riccardo/phd-thesis
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

Kaehler manifolds

Browse Source 
Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
This commit is contained in:
Riccardo
2020-09-02 22:19:52 +02:00
parent 1cca2724dd
commit 7eac6f4191
5 changed files with 197 additions and 71 deletions
Download Patch File Download Diff File Expand all files Collapse all files

125
sec/part1/introduction.tex
View File

@@ -484,11 +484,11 @@ Using the normalisation of the 2-points function $\left\langle X^{\mu}( z, \bz )
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 $\overline{c}(z)$.
The non vanishing components of their stress-energy tensor can be computed as:\footnotemark{}
\footnotetext{%
In general ghosts $b( z )$ and $c( z )$ with weight $(\lambda, 0)$ and $(1 - \lambda, 0)$ can be introduced as a standalone \cft \cite{Friedan:1986:ConformalInvarianceSupersymmetry,Polchinski:1998:StringTheorySuperstring} with action
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{\bz} b( z )\, \ipd{\bz} c( z ).
\end{equation*}
whose equations of motion are $\ipd{\bz} c( z ) = \ipd{\bz} b( z ) = 0$.
The equations of motion are $\ipd{\bz} c( z ) = \ipd{\bz} b( z ) = 0$.
The \ope is
\begin{equation*}
b( z )\, c( z ) = \frac{\varepsilon}{z - w} + \order{1},
@@ -509,7 +509,8 @@ The non vanishing components of their stress-energy tensor can be computed as:\f
T_{\text{ghost}}( z )\, j( w ) = \frac{Q}{( z - w )^3} + \order{(z - w)^{-2}}.
\end{equation*}
This is the case of the worldsheet fermions in~\eqref{eq:super:action} for which $\lambda = \frac{1}{2}$.
For instance the reparametrisation ghosts with $\lambda = 2$ have $Q = -3$.
For instance the reparametrisation ghosts with $\lambda = 2$ have $Q = -3$, while the superghosts with $\lambda = \frac{3}{2}$ present $Q = 2$.
\label{note:conf:ghosts}
}
\begin{equation}
\begin{split}
@@ -615,21 +616,21 @@ The action~\eqref{eq:super:action} is also invariant under the \emph{supersymmet
\begin{equation}
\begin{split}
\sqrt{\frac{2}{\ap}}\,
\delta_{\epsilon, \bepsilon}\,
\delta_{\epsilon, \bepsilon}
X^{\mu}( z, \bz )
& =
\epsilon( z ) \psi^{\mu}( z ) + \bepsilon( \bz ) \bpsi^{\mu}( \bz ),
\epsilon( z )\, \psi^{\mu}( z ) + \bepsilon( \bz )\, \bpsi^{\mu}( \bz ),
\\
\sqrt{\frac{2}{\ap}} \delta_{\epsilon} \psi^{\mu}( z )
& =
- \epsilon( z ) \ipd{z} X^{\mu}( z ),
- \epsilon( z )\, \ipd{z} X^{\mu}( z ),
\\
\sqrt{\frac{2}{\ap}} \delta_{\bepsilon} \bpsi^{\mu}( \bz )
& =
- \bepsilon( \bz ) \ipd{\bz} \bX^{\mu}( \bz )
- \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 $\bepsilon( \bz ) = \left( \epsilon( z ) \right)^*$ are anti-commuting fermions and
\begin{equation}
\begin{split}
T_F( z )
@@ -670,7 +671,7 @@ 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 $\left( \frac{3}{2}, 0 \right)$ and $\left( -\frac{1}{2}, 0 \right)$.
Their central charge becomes $c_{\text{ghost}} = c_{bc} + c_{\beta\gamma} = -26 + 11 = -15$.
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}
\eval{T_{\text{full}}( z )}_{\order{(z - w)^{-4}}}
@@ -686,6 +687,7 @@ When considering the full theory $T_{\text{full}} = T + T_{\text{ghost}}$ and $\
\Leftrightarrow
\quad
D = 10.
\label{eq:super:dimensions}
\end{equation}
@@ -700,6 +702,109 @@ In this section we briefly review for completeness the necessary tools to be abl
These results represent the background knowledge necessary to better understand more complicated scenarios involving strings.
As we will never deal directly with $4$-dimensional physics this is not a complete review and we refer to \cite{Anderson:2018:TASILecturesGeometric,Blumenhagen:2007:FourdimensionalStringCompactifications,Blumenhagen:2013:BasicConceptsString,Grana:2005:FluxCompactificationsString,Grana:2017:StringTheoryCompactifications,Krippendorf:2010:CambridgeLecturesSupersymmetry,Uranga:2005:TASILecturesString} for more in-depth explanations.
