diff --git a/img/branchcuts.pdf b/img/branchcuts.pdf new file mode 100644 index 0000000..8613ad4 Binary files /dev/null and b/img/branchcuts.pdf differ diff --git a/img/branchcuts.pdf_tex b/img/branchcuts.pdf_tex new file mode 100644 index 0000000..bb89f57 --- /dev/null +++ b/img/branchcuts.pdf_tex @@ -0,0 +1,65 @@ +%% Creator: Inkscape 1.0 (4035a4fb49, 2020-05-01), www.inkscape.org +%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010 +%% Accompanies image file 'branchcuts.pdf' (pdf, eps, ps) +%% +%% To include the image in your LaTeX document, write +%% \input{.pdf_tex} +%% instead of +%% \includegraphics{.pdf} +%% To scale the image, write +%% \def\svgwidth{} +%% \input{.pdf_tex} +%% instead of +%% \includegraphics[width=]{.pdf} +%% +%% Images with a different path to the parent latex file can +%% be accessed with the `import' package (which may need to be +%% installed) using +%% \usepackage{import} +%% in the preamble, and then including the image with +%% 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+ \fi% + \global\let\svgwidth\undefined% + \global\let\svgscale\undefined% + \makeatother% + \begin{picture}(1,0.76153711)% + \lineheight{1}% + \setlength\tabcolsep{0pt}% + \put(0,0){\includegraphics[width=\unitlength,page=1]{branchcuts.pdf}}% + \put(0.96250584,0.22924726){\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$x$\end{tabular}}}}% + \put(0.26509366,0.76435918){\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$y$\end{tabular}}}}% + \put(0.63886437,0.19725284){\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$D_{(2)}$\end{tabular}}}}% + \put(0.37615789,0.37851589){\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$D_{(3)}$\end{tabular}}}}% + \put(0.1552996,0.19562747){\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$D_{(4)}$\end{tabular}}}}% + \put(0.8804906,0.32528402){\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$D_{(1)}$\end{tabular}}}}% + \put(-0.0034854,0.32147455){\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$D_{(1)}$\end{tabular}}}}% + \end{picture}% +\endgroup% diff --git a/sec/app/isomorphism.tex b/sec/app/isomorphism.tex new file mode 100644 index 0000000..95a471b --- /dev/null +++ b/sec/app/isomorphism.tex @@ -0,0 +1,176 @@ +In this appendix we explain the conventions used for \SU{2} and show the details of the isomorphism between \SO{4} and a class of equivalence of $\SU{2} \times \SU{2}$. + + +\subsection{Conventions} + +We parameterise \SU{2} matrices $U$ with a vector $\vb{n} \in \R^3$ such that: +\begin{equation} + U(\vb{n}) + = + \cos(2 \pi n)\, \1_2 + + + i\, \frac{\vb{n} \cdot \vb{\sigma}}{n}\, \sin(2 \pi n), + \label{eq:su2parametrisation} +\end{equation} +where $n = \norm{\vb{n}}$ and $0 \le n \le \frac{1}{2}$. +We also identify all $\vb{n}$ when $n=\frac{1}{2}$ since in this case $U(\vb{n})= -\1_2$. +The parametrisation is such that: +\begin{eqnarray} + U^*(\vb{n}) + & = & + \sigma^2\, U(\vb{n})\, \sigma^2 + = + U(\widetilde{\vb{n}}), + \\ + U^{\dagger}(\vb{n}) + & = & + U^T(\widetilde{\vb{n}}) + = + U(-\vb{n}), + \\ + -U(\vb{n}) + & = & + U(\widehat{\vb{n}}) + \label{eq:U_props} +\end{eqnarray} +where $\sigma^2$ is the second Pauli