the primary operators~\eqref{eq:conf:primary} define \begin{split} L_0 \ket{\phi_{\omega, \bomega}} & = \omega \ket{\phi_{\omega, \bomega}}, \\ - \bL_0 \ket{\phi_{\omega, \bomega}} & = \bomega \ket{\phi_{\omega, \bomega}}, + \barL_0 \ket{\phi_{\omega, \bomega}} & = \bomega \ket{\phi_{\omega, \bomega}}, \\ - L_n \ket{\phi_{\omega, \bomega}} & = \bL_n \ket{\phi_{\omega, \bomega}} = 0, + L_n \ket{\phi_{\omega, \bomega}} & = \barL_n \ket{\phi_{\omega, \bomega}} = 0, \quad n \ge 1. \end{split} \label{eq:conf:physical} \end{equation} -From this definition we can build an entire representation of \emph{descendant} states applying any operator $L_{-n}$ (or $\bL_{-n}$) with $n \ge 1$ to $\ket{\phi_{\omega, \bomega}}$. +From this definition we can build an entire representation of \emph{descendant} states applying any operator $L_{-n}$ (or $\barL_{-n}$) with $n \ge 1$ to $\ket{\phi_{\omega, \bomega}}$. Let $\phi_{\omega}( w )$ be a holomorphic field in the \cft for simplicity, and let $\ket{\phi_{\omega}}$ be its corresponding state. The generic state at level $m$ build from such state is \begin{equation} @@ -474,30 +474,30 @@ They are however called \emph{highest weight} states from the mathematical liter The particular case of the \cft in \eqref{eq:conf:polyakov} can be cast in this language. In particular the solutions to the \eom factorise into a holomorphic and an anti-holomorphic contributions: \begin{equation} - \ipd{z} \ipd{\bz} X( z, \bz ) = 0 + \ipd{z} \ipd{\barz} X( z, \barz ) = 0 \qquad \Rightarrow \qquad - X( z, \bz ) = X( z ) + \bX( \bz ), + X( z, \barz ) = X( z ) + \barX( \barz ), \end{equation} and the components of the stress-energy tensor~\eqref{eq:conf:stringT} are \begin{equation} \begin{split} T( z ) & = \ipd{z} X( z ) \cdot \ipd{z} X( z ), \\ - \bT( \bz ) & = \ipd{\bz} \bX( \bz ) \cdot \ipd{\bz} \bX( \bz ). + \barT( \barz ) & = \ipd{\barz} \barX( \barz ) \cdot \ipd{\barz} \barX( \barz ). \end{split} \label{eq:conf:bosonicstringT} \end{equation} -Using the normalisation of the 2-points function $\left\langle X^{\mu}( z, \bz ) X^{\nu}( w, \bw ) \right\rangle = - \frac{1}{2} \eta^{\mu\nu} \ln\left| z - w \right|$ we can prove that $c = D$ in~\eqref{eq:conf:TTexpansion}, where $D$ is the dimensions of spacetime (or equivalently the number of scalar fields $X^{\mu}$ in the action). -It can be shown that in order to cancel the central charge in bosonic string theory we need to introduce a pair of conformal ghosts $b(z)$ and $c(z)$ with conformal weights $(2, 0)$ and $(-1, 0)$ respectively, together with their anti-holomorphic counterparts $\overline{b}(z)$ and $\overline{c}(z)$. +Using the normalisation of the 2-points function $\left\langle X^{\mu}( z, \barz ) X^{\nu}( w, \barw ) \right\rangle = - \frac{1}{2} \eta^{\mu\nu} \ln\left| z - w \right|$ we can prove that $c = D$ in~\eqref{eq:conf:TTexpansion}, where $D$ is the dimensions of spacetime (or equivalently the number of scalar fields $X^{\mu}$ in the action). +It can be shown that in order to cancel the central charge in bosonic string theory we need to