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Up to NS fermions

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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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Riccardo
2020-09-25 17:28:21 +02:00
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116
sec/part1/introduction.tex
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@@ -19,7 +19,7 @@ In particular we recall some results on the symmetries of string theory and how
\subsection{Properties of String Theory and Conformal Symmetry}
Strings are extended one-dimensional objects.
They are curves in spacetime parametrized by a coordinate $\sigma \in \left[0, \ell \right]$.
They are curves in spacetime parametrized by a coordinate $\sigma \in \qty[0, \ell ]$.
When propagating they span a two-dimensional surface, the \emph{worldsheet}, described by the position of the string at given values of $\sigma$ at a time $\tau$, i.e.\ $X^{\mu}(\tau, \sigma)$ where $\mu = 0, 1, \dots, D - 1$ indexes the coordinates.
Such surface can have different topologies according to the nature of the object propagating in spacetime: strings can be \emph{closed} if $X^{\mu}(\tau, 0) = X^{\mu}(\tau, \ell)$ or \emph{open} if the endpoints in $\sigma = 0$ and $\sigma = \ell$ do not coincide.
@@ -46,11 +46,11 @@ The \eom for the string $X^{\mu}(\tau, \sigma)$ is therefore
\begin{equation}
\frac{1}{\sqrt{- \det \gamma}}\,
\ipd{\alpha}
\left(
\qty(
\sqrt{- \det \gamma}\,
\gamma^{\alpha\beta}\,
\ipd{\beta} X^{\mu}
\right)
)
=
0,
\qquad
@@ -66,14 +66,14 @@ In fact
=
- \frac{1}{4 \pi \ap}
\sqrt{- \det \gamma}\,
\left(
\qty(
\ipd{\alpha} X \cdot \ipd{\beta} X
-
\frac{1}{2}
\gamma_{\alpha\beta}\,
\gamma^{\lambda\rho}\,
\ipd{\lambda} X \cdot \ipd{\rho} X
\right)
)
=
0
\label{eq:conf:worldsheetmetric}
@@ -152,13 +152,13 @@ In fact the classical constraint on the tensor is simply
\fdv{S_P[\gamma, X]}{\gamma^{\alpha\beta}}
=
-\frac{1}{\ap}
\left(
\qty(
\ipd{\alpha} X \cdot \ipd{\beta} X
-
\frac{1}{2} \eta_{\alpha\beta}\,
\eta^{\lambda\rho}\,
\ipd{\lambda} X \cdot \ipd{\rho} X
\right)
)
=
0.
\label{eq:conf:stringT}
@@ -188,33 +188,33 @@ 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 \bT_{\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 ),
T_{\xi\xi}( \xi,\, \bxi ) = T_{\xi\xi}( \xi ) = T( \xi ),
\qquad
\bT_{\bxi\bxi}( \xi, \bxi ) = \bT_{\bxi\bxi}( \bxi ) = \bT( \bxi ),
\bT_{\bxi\bxi}( \xi,\, \bxi ) = \bT_{\bxi\bxi}( \bxi ) = \bT( \bxi ),
\end{equation}
which are respectively the holomorphic and the anti-holomorphic components of the 2-dimensional stress energy tensor.
which are respectively the holomorphic and the anti-holomorphic components of the bidimensional stress energy tensor.
The previous properties define what is known as a 2-dimensional \emph{conformal field theory} (\cft).
The previous properties define what is known as a bidimensional \emph{conformal field theory} (\cft).
Ordinary tensor fields
\begin{equation}
\phi_{\omega, \bomega}( \xi, \bxi )
\phi_{\omega, \bomega}( \xi,\, \bxi )
=
\phi_{\underbrace{\xi \dots \xi}\limits_{\omega~\text{times}}\, \underbrace{\bxi \dots \bxi}\limits_{\bomega~\text{times}}}( \xi, \bxi )
\left( \dd{\xi} \right)^{\omega}
\left( \dd{\bxi} \right)^{\bomega}
\phi_{\underbrace{\xi \dots \xi}\limits_{\omega~\text{times}}\, \underbrace{\bxi \dots \bxi}\limits_{\bomega~\text{times}}}( \xi,\, \bxi )
\qty( \dd{\xi} )^{\omega}
\qty( \dd{\bxi} )^{\bomega}
\end{equation}
can be classified according to their weight $\left( \omega, \bomega \right)$ referring to the holomorphic and anti-holomorphic parts respectively.
are classified according to their weight $\qty( \omega,\, \bomega )$ referring to the holomorphic and anti-holomorphic parts respectively.