matrix, $\widetilde{\vb{n}} = \left( -n^1, n^2, -n^3 \right)$ and $\widehat{\vb{n}} = - \left(\frac{1}{2} -n \right)\, \frac{\vb{n}}{n}$. + +The group product of two elements $U(\vb{n} \circ \vb{m} ) = U(\vb{n})\, U(\vb{m})$ has an explicit realisation as: +\begin{equation} + \begin{split} + \cos(2 \pi \norm{\vb{n} \circ \vb{m}}) + & = + \cos(2 \pi n)\, \cos(2 \pi m) + - + \sin(2 \pi n)\, \sin(2\pi m)\, \frac{\vb{n} \cdot \vb{m}}{n\, m}, + \\ + \sin(2 \pi \norm{\vb{n} \circ \vb{m}})\, + \frac{\vb{n} \circ \vb{m}}{\norm{\vb{n} \circ \vb{m}}} + & = + \cos(2 \pi n)\, \sin(2\pi m)\, \frac{\vb{m}}{m} + + + \sin(2 \pi n)\, \cos(2\pi m)\, \frac{\vb{n}}{n}. + \end{split} + \label{eq:product_in_SU2} +\end{equation} + +\subsection{The Isomorphism} + +Let $I = 1,\, 2,\, 3,\, 4$ and define: +\begin{equation} + \tau_I = \left( i\, \1_2,\, \vb{\sigma} \right), +\end{equation} +where $\vb{\sigma} = \left( \sigma^1,\, \sigma^2,\, \sigma^3 \right)$ are the Pauli matrices. +It is possible to show that: +\begin{equation} + \begin{split} + \left( \tau_I \right)^{\dagger} + & = + \eta_{IJ}\, {\tau}^I, + \\ + \left( \tau^I \right)^* + & = + -\sigma_2\, \tau_I\, \sigma_2, + \end{split} + \label{eq:tau_props} +\end{equation} +where $\eta_{IJ} = \mathrm{diag}(-1,1,1,1)$. +The following relations are then a natural consequence: +\begin{eqnarray} + \tr(\tau_I) + & = & + 2\, i\, \delta_{I1}, + \\ + \tr(\tau_I \tau_J) + & = & + 2\, \eta_{IJ}, + \\ + \tr(\tau_I \left( \tau_J \right)^{\dagger}) + & = & + 2\, \delta_{IJ}. +\end{eqnarray} + +Now consider a vector in the spinor representation: +\begin{equation} + X_{(s)} = X^I\, \tau_I. +\end{equation} +We can recover the components using the previous properties: +\begin{equation} + X^I + = + \frac{1}{2}\, \delta^{IJ}\, + \tr(X_{(s)} \left( \tau_J \right)^{\dagger}) + = + \frac{1}{2}\, \eta^{IJ}\, \tr(X_{(s)} \tau_J), +\end{equation} +where the trace acts on the space of the $\tau$ matrices. +If the vector $X^I$ is real, using~\eqref{eq:tau_props} we have: +\begin{equation} + \begin{split} + X_{(s)}^{\dagger} + & = + X^I\, \eta_{IJ}\, \tau^J + = + \frac{1}{2} \tr(X_{(s)} \tau_I)\, \tau^I, + \\ + X_{(s)}^* + & = + - \sigma_2\, X_{(s)}\, \sigma_2. + \end{split} + \label{eq:X_dagger} +\end{equation} + +A rotation in spinor representation is defined as: +\begin{equation} + X'_{(s)} = U_{L}(\vb{n})\, X_{(s)}\, U_{R}^{\dagger}(\vb{m}) +\end{equation} +and it is equivalent to: +\begin{equation} + \left( X' \right)^I + = + \tensor{R}{^I_J}\, + X^J +\end{equation} +through +\begin{equation} + R_{IJ} + = + \frac{1}{2} + \tr( + \left( \tau_I \right)^{\dagger}\, + U_{L}(\vb{n})\, + \tau_J\, + U_{R}^{\dagger}(\vb{m}) + ). +\end{equation} +The matrix $R$ is the $4$-dimensional rotation matrix we are looking for since: +\begin{equation} + \tr(X'_{(s)}\, (X')^{\dagger}_{(s)}) + = + \tr(X_{(s)}\, X^{\dagger}_{(s)}) + \qquad + \Rightarrow + \qquad + \finitesum{K}{1}{4} R_{IK} R^*_{JK} = \delta_{I \,J}. +\end{equation} +From the second equation in \eqref{eq:tau_props} and the first equation in \eqref{eq:U_props} we then get the reality condition on $R$: +\begin{equation} + R_{NM} + = + \frac{1}{2}\, \eta_{NI}\, \eta_{MJ}\, + \tr(\tau_I ^{\dagger}\, U_{R}\, \tau_J\, U_{L}^{\dagger}) + = + \frac{1}{2} + \tr(\tau_N\, U_{R}\, \tau_M^\dagger\, U_{L}^{\dagger}) + = + R_{NM}^*. +\end{equation} +Furthermore the direct computation of the determinant of $R$ using the parametrisation~\eqref{eq:su2parametrisation} shows that $\det R = 1$. +Finally the explicit choice of the basis $\tau$ ensures $R$ to be a real matrix which ensures $R \in \SO{4}$. +Since $\left\lbrace U_{L},\, U_{R} \right\rbrace$ and $\left\lbrace -U_{L},\, -U_{R} \right\rbrace$ generate the same \SO{4} matrix then the correct isomorphism takes the form: +\begin{equation} + \SO{4} + \cong + \frac{\SU{2} \times \SU{2}}{\Z_2}. +\end{equation} + diff --git a/sec/part1/dbranes.tex b/sec/part1/dbranes.tex index 9c9456b..ea07c54 100644 --- a/sec/part1/dbranes.tex +++ b/sec/part1/dbranes.tex @@ -142,4 +142,440 @@ The superscript $\parallel$ represents any of the coordinates parallel to the D- Notice that the additional $\Z_2$ factor in $\rS\left( \OO{2} \times \OO{2} \right)$ with respect to $\SO{2} \times \SO{2}$ can be used to set $g_{(t)}^{\perp} \ge 0$. +\subsubsection{Boundary Conditions for Branes at Angles} + +The peculiar embedding of the D-branes has natural consequences on the boundary conditions of the open strings. +Let $\tau_E = i \tau$ be the Wick rotated time direction. +We define the usual upper plane coordinates: +\begin{eqnarray} + u + = + x + i y + = + e^{\tau_E + i \sigma} + & \in & + \ccH \cup \left\lbrace z \in \C \mid \Im z = 0 \right\rbrace, + \\ + \bu + = + x - i y + = + e^{\tau_E - i \sigma} + & \in & + \overline{\ccH} \cup \left\lbrace z \in \C \mid \Im z = 0 \right\rbrace, +\end{eqnarray} +where $\ccH = \left\lbrace z \in \C \mid \Im z > 0 \right\rbrace$ is the upper complex plane and $\overline{\ccH} = \left\lbrace z \in \C \mid \Im z < 0 \right\rbrace$ 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 $\Im z = 0$. +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)} = \left[ x_{(t)}, x_{(t-1)} \right], + \qquad + t = 2,\, 3,\, \dots,\, N_B, + \qquad + x_{(t)} < x_{(t-1)}. +\end{equation} +The points $x_{(t)}$ and $x_{(t-1)}$ represent the worldsheet intersection points of the brane $D_{(t)}$ with the branes $D_{(t+1)}$ and $D_{(t-1)}$ respectively. +The choice of the intervals must be carefully considered: since the D-branes are defined modulo $N_B$, the shorthand for the interval $D_{(1)} = \left[ x_{(1)}, x_{(N_B)} \right]$ should actually be: +\begin{equation} + D_{(1)} + = + \left[ x_{(1)}, +\infty \right) \cup \left( -\infty, x_{(N_B)} \right]. +\end{equation} + +In the global coordinates system associated to the subspace $\R^4 \subset \ccM^{1, d+4}$ where D-branes intersect, the relevant part of the action in conformal gauge is: +\begin{equation} + \begin{split} + S_{\R^4} + & = + \frac{1}{2 \pi \ap} + \iint\limits_{\ccH} + \dd{u} \dd{\bu}\, + \ipd{u} X^I\, \ipd{\bu} X^J\, + \eta_{IJ} + \\ + & = + \frac{1}{4 \pi \ap} + \iint\limits_{\R \times \R^+} + \dd{x}\dd{y}\, + \left( + \ipd{x} X^I\, \ipd{x} X^J + + + \ipd{y} X^I\, \ipd{y} X^J + \right)\, + \eta_{IJ}, + \end{split} + \label{eq:string_action} +\end{equation} +where $2\, \ipd{u} = \ipd{x} - i\, \ipd{y}$ and $2\, \ipd{\bu} = \ipd{x} + i\, \ipd{y}$. +The \eom in these coordinates are: +\begin{equation} + \ipd{u} \ipd{\bu} X^I( u, \bu ) + = + \frac{1}{4} + \left( \ipd{x}^2 + \ipd{y}^2 \right) X^I( x+iy, x-iy ) + = + 0. + \label{eq:string_equation_of_motion} +\end{equation} +Their solution factorises as usual in holomorphic and anti-holomorphic components $X^I( u, \bu ) = X^I( u ) + \bX^I( \bu )$. + +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} + & = & + 0, + \qquad + i = 1,\, 2, + \label{eq:neumann_bc} + \\ + X^m_{(t)}( \tau, 0 ) + = + X^m_{(t)}( x, x ) + & = & + 0, + \qquad + m = 3,\, 4, + \label{eq:dirichlet_bc} +\end{eqnarray} +where $x \in D_{(t)} = \left[ x_t, x_{t-1} \right]$ and the index $i$ labels the Neumann boundary conditions while $m$ labels the Dirichlet coordinates associated to the direction orthogonal to the D-branes. + +As the presence of $g_{(t)}^m$ in \eqref{eq:brane_rotation} and \eqref{eq:dirichlet_bc} may complicate the analysis, we consider the derivative along the boundary direction of \eqref{eq:dirichlet_bc} to remove the dependence on the translation vector. +This procedure produces simpler boundary conditions which are nevertheless not equivalent to the original~\eqref{eq:neumann_bc} and \eqref{eq:dirichlet_bc}: they will be recovered later by adding further constraints. +The simpler boundary conditions we consider in the global coordinates are: +\begin{eqnarray} + \tensor{\left( R_{(t)} \right)}{^i_J} + \eval{\ipd{\sigma} X^J( \tau, \sigma )}_{\sigma = 0} + & = & + i\, \tensor{\left( R_{(t)} \right)}{^i_J} + \left( + \ipd{u} X^J( x + i\, 0^+ ) - \ipd{\bu} \bX^J( x - i\, 0^+ ) + \right) + = + 0, + \\ + \tensor{\left( R_{(t)} \right)}{^m_J} + \eval{\ipd{\tau} X^J( \tau, \sigma )}_{\sigma = 0} + & = & + i\, \tensor{\left( R_{(t)} \right)}{^m_J} + \left( + \ipd{u} X^J( x + i\, 0^+ ) + \ipd{\bu} \bX^J( x - i\, 0^+ ) + \right) + = + 0, +\end{eqnarray} +where $i = 1,\, 2$, $m = 3,\, 4$ and $x \in D_{(t)}$. + +With the introduction of the target space embedding of the worldsheet interaction point between D-branes $D_{(t)}$ and $D_{(t+1)}$, $f_{(t)}$, we recover the full boundary conditions in terms of discontinuities on the real axis: +\begin{equation} + \begin{cases} + \ipd{u} X^I( x + i\, 0^+ ) + & = + \tensor{\left( U_{(t)} \right)}{^I_J} + \ipd{\bu} \bX^J( x - i\, 0^+ ), + \qquad + x \in D_{(t)} + \\ + X^I( x_{(t)}, x_{(t)} ) + & = + f_{(t)} + \end{cases}. + \label{eq:discontinuity_bc} +\end{equation} +In the last expression we introduced the matrix +\begin{equation} + U_{(t)} + = + \left( R_{(t)} \right)^{-1}\, + \cS\, + R_{(t)} + \in + \frac{\SO{4}}{\rS\left( \OO{2} \times \OO{2} \right)}, + \label{eq:Umatrices} +\end{equation} +where +\begin{equation} + \cS + = + \mqty( \dmat{ 1, 1, -1, -1 } ) + \label{eq:reflection_S} +\end{equation} +embeds the difference between Neumann and Dirichlet conditions. +Given its definition $U_{(t)}$ is such that $U_{(t)} = \left( U_{(t)} \right)^{-1} = \left( U_{(t)} \right)^T$. + +The target space vector $f_{(t)}$ recovers the apparent loss of information suffered when losing $g_{(t)}$. +Consider for instance the embedding equations~\eqref{eq:dirichlet_bc} for