introduce a pair of conformal ghosts $b(z)$ and $c(z)$ with conformal weights $(2, 0)$ and $(-1, 0)$ respectively, together with their anti-holomorphic counterparts $\barb(z)$ and $\barc(z)$. The non vanishing components of their stress-energy tensor can be computed as:\footnotemark{} \footnotetext{% In general a system of ghosts $b( z )$ and $c( z )$ with weight $(\lambda, 0)$ and $(1 - \lambda, 0)$ can be introduced as a standalone \cft with action~\cite{Friedan:1986:ConformalInvarianceSupersymmetry,Polchinski:1998:StringTheorySuperstring} \begin{equation*} - S = \frac{1}{2 \pi} \iint \dd{z} \dd{\bz} b( z )\, \ipd{\bz} c( z ). + S = \frac{1}{2 \pi} \iint \dd{z} \dd{\barz} b( z )\, \ipd{\barz} c( z ). \end{equation*} - The equations of motion are $\ipd{\bz} c( z ) = \ipd{\bz} b( z ) = 0$. + The equations of motion are $\ipd{\barz} c( z ) = \ipd{\barz} b( z ) = 0$. The \ope is \begin{equation*} b( z )\, c( z ) = \frac{\varepsilon}{z - w} + \order{1}, @@ -527,9 +527,9 @@ The non vanishing components of their stress-energy tensor can be computed as:\f & = c( z )\, \ipd{z} b( z ) - 2\, b( z )\, \ipd{z} c( z ), \\ - \bT_{\text{ghost}}( \bz ) + \barT_{\text{ghost}}( \barz ) & = - \overline{c}( \bz )\, \ipd{\bz} \overline{b}( \bz ) - 2\, \overline{b}( \bz )\, \ipd{\bz} \overline{c}( \bz ). + \barc( \barz )\, \ipd{\barz} \barb( \barz ) - 2\, \barb( \barz )\, \ipd{\barz} \barc( \barz ). \end{split} \end{equation} @@ -537,7 +537,7 @@ From their 2-points functions \begin{equation} \left\langle b(z)\, c(w) \right\rangle = \frac{1}{z - w}, \qquad - \left\langle \overline{b}(\bz)\, \overline{c}(\bw) \right\rangle = \frac{1}{\bz - \bw}, + \left\langle \barb(\barz)\, \barc(\barw) \right\rangle = \frac{1}{\barz - \barw}, \end{equation} we get the \ope of the components of their stress-energy tensor: \begin{equation} @@ -550,22 +550,22 @@ we get the \ope of the components of their stress-energy tensor: + \frac{1}{z - w}\, \ipd{z} T_{\text{ghost}}(z), \\ - \bT_{\text{ghost}}(\bz)\, \bT_{\text{ghost}}(\bw) + \barT_{\text{ghost}}(\barz)\, \barT_{\text{ghost}}(\barw) & = - \frac{-13}{(\bz - \bw)^4} + \frac{-13}{(\barz - \barw)^4} + - \frac{2}{(\bz - \bw)^2}\, \bT_{\text{ghost}}(\bz) + \frac{2}{(\barz - \barw)^2}\, \barT_{\text{ghost}}(\barz) + - \frac{1}{\bz - \bw}\, \ipd{\bz} \bT_{\text{ghost}}(\bz), + \frac{1}{\barz - \barw}\, \ipd{\barz} \barT_{\text{ghost}}(\barz), \end{split} \end{equation} which show that $c_{\text{ghost}} = - 26$. The central charge is therefore cancelled in the full theory (bosonic string and ghosts) when the spacetime dimensions are $D = 26$. -In fact let $T_{\text{full}} = T + T_{\text{ghost}}$ and $\bT_{\text{full}} = \bT + \bT_{\text{ghost}}$, then: +In fact let $T_{\text{full}} = T + T_{\text{ghost}}$ and $\barT_{\text{full}} = \barT + \barT_{\text{ghost}}$, then: \begin{equation} \eval{T_{\text{full}}( z )}_{\order{(z - w)^{-4}}} = - \eval{\bT_{\text{full}}( \bz )}_{\order{(\bz - \bw)^{-4}}} + \eval{\barT_{\text{full}}( \barz )}_{\order{(\barz - \barw)^{-4}}} = c + c_{\text{ghost}} = @@ -591,11 +591,11 @@ In complex coordinates on the plane it is: S[ X, \psi ] = - \frac{1}{4 \pi} - \iint \dd{z} \dd{\bz} + \iint \dd{z} \dd{\barz} \qty( - \frac{2}{\ap}\, \ipd{\bz} X^{\mu}\, \ipd{z} X^{\nu} + \frac{2}{\ap}\, \ipd{\barz} X^{\mu}\, \ipd{z} X^{\nu} + - \psi^{\mu}\, \ipd{\bz} \psi^{\nu} + \psi^{\mu}\, \ipd{\barz} \psi^{\nu} + \bpsi^{\mu}\, \ipd{z} \bpsi^{\nu} ) @@ -606,7 +606,7 @@ In the last expression $\psi^{\mu}$ are $D$ two-dimensional holomorphic fermion \begin{equation} \psi^{\mu}( z )\, \psi^{\nu}( w ) = \frac{\eta^{\mu\nu}}{z - w}, \qquad - \bpsi^{\mu}( \bz )\, \bpsi^{\nu}( \bw ) = \frac{\eta^{\mu\nu}}{\bz - \bw}. + \bpsi^{\mu}( \barz )\, \bpsi^{\nu}( \barw ) = \frac{\eta^{\mu\nu}}{\barz - \barw}. \end{equation} In this case the components of the stress-energy tensor of the theory are: \begin{equation} @@ -615,9 +615,9 @@ In this case the components of the stress-energy tensor of the theory are: & = -\frac{1}{\ap}\, \ipd{z} X( z ) \cdot \ipd{z} X( z ) - \frac{1}{2}\, \psi( z ) \cdot \ipd{z} \psi( z ), \\ - \bT( \bz ) + \barT( \barz ) & = - -\frac{1}{\ap}\, \ipd{\bz} \bX( \bz ) \cdot \ipd{\bz} \bX( \bz ) - \frac{1}{2}\, \bpsi( \bz ) \cdot \ipd{\bz} \bpsi( \bz ). + -\frac{1}{\ap}\, \ipd{\barz} \barX( \barz ) \cdot \ipd{\barz} \barX( \barz ) - \frac{1}{2}\, \bpsi( \barz ) \cdot \ipd{\barz} \bpsi( \barz ). \end{split} \end{equation} @@ -626,9 +626,9 @@ The action~\eqref{eq:super:action} is also invariant under the \emph{supersymmet \begin{split} \sqrt{\frac{2}{\ap}}\, \delta_{\epsilon, \bepsilon} - X^{\mu}( z, \bz ) + X^{\mu}( z, \barz ) & = - \epsilon( z )\, \psi^{\mu}( z ) + \bepsilon( \bz )\, \bpsi^{\mu}( \bz ), + \epsilon( z )\, \psi^{\mu}( z ) + \bepsilon( \barz )\, \bpsi^{\mu}( \barz ), \\ \sqrt{\frac{2}{\ap}}\, \delta_{\epsilon} \psi^{\mu}( z ) @@ -636,21 +636,21 @@ The action~\eqref{eq:super:action} is also invariant under the \emph{supersymmet - \epsilon( z )\, \ipd{z} X^{\mu}( z ), \\ \sqrt{\frac{2}{\ap}}\, - \delta_{\bepsilon} \bpsi^{\mu}( \bz ) + \delta_{\bepsilon} \bpsi^{\mu}( \barz ) & = - - \bepsilon( \bz )\, \ipd{\bz} \bX^{\mu}( \bz ) + - \bepsilon( \barz )\, \ipd{\barz} \barX^{\mu}( \barz ) \end{split} \end{equation} -generated by the currents $J( z ) = \epsilon( z )\, T_F( z )$ and $\overline{J}( \bz ) = \bepsilon( \bz )\, \bT_F( \bz )$, where $\epsilon( z )$ and $\bepsilon( \bz ) = \qty( \epsilon( z ) )^*$ are anti-commuting fermions and +generated by the currents $J( z ) = \epsilon( z )\, T_F( z )$ and $\barJ( \barz ) = \bepsilon( \barz )\, \barT_F( \barz )$, where $\epsilon( z )$ and $\bepsilon( \barz ) = \qty( \epsilon( z ) )^*$ are anti-commuting fermions and \begin{equation} \begin{split} T_F( z ) & = i\, \sqrt{\frac{2}{\ap}}\, \psi( z ) \cdot \ipd{z} X( z ), \\ - \bT_F( \bz ) + \barT_F( \barz ) & = - i\, \sqrt{\frac{2}{\ap}}\, \bpsi( \bz ) \cdot \ipd{\bz} \bX( \bz ) + i\, \sqrt{\frac{2}{\ap}}\, \bpsi( \barz ) \cdot \ipd{\barz} \barX( \barz ) \end{split} \end{equation} are the \emph{supercurrents}. @@ -667,13 +667,13 @@ The central charge associated to the Virasoro algebra is in this case given by