In general we consider Minkowski space in $10$ dimensions $\ccM^{1,9}$.
To recover $4$-dimensional spacetime we let it be defined as a product
\begin{equation}
\ccM^{1,9}
=
\ccM^{1,3} \otimes \ccX_6,
\end{equation}
where $\ccX_6$ is a generic $6$-dimensional manifold at this stage.
This \emph{internal} manifold $\ccX_6$ is however subject to very stringent restrictions due to mathemtical consistency conditions and physical requests.
In particular $\ccX_6$ should be a compact manifold to ``hide'' the 6 extra-dimensions computed in~\eqref{eq:super:dimensions}.
Moreover the geometry of $\ccM^{1,3}$ should be a maximally symmetric space and there should be a $N = 1$ unbroken supersymmetry in $4$ dimensions.
Finally the gauge group of and the spectrum of fermions should be realistic (e.g.\ it should be possible to define chiral fermion states) \cite{Candelas:1985:VacuumConfigurationsSuperstrings}.
These manifolds were first conjectured to exist by Eugenio Calabi~\cite{Calabi:1957:KahlerManifoldsVanishing}.
Their existence was later proved by Shing-Tung Yau~\cite{Yau:1977:CalabiConjectureNew}.
They are defined as complex Ricci-flat Kähler manifolds $M$ of dimensions $2m$ and with holonomy \SU{3} (see for instance \cite{Joyce:2000:CompactManifoldsSpecial,Joyce:2002:LecturesCalabiYauSpecial}).
\subsubsection{Complex and Kähler Manifolds}
In general an \emph{almost complex structure} $J$ is a tensor such that $\tensor{J}{^a_b}\, \tensor{J}{^b_c} = - \tensor{\delta}{^a_c}$.
For any vector field $v_p \in \rT_p M$ defined in $p \in M$ we then define $(J v)^a = \tensor{J}{^a_b} v^b$, thus the tangent space $\rT_p M$ has the structure of a \emph{complex vector space}.
The tensor $J$ is called \emph{complex structure} if there exist a tensor $N$ such that
\begin{equation}
\tensor{N}{^a_{bc}} v_p^b w_p^c
=
\left(
[ v_p, w_p ]
+
J
\left( [ J\, v_p, w_p ] + [ v_p, J\, w_p ] \right)
-
[ J\, v_p, J\, w_p ]
\right)^a
=
0
\end{equation}
for any $v_p,\, w_p \in \rT_p M$, where $[ \cdot, \cdot ]$ is the Lie braket of vector fields.
A manifold $M$ is a \emph{complex} manifold if it is possible to define a complex structure $J$ on it.\footnotemark{}
\footnotetext{%
Notice that a smooth function $f\colon\, M \to \C$ whose pushforward of $v_p \in \rT_p M$ is $f_{*\, p}\colon\, \rT_p M \to \rT_{f(p)} \C$ is called holomorphic if $(J\, f_{*\, p}( v_p ))^a = i ( f_{*\, p}( v_p ) )^a$ as such expression encodes the Cauchy-Riemann equations.
Let in fact $f( x, y ) = f_1( x, y ) + i\, f_2( x, y )$, then the expression implies
\begin{equation*}
\begin{cases}
\ipd{x} f_1( x, y )
& =
\ipd{y} f_2( x, y )
\\
\ipd{x} f_2( x, y )
& =
-\ipd{y} f_1( x, y )
\end{cases}
\Rightarrow
\ipd{x} f( x, y ) = -i \ipd{y} f( x, y )
\Rightarrow
\ipd{\bz} f( z, \bz ) = 0
\Rightarrow
f( z, \bz ) = f( z ).
\end{equation*}
}
Let then $(M, J, g)$ be a complex manifold with a Riemannian metric $g$.
The metric is \emph{Hermitian} if
\begin{equation}
g( v_p, w_p ) = g( J\, v_p, J\, w_p )
\quad
\forall v_p,\, w_p \in \rT_p M
\quad
\Leftrightarrow
\quad
\tensor{g}{_{ab}}
=
\tensor{J}{_a^c}\,
\tensor{J}{_b^d}\,
\tensor{g}{_{cd}}.
\end{equation}
In this case we can define a $(1, 1)$-form $\omega$ as
\begin{equation}
\omega( v_p, w_p )
=
g( J\, v_p, w_p )
\quad
\forall v_p,\, w_p \in \rT_p M.
\quad
\Leftrightarrow
\quad
\tensor{\omega}{_{ab}}
=
\tensor{J}{_a^c}\,
\tensor{g}{_{cb}}.
\end{equation}
$(M, J, g)$ is a \emph{Kähler} manifold if ~\cite{Joyce:2002:LecturesCalabiYauSpecial}:
\begin{equation}
\dd{\omega}
=
\left( \ipd{z} + \ipd{\bz} \right)
\omega(z, \bz)
=
0,
\end{equation}
or equivalently $\nabla J = 0$ or $\nabla \omega = 0$, where $\nabla$ is the connection of $g$.
The covariant conservation of $J$ and $\omega$ implies that the holonomy group must preserve these objects in $\R^{2m}$.
Thus we have $\mathrm{Hol}(g) \subseteq \U{m} \subset \mathrm{O}(2m)$.
\subsection{D-branes and Open Strings}