any two intersecting D-branes $D_{(t)}$ and $D_{(t+1)}$. +Introducing the auxiliary quantities +\begin{eqnarray} + \cR_{(t,\, t+1)} + = + \mqty( R_{(t)}^m \\ R_{(t+1)}^n ) + & \in & + \GL{4}{\R}, + \qquad + m, n = 3, 4, + \\ + \cG_{(t,\, t+1)} + = + \mqty( g_{(t)}^m \\ g_{(t+1)}^n ) + & \in & + \R^4, + \qquad + m, n = 3, 4, +\end{eqnarray} +we can compute the intersection point as: +\begin{equation} + f_{(t)} + = + \left( \cR_{(t,\, t+1)} \right)^{-1}\, + \cG_{(t,\, t+1)}. +\end{equation} +Information on $g_{(t)}$ is thus recovered through the global boundary conditions in the second equation in \eqref{eq:discontinuity_bc}. + + +\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 $D_{(\bt)}$: +\begin{equation} + \ipd{z} \cX(z) = + \begin{cases} + \ipd{u} X(u) + & + \qif + z = u \qand \Im z > 0 \qor z \in D_{(\bt)} + \\ + U_{(\bt)}\, + \ipd{\bu} \bX(\bu) + & \qif z = \bu \qand \Im z < 0 \qor z \in D_{(\bt)} + \end{cases}. + \label{eq:real_doubling_trick} +\end{equation} +Let then $\cU_{(t,\, t+1)} = U_{(t+1)}\, U_{(t)}$ and $\widetilde{\cU}_{(t,\, t+1)} = U_{(\bt)}\, U_{(t)}\, U_{(t+1)}\, U_{(\bt)}$. +The boundary conditions in terms of the doubling field are: +\begin{eqnarray} + \ipd{z} \cX( x_t + e^{2 \pi i}( \eta + i\, 0^+ ) ) + & = & + \cU_{(t,\, t+1)} + \ipd{z} \cX( x_t + \eta + i\, 0^+ ), + \label{eq:top_monodromy} + \\ + \partial \cX( x_t + e^{2 \pi i}( \eta - i\, 0^+ ) ) + & = & + \widetilde{\cU}_{(t,\, t+1)} + \ipd{z} \cX( x_t + \eta - i\, 0^+ ), + \label{eq:bottom_monodromy} +\end{eqnarray} +for $0 < \eta < \min\left( \abs{x_{(t-1)} - x_{(t)}}, \abs{x_{(t)} - x_{(t+1)}} \right)$ 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. + +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 $\overline{\ccH}$. +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)} + = + \finiteprod{t}{1}{N_B}\, + \widetilde{\cU}_{(\bt + t, \bt + 1 + t)} + = + \1_4. +\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. + +\begin{figure}[tbp] + \centering + \def\svgwidth{0.5\textwidth} + \import{img/}{branchcuts.pdf_tex} + \caption{% + Branch cut structure of the complex plane with $N_B = 4$. + Cuts are pictured as solid coloured blocks running from one intersection point to another at finite. + } + \label{fig:finite_cuts} +\end{figure} + +As a consistency check, the action~\eqref{eq:string_action} can be computed in terms of the doubling field \cX. +The map +\begin{equation} + x_{(t)} + \eta \pm i\, 0^+ + \quad + \mapsto + \quad + x_{(t)} + e^{2 \pi i}( \eta \pm i\, 0^+) +\end{equation} +must leave the action untouched since it does not depend on the branch cut structure. +In fact we can show that +\begin{equation} + S_{\R^4} + = + \frac{1}{4 \pi \ap} + \iint\limits_{\C} + \dd{z} \dd{\bz}\, + \ipd{z} \cX^T(z)\, + U_{(\bt)}\, + \ipd{\bz} \cX(\bz). +\end{equation} +As a matter of fact the action does not depend on the branch structure of the complex plane. + + +\subsection{D-branes at Angles in Spinor Representation} + +In the previous section we showed that it is possible to map the information on the rotations of the D-branes to non trivial monodromies of the doubling field. +We thus recast the issue of solving