In fact a transformation $\xi \mapsto \chi(\xi)$ and $\bxi \mapsto \bchi(\bxi)$ maps the conformal fields to
\begin{equation}
\phi_{\omega, \bomega}( \chi, \bchi )
=
\left( \dv{\chi}{\xi} \right)^{\omega}\,
\left( \dv{\bchi}{\bxi} \right)^{\bomega}\,
\phi_{\omega, \bomega}( \xi, \bxi ).
\qty( \dv{\chi}{\xi} )^{\omega}\,
\qty( \dv{\bchi}{\bxi} )^{\bomega}\,
\phi_{\omega, \bomega}( \xi,\, \bxi ).
\end{equation}
\begin{figure}[tbp]
@@ -238,9 +238,9 @@ In fact a transformation $\xi \mapsto \chi(\xi)$ and $\bxi \mapsto \bchi(\bxi)$
An additional conformal transformation
\begin{equation}
z = e^{\xi} = e^{\tau_e + i \sigma} \in \left\lbrace z \in \C | \Im z \ge 0 \right\rbrace,
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 \left\lbrace z \in \C | \Im z \le 0 \right\rbrace
\bz = 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}).
@@ -265,14 +265,14 @@ The transformation on a field $\phi_{\omega, \bomega}$ of weight $(\omega, \bome
\liebraket{Q_{\epsilon, \bepsilon}}{\phi_{\omega, \bomega}( w, \bw )}
\\
& =
\cint{0} \ddz \epsilon(z) \left[ T(z), \phi_{\omega, \bomega}( w, \bw ) \right]
\cint{0} \ddz \epsilon(z) \qty[ T(z), \phi_{\omega, \bomega}( w, \bw ) ]
+
\cint{0} \ddbz \bepsilon(\bz) \left[ \bT(\bz), \phi_{\omega, \bomega}( w, \bw ) \right]
\cint{0} \ddbz \bepsilon(\bz) \qty[ \bT(\bz), \phi_{\omega, \bomega}( w, \bw ) ]
\\
& =
\cint{w} \ddz \epsilon(z)\, \rR\!\left( T(z)\, \phi_{\omega, \bomega}( w, \bw ) \right)
\cint{w} \ddz \epsilon(z)\, \rR\!\qty( T(z)\, \phi_{\omega, \bomega}( w, \bw ) )
+
\cint{\bw} \ddbz \bepsilon(\bz)\, \rR\!\left( \bT(\bz)\, \phi_{\omega, \bomega}( w, \bw ) \right),
\cint{\bw} \ddbz \bepsilon(\bz)\, \rR\!\qty( \bT(\bz)\, \phi_{\omega, \bomega}( w, \bw ) ),
\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 $w$.
@@ -393,7 +393,7 @@ This ultimately leads to the quantum algebra
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{}
\footnotetext{%
Notice that the subset of Virasoro operators $\left\lbrace L_{-1}, L_0, L_1 \right\rbrace$ forms a closed subalgebra generating the group $\SL{2}{\R}$.
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.
@@ -592,13 +592,13 @@ In complex coordinates on the plane it is:
=
- \frac{1}{4 \pi}
\iint \dd{z} \dd{\bz}
\left(
\qty(
\frac{2}{\ap}\, \ipd{\bz} X^{\mu}\, \ipd{z} X^{\nu}
+
\psi^{\mu}\, \ipd{\bz} \psi^{\nu}
+
\bpsi^{\mu}\, \ipd{z} \bpsi^{\nu}
\right)
)
\eta_{\mu\nu}.