b + \order{1}, \\ - \bT( \bz )\, \bT( \bw ) + \barT( \barz )\, \barT( \barw ) & = - \frac{\frac{3 D}{4}}{( \bz - \bw )^4} + \frac{\frac{3 D}{4}}{( \barz - \barw )^4} + - \frac{2}{( \bz - \bw )^2} \bT( \bw ) + \frac{2}{( \barz - \barw )^2} \barT( \barw ) + - \frac{1}{\bz - \bw} \ipd{\bw} \bT( \bw ) + \frac{1}{\barz - \barw} \ipd{\barw} \barT( \barw ) + \order{1}. \end{split} @@ -683,11 +683,11 @@ The central charge is therefore $c = \frac{3}{2} D$ for the \cft defined in~\eqr As in the case of the bosonic string, in order to cancel the central charge we introduce the reparametrisation anti-commuting ghosts $b( z )$ and $c( z )$ and their anti-holomorphic components as well as the commuting \emph{superghosts} $\beta( z )$ and $\gamma( z )$ and their anti-holomorphic counterparts. These are conformal fields with conformal weights $\qty( \frac{3}{2}, 0 )$ and $\qty( -\frac{1}{2}, 0 )$. Their central charge becomes $c_{\text{ghost}} = c_{bc} + c_{\beta\gamma} = -26 + 11 = -15$ (see \cref{note:conf:ghosts} for the general computation). -When considering the full theory $T_{\text{full}} = T + T_{\text{ghost}}$ and $\bT_{\text{full}} = \bT + \bT_{\text{ghost}}$ the central charge vanishes only in 10-dimensional spacetime: +When considering the full theory $T_{\text{full}} = T + T_{\text{ghost}}$ and $\barT_{\text{full}} = \barT + \barT_{\text{ghost}}$ the central charge vanishes only in 10-dimensional spacetime: \begin{equation} \eval{T_{\text{full}}( z )}_{\order{(z - w)^{-4}}} = - \eval{\bT_{\text{full}}( \bz )}_{\order{(\bz - \bw)^{-4}}} + \eval{\barT_{\text{full}}( \barz )}_{\order{(\barz - \barw)^{-4}}} = c + c_{\text{ghost}} = @@ -767,9 +767,9 @@ A manifold $M$ is a \emph{complex} manifold if it is possible to define a comple \Rightarrow \ipd{x} f( x, y ) = -i \ipd{y} f( x, y ) \Rightarrow - \ipd{\bz} f( z, \bz ) = 0 + \ipd{\barz} f( z, \barz ) = 0 \Rightarrow - f( z, \bz ) = f( z ). + f( z, \barz ) = f( z ). \end{equation*} } @@ -808,7 +808,7 @@ $(M, J, g)$ is a \emph{Kähler} manifold if: \dd{\omega} = \qty( \pd + \bpd ) - \omega(z, \bz) + \omega(z, \barz) = 0, \label{eq:cy:kaehler} @@ -826,58 +826,58 @@ In local coordinates a Hermitian metric is such that \begin{equation} g = - g_{a\overline{b}}\, \dd{z}^a \otimes \dd{\bz}^{\overline{b}} + g_{a \barb}\, \dd{z}^a \otimes \dd{\barz}^{\barb} + - g_{\overline{a}b}\, \dd{\bz}^{\overline{a}} \otimes \dd{z}^b, + g_{\bara b}\, \dd{\barz}^{\bara} \otimes \dd{z}^b, \end{equation} -thus the Kähler form becomes $\omega = i g_{a\overline{b}}\, \dd{z}^a \wedge \dd{\bz}^{\overline{b}}$. +thus the Kähler form becomes $\omega = i g_{a\barb}\, \dd{z}^a \wedge \dd{\barz}^{\barb}$. The relation~\eqref{eq:cy:kaehler} then translates into: \begin{equation} - \dd{\omega} = i\, \qty( \pd + \bpd )\, g_{a\overline{b}}\, \dd{z}^a \wedge \dd{\bz}^{\overline{b}} + \dd{\omega} = i\, \qty( \pd + \bpd )\, g_{a\barb}\, \dd{z}^a \wedge \dd{\barz}^{\barb} = 0 \quad \Leftrightarrow \quad \begin{cases} - \ipd{z^c} g_{a\overline{b}} & = \ipd{z^a} g_{c\overline{b}} + \ipd{z^c} g_{a\barb} & = \ipd{z^a} g_{c\barb} \\ - \ipd{\bz^c} g_{\overline{a}b} & = \ipd{\bz^a} g_{\overline{c}b} + \ipd{\barz^c} g_{\bara b} & = \ipd{\barz^a} g_{\barc b} \end{cases}. \end{equation} -The $(1,1)$-form $\omega$ can locally be written as $\omega = i\, \pd \bpd\, \phi( z, \bz )$ up to a constant. +The $(1,1)$-form $\omega$ can locally be written as $\omega = i\, \pd \bpd\, \phi( z, \barz )$ up to a constant. This ultimately leads to \begin{equation} - g_{a\overline{b}} + g_{a\barb} = - \pdv{\phi( z, \bz )}{z^a}{\bz^{\overline{b}}} + \pdv{\phi( z, \barz )}{z^a}{\barz^{\barb}} = - \ipd{a} \ipd{\overline{b}}\, \phi( z, \bz ), + \ipd{a} \ipd{\barb}\, \phi( z, \barz ), \end{equation} Since $\omega$ is the Kähler form then the Levi-Civita connection has only fully holomorphic and anti-holomorphic components: \begin{equation} \tensor{\Gamma}{^a_{bc}} = - \tensor{g}{^{a\overline{d}}}\, + \tensor{g}{^{a\bard}}\, \ipd{b}\, - \tensor{g}{_{\overline{d}c}}, + \tensor{g}{_{\bard c}}, \qquad -\tensor{\Gamma}{^{\overline{a}}_{\overline{b}\overline{c}}} +\tensor{\Gamma}{^{\bara}_{\barb\barc}} = - \tensor{g}{^{\overline{a}d}}\, - \ipd{\overline{b}}\, - \tensor{g}{_{d\overline{c}}}. + \tensor{g}{^{\bara d}}\, + \ipd{\barb}\, + \tensor{g}{_{d\barc}}. \end{equation} As a consequence the Ricci tensor becomes \begin{equation} - \tensor{R}{_{\overline{a}b}} + \tensor{R}{_{\bara b}} = - - \pdv{\tensor{\Gamma}{^{\overline{c}}_{\overline{a}\overline{c}}}}{z^b}. + \pdv{\tensor{\Gamma}{^{\barc}_{\bara\barc}}}{z^b}. \end{equation} Since \cy manifolds have \SU{m} holonomy, the trace part of the coefficients of the connection vanishes. -\cy manifolds thus have $\tensor{R}{_{\overline{a}b}} = 0$, that is they are complex Ricci-flat Kähler manifolds. +\cy manifolds thus have $\tensor{R}{_{\bara b}} = 0$, that is they are complex Ricci-flat Kähler manifolds. \subsubsection{Cohomology and Hodge Numbers} @@ -987,7 +987,7 @@ and naturally to the \emph{Neumann} boundary conditions:\footnotemark{} \end{equation} Closed strings are such that $X^{\mu}( \tau, \sigma + \ell ) = X^{\mu}( \tau, \sigma )$. -The usual mode expansion in conformal coordinates $X^{\mu}( z, \bz ) = X( z ) + \bX( \bz )$ leads to +The usual mode expansion in conformal coordinates $X^{\mu}( z, \barz ) = X( z ) + \barX( \barz )$ leads to \begin{equation} \begin{split} X^{\mu}( z ) @@ -1000,14 +1000,14 @@ The usual mode expansion in conformal coordinates $X^{\mu}( z, \bz ) = X( z ) + + \sum\limits_{n \in \Z \setminus \{0\}} \frac{\alpha_n^{\mu}}{n} z^{-n} ), \\ - \bX^{\mu}( \bz ) + \barX^{\mu}( \barz ) & = - \overline{x}_0^{\mu} + \barx_0^{\mu} + i\, \sqrt{\frac{\ap}{2}}\, \qty( - - \balpha_0^{\mu}\, \ln{\bz} - + \sum\limits_{n \in \Z \setminus \{0\}} \frac{\balpha_n^{\mu}}{n} \bz^{-n} + - \balpha_0^{\mu}\, \ln{\barz} + + \sum\limits_{n \in \Z \setminus \{0\}} \frac{\balpha_n^{\mu}}{n} \barz^{-n} ), \end{split} \label{eq:tduality:modes} @@ -1022,9 +1022,9 @@ Now let \end{equation} where $S^1( R )$ is a compact $1$-dimensional circle of radius $R$ such that the boundary conditions for the compact coordinate are \begin{equation} - X^{D - 1}( z\, e^{2\pi i}, \bz\, e^{-2\pi i} ) + X^{D - 1}( z\, e^{2\pi i}, \barz\, e^{-2\pi i} ) = - X^{D - 1}( z, \bz ) + 2 \pi\, m\, R, + X^{D - 1}( z, \barz ) + 2 \pi\, m\, R, \qquad m \in \Z. \label{eq:dbranes:winding} @@ -1066,7 +1066,7 @@ From~\eqref{eq:conf:Texpansion} and \eqref{eq:conf:bosonicstringT} we find \sum\limits_{n = 1}^{+\infty}\, \qty( 2 \alpha_{-n}^{\mu} \alpha_n^{\nu}\, \eta_{\mu\nu} + a ) ), \\ - \bL_0 + \barL_0 &= \frac{\ap}{2}\, \qty( @@ -1079,23 +1079,23 @@ From~\eqref{eq:conf:Texpansion} and \eqref{eq:conf:bosonicstringT} we find \end{split} \end{equation} where $a$ is constant given by normal ordering, representing the zero point energy of the theory. -Imposing physical conditions~\eqref{eq:conf:physical} and the \emph{level matching} $(L_0 - \bL_0) \ket{\phi} = 0$ for closed strings, we find +Imposing physical conditions~\eqref{eq:conf:physical} and the \emph{level matching} $(L_0 - \barL_0) \ket{\phi} = 0$ for closed strings, we find \begin{equation} \begin{split} M^2 & = - \frac{1}{\ap^2}\, \qty( n\, \frac{\ap}{R} + m\, R )^2 + \frac{1}{(\ap)^2}\, \qty( n\, \frac{\ap}{R} + m\, R )^2 + \frac{4}{\ap}\, \qty( \rN + a ) \\ & = - \frac{1}{\ap^2}\, \qty( n\, \frac{\ap}{R} - m\, R )^2 + \frac{1}{(\ap)^2}\, \qty( n\, \frac{\ap}{R} - m\, R )^2 + - \frac{4}{\ap}\, \qty( \overline{\rN} + a ), + \frac{4}{\ap}\, \qty( \brN + a ), \end{split} \label{eq:dbranes:closedspectrum} \end{equation} -where $\rN = \sum\limits_{n = 1}^{+\infty}\, \alpha_{-n} \cdot \alpha_n$ and $\overline{\rN} = \sum\limits_{n = 1}^{+\infty}\, \balpha_{-n} \cdot \balpha_n$. +where $\rN = \sum\limits_{n = 1}^{+\infty}\, \alpha_{-n} \cdot \alpha_n$ and $\brN = \sum\limits_{n = 1}^{+\infty}\, \balpha_{-n} \cdot \balpha_n$. We then notice that as $R \to \infty$ all states with $m \neq 0$ become infinitely massive while the states for $m = 0$ and all values of $n$ become a continuum. Conversely, as $R \to 0$ all states with $n \neq 0$ become infinitely heavy. In field theory this would translate into a reduction of the number of dimensions since the remaining fields would be independent of the compact coordinate~\cite{Polchinski:1996:TASILecturesDBranes,Zwiebach::FirstCourseString}. @@ -1110,7 +1110,7 @@ At the level of the modes this \emph{T-duality} acts by swapping the sign of the \end{equation} defining the dual coordinate \begin{equation} - Y^{D-1}( z, \bz ) = Y^{D-1}( z ) + \overline{Y}^{D-1}( \bz ) = X^{D-1}( z ) - \bX^{D-1}( \bz ). + Y^{D-1}( z, \barz ) = Y^{D-1}( z ) + \barY^{D-1}( \barz ) = X^{D-1}( z ) - \barX^{D-1}( \barz ). \label{eq:tduality:compactdirection} \end{equation} @@ -1121,15 +1121,15 @@ defining the dual coordinate Consider the case of open strings satisfying the \eom~\eqref{eq:tduality:eom} and the coundary conditions~\eqref{eq:tduality:bc}. The usual mode expansion~\eqref{eq:tduality:modes} here leads to \begin{equation} - X^{\mu}( z, \bz ) + X^{\mu}( z, \barz ) = x_0^{\mu} - - i\, \ap\, p^{\mu}\, \ln( z \bz ) + i\, \ap\, p^{\mu}\, \ln( z \barz ) + i\, \sqrt{\frac{\ap}{2}}\, \sum\limits_{n \in \Z \setminus \{0\}} - \frac{\alpha_n^{\mu}}{n} \qty( z^{-n} + \bz^{-n} ) + \frac{\alpha_n^{\mu}}{n} \qty( z^{-n} + \barz^{-n} ) \end{equation} and $\ell = \pi$. @@ -1148,13 +1148,13 @@ In fact the original Neumann boundary condition~\eqref{eq:tduality:bc} becomes a \begin{split} \eval{\ipd{\sigma} X^{D-1}( \tau, \sigma )}^{\sigma = \pi}_{\sigma = 0} & = - \eval{\ipd{\sigma} X^{D-1}( e^{\tau_E + i \sigma} ) + \ipd{\sigma} \bX^{D-1}( e^{\tau_E - i \sigma} )}^{\sigma = \pi}_{\sigma = 0} + \eval{\ipd{\sigma} X^{D-1}( e^{\tau_E + i \sigma} ) + \ipd{\sigma} \barX^{D-1}( e^{\tau_E - i \sigma} )}^{\sigma = \pi}_{\sigma = 0} \\ & = - \eval{i\, \ipd{\xi} X^{D-1}( e^{\xi} ) - i\, \ipd{\bxi} \bX^{D-1}( e^{\bxi} )}^{\Im \xi = \pi}_{\Im \xi = 0} + \eval{i\, \ipd{\xi} X^{D-1}( e^{\xi} ) - i\, \ipd{\bxi} \barX^{D-1}( e^{\bxi} )}^{\Im \xi = \pi}_{\Im \xi = 0} \\ & = - \eval{i\, \ipd{\tau_E} Y^{D-1}( e^{\tau_E + i \sigma} ) + i\, \ipd{\tau_E} \overline{Y}^{D-1}( e^{\tau_E - i \sigma} )}^{\sigma = \pi}_{\sigma = 0} + \eval{i\, \ipd{\tau_E} Y^{D-1}( e^{\tau_E + i \sigma} ) + i\, \ipd{\tau_E} \barY^{D-1}( e^{\tau_E - i \sigma} )}^{\sigma = \pi}_{\sigma = 0} \\ & = \eval{i\, \ipd{\tau} Y^{D-1}( \tau, \sigma )}^{\sigma = \pi}_{\sigma = 0} diff --git a/thesis.cls b/thesis.cls index cc4dbfd..9d4a43c 100644 --- a/thesis.cls +++ b/thesis.cls @@ -36,9 +36,10 @@ \RequirePackage{titlesec} %--------------------- custom title section \RequirePackage{changepage} %------------------- change page layout \RequirePackage{lastpage} %--------------------- reference to last page +\RequirePackage{tocloft} %---------------------- modify table of contents \RequirePackage[type={CC}, modifier={by-nc-nd}, - version={4.0}]{doclicense} %- licence + version={4.0}]{doclicense} %---- licence \RequirePackage[nottoc]{tocbibind} %------------ put bibliography in TOC \RequirePackage[backend=biber, citestyle=numeric-comp, @@ -75,10 +76,15 @@ \newlength{\blockskip} \setlength\blockskip{1em} + \newlength{\sepwidth} \setlength\sepwidth{1pt} \setlength\parskip{1em} +%---- table of contents + +\setlength\cftbeforesubsecskip{5pt} + %---- metadata \makeatletter diff --git a/thesis.tex b/thesis.tex index 2ad8fc7..35b2a41 100644 --- a/thesis.tex +++ b/thesis.tex @@ -23,64 +23,46 @@ pdfauthor={Riccardo Finotello} } -%---- additional commands -\newcommand{\sm}{\textsc{sm}\xspace} -\newcommand{\eom}{\textsc{e.o.m.}\xspace} -\newcommand{\cft}{\textsc{cft}\xspace} -\newcommand{\ope}{\textsc{ope}\xspace} -\newcommand{\ap}{\ensuremath{\alpha'}} -\newcommand{\cy}{\textsc{CY}\xspace} - %---- functions \newcommand{\hyp}[4]{\ensuremath{\mathrm{F}\left( #1,\, #2;\, #3;\, #4 \right)}} \newcommand{\poch}[2]{\ensuremath{\left( #1 \right)_{#2}}} \newcommand{\gfun}[1]{\ensuremath{\Gamma\left( #1 \right)}} %---- derivatives -\newcommand{\pd}{\ensuremath{\partial}} -\newcommand{\bpd}{\ensuremath{\overline{\partial}}} -\newcommand{\lrpartial}[1]{\overset{\leftrightarrow} - {\partial_{ #1 }} - } \newcommand{\consprod}[2]{\left\langle #1, #2 \right\rangle} \newcommand{\lconsprod}[2]{\left\langle\hspace{-0.25em}\left\langle #1\right.,\, #2 \right\rangle} -\newcommand{\lfdv}[2]{\frac{\overset{\leftarrow}{\delta} #1}{\delta #2}} -\newcommand{\rfdv}[2]{\frac{\overset{\rightarrow}{\delta} #1}{\delta #2}} \newcommand{\dual}[1]{\tensor[^*]{#1}{}} %---- integrals \newcommand{\ddz}{\ensuremath{\frac{\mathrm{d}z}{2 \pi i}}} \newcommand{\ddbz}{\ensuremath{\frac{\mathrm{d}\overline{z}}{2 \pi i}}} +\newcommand{\ddw}{\ensuremath{\frac{\mathrm{d}w}{2 \pi i}}} +\newcommand{\ddbw}{\ensuremath{\frac{\mathrm{d}\overline{w}}{2 \pi i}}} \newcommand{\cint}[1]{\ensuremath{\oint\limits_{\ccC_{#1}}}} -%---- operators -\newcommand{\bT}{\ensuremath{\overline{T}}} -\newcommand{\bL}{\ensuremath{\overline{L}}} - %---- states -\newcommand{\regvacuum}{\ensuremath{\ket{0}_{\SL{2}{R}}}} +\newcommand{\regvacuum}{\ensuremath{\ket{0}_{\SL{2}{R}}}\xspace} +\newcommand{\regvacuumin}{\ensuremath{\ket{0_{(\text{in})}}_{\SL{2}{R}}}\xspace} +\newcommand{\regvacuumout}{\ensuremath{\ket{0_{(\text{out})}}_{\SL{2}{R}}}\xspace} +\newcommand{\twsvacket}{\ensuremath{\ket{\mathrm{T}}}\xspace} +\newcommand{\twsvacbra}{\ensuremath{\bra{\mathrm{T}}}\xspace} +\newcommand{\excvacket}{\ensuremath{\ket{T_{\rE,\, \brE}}}\xspace} +\newcommand{\excvacbra}{\ensuremath{\bra{T_{\rE,\, \brE}}}\xspace} +\newcommand{\eexcvacket}{\ensuremath{\ket{T_{\rE}}}\xspace} +\newcommand{\eexcvacbra}{\ensuremath{\bra{T_{\rE}}}\xspace} +\newcommand{\Gexcvac}{\Omega_{\qty{x_{(t)},\, \rE_{(t)},\, \brE_{(t)}}}} +\newcommand{\Gexcvacket}{\ket{\Gexcvac}} +\newcommand{\Gexcvacbra}{\bra{\Gexcvac}} +\newcommand{\GGexcvac}{\Omega_{\qty{x_{(t)},\, \rE_{(t)}}}} +\newcommand{\GGexcvacket}{\ket{\GGexcvac}} +\newcommand{\GGexcvacbra}{\bra{\GGexcvac}} +\newcommand{\cmode}[4]{\mathfrak{C}_{#1}\qty(#2,\, \qty{#4,\, #3})} + +%---- operators +\newcommand{\noE}[1]{\ccN_{\rE,\, \brE}\qty[ #1 ]} %---- coordinates -\newcommand{\dX}{\ensuremath{\dot{X}}} -\newcommand{\pX}{\ensuremath{X'}} -\newcommand{\bX}{\ensuremath{\overline{X}}} -\newcommand{\bpsi}{\ensuremath{\overline{\psi}}} -\newcommand{\bxi}{\ensuremath{\overline{\xi}}} -\newcommand{\bchi}{\ensuremath{\overline{\chi}}} -\newcommand{\bz}{\ensuremath{\overline{z}}} -\newcommand{\bu}{\ensuremath{\overline{u}}} -\newcommand{\bt}{\ensuremath{\overline{t}}} -\newcommand{\bw}{\ensuremath{\overline{w}}} -\newcommand{\bomega}{\ensuremath{\overline{\omega}}} -\newcommand{\bepsilon}{\ensuremath{\overline{\epsilon}}} -\newcommand{\balpha}{\ensuremath{\overline{\alpha}}} -\newcommand{\bzeta}{\ensuremath{\overline{\zeta}}} -\newcommand{\halpha}{\ensuremath{\widehat{\alpha}}} -\newcommand{\hbeta}{\ensuremath{\widehat{\beta}}} -\newcommand{\htau}{\ensuremath{\widehat{\tau}}} -\newcommand{\hpsi}{\ensuremath{\widehat{\psi}}} -\newcommand{\Hpsi}{\ensuremath{\widehat{\Psi}}} - +\newcommand{\pX}{\ensuremath{X'}\xspace} %---- BEGIN DOCUMENT @@ -142,6 +124,10 @@ \label{sec:parameters} \input{sec/app/parameters.tex} +\section{Reflection Conditions on the Vacuum} +\label{sec:details_reflection} +\input{sec/app/reflection.tex} + %---- BIBLIOGRAPHY \cleardoubleplainpage{}