the \eom of the string in the presence of rotated boundary conditions to the search for an explicit solution $\ipd{z} \cX( z )$ reproducing the non trivial monodromies in~\eqref{eq:top_monodromy} and \eqref{eq:bottom_monodromy}. + +The field $\ipd{z} \cX( z )$ is technically a $4$-dimensional \emph{real} vector which has $N_B$ non trivial monodromy factors represented by $4 +\times 4$ real matrices, one for each interaction point $x_{(t)}$. +A solution of the \eom is encoded in four linearly independent functions with $N_B$ branch points. +In principle we could try to write it as a solution to fourth order differential equations with $N_B$ finite Fuchsian points. +This is however an open mathematical debate. +In fact the basis of such functions around each branch point are usually complicated and defined up to several free parameters. +Moreover the explicit connection formulae between any two of them is an unsolved mathematical problem. +Using contour integrals and writing the functions as Mellin--Barnes integrals it might be possible to solve the issue in the case $N_B = 3$ but it is certainly not the best course of action. + +On the other hand $N_B = 3$ is exactly the case we are investigating. +In what follows we use the isomorphism +\begin{equation} + \SO{4} + \cong + \frac{\SU{2} \times \SU{2}}{\Z_2} + \label{eq:su2isomorphism} +\end{equation} +to map the problem of finding a $4$-dimensional real solution to the \eom to a quest for a $2 \times 2$ complex matrix. +Such matrix is a linear superposition of tensor products of vectors in the fundamental representation of two different \SU{2} groups. +These vectors are solutions to second order differential equations with three Fuchsian points, that is the hypergeometric equation. +The task is then to find the parameters of the hypergeometric functions producing the spinor representation of the monodromies in~\eqref{eq:top_monodromy} and \eqref{eq:bottom_monodromy}. + + +\subsubsection{Doubling Trick and Rotations in Spinor Representation} + +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, +\end{equation} +where $\tau = \left( i\, \1_2,\, \vb{\sigma} \right)$ and $\vb{\sigma}$ is the vector of the Pauli matrices. +Consider then: +\begin{equation} + \ipd{z} \cX_{(s)}( z ) + = + \begin{cases} + \ipd{u} X_{(s)}(u) + & \qif + z \in \ccH \qor z \in D_{(\bt)} + \\ + 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)} + \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. +Let $0 < \eta < \min\left( \abs{x_{(t)} - x_{(t-1)}}, \abs{x_{(t+1)} - x_{(t)}} \right)$. +We find: +\begin{eqnarray} + \ipd{z} \cX_{(s)}( x_t + e^{2 \pi i}( \eta + i\, 0^+) ) + & = & + \cL_{(t,\, t+1)} \ipd{z}\, + \cX_{(s)}( x_t + \eta + i\, 0^+ )\, + \cR_{(t,\, t+1)}^{\dagger}, + \label{eq:top_spinor_monodromy} + \\ + \ipd{z} \cX_{(s)}( x_t + e^{2 \pi i}( \eta - i\, 0^+) ) + & = & + \widetilde{\cL}_{(t,\, t+1)}\, + \ipd{z} \cX_{(s)}( x_t + \eta - i\, 0^+ )\, + \widetilde{\cR}_{(t,\, t+1)}^{\dagger}, + \label{eq:bottom_spinor_monodromy} +\end{eqnarray} + where: +\begin{eqnarray} + \cL_{(t,\, t+1)} + & = & + U_{L}(\vb{n}_{(t+1)})\, + U_{L}^{\dagger}(\vb{n}_{(t)}), + \\ + \widetilde{\cL}_{(t,\, t+1)} + & = & + U_{L}(\vb{n}_{(\bt)})\, + U_{L}^{\dagger}(\vb{n}_{(t)})\, + U_{L}(\vb{n}_{(t+1)})\, + U_{L}^{\dagger}(\vb{n}_{(\bt)}), + \\ + \cR_{(t,\, t+1)} + & = & + U_{R}(\vb{m}_{(t+1)})\, + U_{R}^{\dagger}(\vb{m}_{(t)}), + \\ + \widetilde{\cR}_{(t,\, t+1)} + & = & + U_{R}(\vb{m}_{(\bt)})\, + U_{R}^{\dagger}(\vb{m}_{(t)})\, + U_{R}(\vb{m}_{(t+1)})\, + U_{R}^{\dagger}(\vb{m}_{(\bt)}). +\end{eqnarray} + +In spinor representation the action~\eqref{eq:string_action} becomes +\begin{equation} + \begin{split} + S_{\R^4} + & = + \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)) + \\ + & = + \frac{1}{8 \pi \ap} + \iint\limits_{\C} + \dd{z} \dd{\bz}\, + \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) + ). + \end{split} + \label{eq:action_doubling_fields_spinor_representation} +\end{equation} +It is possible to show that the closed loop $x_t + \eta \pm i\, 0^+ \mapsto x_t + e^{2 \pi i}( \eta \pm i\, 0^+ )$ does not generate additional contributions in the action. + + +\subsubsection{Special Form of Matrices for D-Branes at Angles} +\label{sect:special_SO4} + +The $\SU{2}$ matrices involved in this scenario with D-branes intersecting at angles have a particular form. +In the left sector (i.e.\ $\SU{2}_L$ matrices) we have: +\begin{equation} + \cL_{(t,\, t+1)} + = + U_{L}(\vb{n}_{(t+1)})\, + U_{L}^{\dagger}(\vb{n}_{(t)})\, + = + -\vb{n}_{(t+1)} \cdot \vb{n}_{(t)} + + + i\, (\vb{n}_{(t+1)} \times \vb{n}_{(t)}) \cdot \vb{\sigma} , +\end{equation} +with $\vb{n}_{(t)}^2 = 1$. +This is a consequence of the peculiar properties of the \SO{4} matrices $U_{(t)}$ defined in \eqref{eq:Umatrices}. +Hence the corresponding $\SU{2}_L \times \SU{2}_R$ element $(U_{L}(\vb{n}_{(t)}),\, U_{R}(\vb{m}_{(t)}))$ reflects such characteristics. +In particular for the left part we have +\begin{equation} + U_{L}(\vb{n}_{(t)}) + = + i\, \vb{n}_{(t)} \cdot \vb{\sigma}, + \qquad + \vb{n}_{(t)}^2 = 1, + \label{eq:special_UL_brane_t} +\end{equation} +since $U_{(t)}^2 = \1_4$ implies that $U_{L}^2 = \pm \1_2$. +The right sector clearly follows the same discussion. + +In fact \cS in~\eqref{eq:reflection_S} can be represented as $U_{L} = U_{R} = i\, \sigma_1$. +Then any matrix $U_{L}(\vb{n}_{(t)})$ is of the form $U_{L}(\vb{n}_{(t)}) = i\, U(\vb{r}_{(t)}) \cdot \sigma_1 \cdot U^\dagger(\vb{r}_{(t)})$, for some $\vb{r}_{(t)}$ as follows from~\eqref{eq:Umatrices}. +Such matrix has vanishing trace and squares to $-\1_2$ hence the term proportional to two-dimensional unit matrix in the expression of the generic $\mathrm{SU}(2)$ element given in \Cref{sec:isomorphism} vanishes. +As a consequence $n_{(t)} = \frac{1}{4}$ such that \eqref{eq:special_UL_brane_t} follows. + % vim ft=tex diff --git a/thesis.tex b/thesis.tex index 79d1155..1420e08 100644 --- a/thesis.tex +++ b/thesis.tex @@ -55,6 +55,8 @@ \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}}} @@ -111,10 +113,16 @@ \input{sec/part3/introduction.tex} %---- APPENDIX +\cleardoubleplainpage{} \appendix +\section{The Isomorphism in Details} +\label{sec:isomorphism} +\input{sec/app/isomorphism.tex} + %---- BIBLIOGRAPHY \cleardoubleplainpage{} +\small \printbibliography[heading=bibintoc] \end{document}