\label{eq:super:action}
\end{equation}
@@ -641,7 +641,7 @@ The action~\eqref{eq:super:action} is also invariant under the \emph{supersymmet
- \bepsilon( \bz )\, \ipd{\bz} \bX^{\mu}( \bz )
\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 $\bepsilon( \bz ) = \left( \epsilon( z ) \right)^*$ are anti-commuting fermions and
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
\begin{equation}
\begin{split}
T_F( z )
@@ -681,7 +681,7 @@ The central charge associated to the Virasoro algebra is in this case given by b
The central charge is therefore $c = \frac{3}{2} D$ for the \cft defined in~\eqref{eq:super:action}.
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 $\left( \frac{3}{2}, 0 \right)$ and $\left( -\frac{1}{2}, 0 \right)$.
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:
\begin{equation}
@@ -738,14 +738,14 @@ The tensor $J$ is called \emph{complex structure} if there exist a tensor $N$ su
\begin{equation}
\tensor{N}{^a_{bc}}\, v_p^b\, w_p^c
=
\left(
\qty(
\liebraket{v_p}{w_p}
+
J
\left( \liebraket{J\, v_p}{w_p} + \liebraket{v_p}{J\, w_p} \right)
\qty( \liebraket{J\, v_p}{w_p} + \liebraket{v_p}{J\, w_p} )
-
\liebraket{J\, v_p}{J\, w_p}
\right)^a
)^a
=
0
\end{equation}
@@ -807,7 +807,7 @@ $(M, J, g)$ is a \emph{Kähler} manifold if:
\begin{equation}
\dd{\omega}
=
\left( \pd + \bpd \right)
\qty( \pd + \bpd )
\omega(z, \bz)
=
0,
@@ -833,7 +833,7 @@ In local coordinates a Hermitian metric is such that
thus the Kähler form becomes $\omega = i g_{a\overline{b}}\, \dd{z}^a \wedge \dd{\bz}^{\overline{b}}$.
The relation~\eqref{eq:cy:kaehler} then translates into:
\begin{equation}
\dd{\omega} = i\, \left( \pd + \bpd \right)\, g_{a\overline{b}}\, \dd{z}^a \wedge \dd{\bz}^{\overline{b}}
\dd{\omega} = i\, \qty( \pd + \bpd )\, g_{a\overline{b}}\, \dd{z}^a \wedge \dd{\bz}^{\overline{b}}
=
0
\quad
@@ -995,20 +995,20 @@ The usual mode expansion in conformal coordinates $X^{\mu}( z, \bz ) = X( z ) +
x_0^{\mu}
+
i\, \sqrt{\frac{\ap}{2}}\,
\left(
\qty(
- \alpha_0^{\mu}\, \ln{z}
+ \sum\limits_{n \in \Z \setminus \{0\}} \frac{\alpha_n^{\mu}}{n} z^{-n}
\right),
),
\\
\bX^{\mu}( \bz )
& =
\overline{x}_0^{\mu}
+
i\, \sqrt{\frac{\ap}{2}}\,
\left(
\qty(
- \balpha_0^{\mu}\, \ln{\bz}
+ \sum\limits_{n \in \Z \setminus \{0\}} \frac{\balpha_n^{\mu}}{n} \bz^{-n}
\right),
),
\end{split}
\label{eq:tduality:modes}
\end{equation}
@@ -1045,9 +1045,9 @@ respectively encoding the quantisation of the momentum for a compact coordinate
We finally have
\begin{equation}
\begin{split}
\alpha_0^{D-1} &= \frac{1}{\sqrt{2 \ap}}\, \left( n\, \frac{\ap}{R} + m\, R \right),
\alpha_0^{D-1} &= \frac{1}{\sqrt{2 \ap}}\, \qty( n\, \frac{\ap}{R} + m\, R ),
