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Progress on fermions with defects

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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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Riccardo
2020-09-29 18:48:42 +02:00
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1008
sciencestuff.sty
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6
sec/app/parameters.tex
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@@ -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}

95
sec/app/reflection.tex Normal file
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@@ -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

204
sec/part1/dbranes.tex
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@@ -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 $f^i_{(t)} = Z^i_L(x_{(t)} + i\, 0^+)$.

1378
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248
sec/part1/introduction.tex
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@@ -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}

8
thesis.cls
View File

@@ -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

66
thesis.tex
View File

@@ -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{}