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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
parent 28c749e568
commit c81838154a
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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}