\\
\balpha_0^{D-1} &= \frac{1}{\sqrt{2 \ap}}\, \left( n\, \frac{\ap}{R} - m\, R \right),
\balpha_0^{D-1} &= \frac{1}{\sqrt{2 \ap}}\, \qty( n\, \frac{\ap}{R} - m\, R ),
\end{split}
\end{equation}
@@ -1058,24 +1058,24 @@ From~\eqref{eq:conf:Texpansion} and \eqref{eq:conf:bosonicstringT} we find
L_0
&=
\frac{\ap}{2}\,
\left(
\left( \alpha_0^{D-1} \right)^2
\qty(
\qty( \alpha_0^{D-1} )^2
+
\sum\limits_{i = 0}^{D-2}\, \left( \alpha_0^i \right)^2
\sum\limits_{i = 0}^{D-2}\, \qty( \alpha_0^i )^2
+
\sum\limits_{n = 1}^{+\infty}\, \left( 2 \alpha_{-n}^{\mu} \alpha_n^{\nu}\, \eta_{\mu\nu} + a \right)
\right),
\sum\limits_{n = 1}^{+\infty}\, \qty( 2 \alpha_{-n}^{\mu} \alpha_n^{\nu}\, \eta_{\mu\nu} + a )
),
\\
\bL_0
&=
\frac{\ap}{2}\,
\left(
\left( \balpha_0^{D-1} \right)^2
\qty(
\qty( \balpha_0^{D-1} )^2
+
\sum\limits_{i = 0}^{D-2}\, \left( \balpha_0^i \right)^2
\sum\limits_{i = 0}^{D-2}\, \qty( \balpha_0^i )^2
+
\sum\limits_{n = 1}^{+\infty}\, \left( 2 \balpha_{-n}^{\mu} \balpha_n^{\nu}\, \eta_{\mu\nu} + a \right)
\right),
\sum\limits_{n = 1}^{+\infty}\, \qty( 2 \balpha_{-n}^{\mu} \balpha_n^{\nu}\, \eta_{\mu\nu} + a )
),
\end{split}
\end{equation}
where $a$ is constant given by normal ordering, representing the zero point energy of the theory.
@@ -1084,14 +1084,14 @@ Imposing physical conditions~\eqref{eq:conf:physical} and the \emph{level matchi
\begin{split}
M^2
& =
\frac{1}{\ap^2}\, \left( n\, \frac{\ap}{R} + m\, R \right)^2
\frac{1}{\ap^2}\, \qty( n\, \frac{\ap}{R} + m\, R )^2
+
\frac{4}{\ap}\, \left( \rN + a \right)
\frac{4}{\ap}\, \qty( \rN + a )
\\
& =
\frac{1}{\ap^2}\, \left( n\, \frac{\ap}{R} - m\, R \right)^2
\frac{1}{\ap^2}\, \qty( n\, \frac{\ap}{R} - m\, R )^2
+
\frac{4}{\ap}\, \left( \overline{\rN} + a \right),
\frac{4}{\ap}\, \qty( \overline{\rN} + a ),
\end{split}
\label{eq:dbranes:closedspectrum}
\end{equation}
@@ -1129,7 +1129,7 @@ The usual mode expansion~\eqref{eq:tduality:modes} here leads to
+
i\, \sqrt{\frac{\ap}{2}}\,
\sum\limits_{n \in \Z \setminus \{0\}}
\frac{\alpha_n^{\mu}}{n} \left( z^{-n} + \bz^{-n} \right)
\frac{\alpha_n^{\mu}}{n} \qty( z^{-n} + \bz^{-n} )
\end{equation}
and $\ell = \pi$.
@@ -1217,7 +1217,7 @@ The mass shell condition for open strings then becomes:\footnotemark{}
The constant $a$ in~\eqref{eq:dbranes:closedspectrum} takes here the value $-1$ from the imposition of the canonical commutation relations and a $\zeta$-regularisation.
}
\begin{equation}
M^2 = \frac{1}{\ap} \left( N - 1 \right).
M^2 = \frac{1}{\ap} \qty( N - 1 ).
\end{equation}
Consider for a moment bosonic string theory and define the usual vacuum as