% File: cft2_symm.tex
% Section: CFT2
% Title: Conformal symmetry
% Last modified: 19.10.2003
%
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\begin{center}
\large\textbf{CONFORMAL SYMMETRY}
\end{center}

\vspace{.0cm}

\begin{center}
\large\textbf{Conformal group: $d>2$ vs $d=2$}
\end{center}

\vspace{.1cm}

Consider a flat $d$-dimensional Euclidean space. Conformal transformations
are those which at most scale the metric tensor $\delta_{\alpha\beta}$
by an arbitrary factor $\sigma(x)$\,:
\begin{equation}
\label{tra}
\tilde{x}^\mu = x^\mu + \varepsilon^\mu(x) \ \ \ \ \
\Rightarrow \ \ \ \ h_{\mu\nu}(\tilde{x})
\equiv \frac{\partial x^\alpha}{\partial\tilde{x}^\mu}
 \frac{\partial x^\beta}{\partial\tilde{x}^\nu}\,\delta_{\alpha\beta}
= \sigma(x)\,\delta_{\mu\nu}\,.
\end{equation}
This implies
\begin{equation}
\label{cond}
\partial_\mu\varepsilon_\nu + \partial_\nu\varepsilon_\mu
 = (1-\sigma)\,\delta_{\mu\nu}
\end{equation}
or
\begin{equation}
\label{}
\partial_\mu\varepsilon_\nu = p\,\delta_{\mu\nu} + \theta_{\mu\nu}\,,
\ \ \ \ 2p(x) = 1-\sigma(x)\,, \ \ \ \
\theta_{\mu\nu}(x) = -\theta_{\nu\mu}(x) = \frac{1}{2}\,
(\partial_\mu\varepsilon_\nu - \partial_\nu\varepsilon_\mu)\,.
\end{equation}
As a direct consequence, we obtain
\begin{equation}
\label{dde}
\partial_\lambda\partial_\mu\varepsilon_\nu
= \partial_\lambda p\,\delta_{\mu\nu}
+ \partial_\lambda\theta_{\mu\nu}
= \partial_\mu p\,\delta_{\lambda\nu}
+ \partial_\mu\theta_{\lambda\nu}\,.
\end{equation}

First, we consider the case $d>2$ where at least 3 different
values are allowed for all indices. It follows from (\ref{dde})
that for $\mu,\nu,\lambda$ being unequal to each other
\begin{equation}
\label{}
\partial_\lambda\theta_{\mu\nu} = \partial_\mu\theta_{\lambda\nu}
\,, \ \ \ \ \ \ \partial_\lambda p = \partial_\mu\theta_{\lambda\mu}
\ \ \ \ \ \ \ \ \text{(no summation!)}
\end{equation}
The following calculations
\begin{gather}
\partial_1\theta_{23}=\partial_2\theta_{13}=-\partial_2\theta_{31}
 =-\partial_3\theta_{21}=\partial_3\theta_{12}=\partial_1\theta_{32}
 =-\partial_1\theta_{23}=0\,, \\
\partial_1\partial_2 p = \partial_1\partial_1\theta_{21}
 = \partial_1\partial_3\theta_{23} = 0\,, \\
\partial_1\partial_1 p = \partial_1\partial_2\theta_{12}
 = -\partial_2\partial_1\theta_{21} = - \partial_2\partial_2 p
 = \ldots = \partial_3\partial_3 p = \ldots
 = -\partial_1\partial_1 p = 0
\end{gather}
enables one to conclude that
\begin{equation}
\label{}
\partial_\mu\partial_\nu p = 0\,, \ \ \ \ \
\partial_\mu\partial_\nu\theta_{\alpha\beta} = 0\,,
\end{equation}
which means that $p\,(x)$ and $\theta_{\mu\nu}(x)$ are at most linear
in $x$, and $\varepsilon_\mu(x)$ at most quadratic.
Substituting an appropriate Ansatz into (\ref{cond}) yields the
well known general form of conformal transformation,
\begin{equation}
\label{d>2}
\varepsilon_\mu(x) = a_\mu + \omega_{\mu\nu}x_\nu + b\,x_\mu
 + c_\mu x^2 - 2(cx)\,x_\mu\,, \ \ \ \ \omega_{\mu\nu}=-\omega_{\nu\mu}\,,
\end{equation}
and so the dimensionality of conformal group proves to be
$(d+1)(d+2)/2$ for $d>2$\,.

For $d=2$, only two different values can be taken by indices, so we
cannot freely use the third independent value to repeat the derivation
given above. We remain with
\begin{equation}
\label{d=2}
\partial_1\varepsilon_1 = \partial_2\varepsilon_2 = p\,, \ \ \ \
\partial_1\varepsilon_2 = -\partial_2\varepsilon_1 = \theta_{12}\,,
\end{equation}
and no additional relations for $p$ and $\theta$\,.
But the first equalities in (\ref{d=2}) are exactly
the Cauchy-Riemann conditions of analyticity of
$\varepsilon_1 +i\varepsilon_2$. Therefore, conformal transformations
in 2-dimensional space can be parametrized by arbitrary analytic functions,
and the conformal group for $d=2$ becomes infinite dimensional.

\newpage

\begin{center}
\large\textbf{Conformally flat metric in complex coordinates}
\end{center}

\vspace{.1cm}

Consider a 2-dimensional space with `conformally Euclidean' metric
\begin{equation}
\label{}
h_{\mu\nu} = \rho(x)\,\delta_{\mu\nu}\,, \ \ \
h^{\mu\nu} = \rho^{-1}\,\delta_{\mu\nu}\,, \ \ \
h = \text{det}(h_{\mu\nu})\,, \ \ \
\sqrt h = \rho\,, \ \ \ A_\mu = \rho A^\mu\,,
\end{equation}
with $x_\mu$ and $\rho$ real. Introduce (formally independent) complex
coordinates $z,\,\bar{z}$\,:
\begin{align}
z &= x^1+ix^2 & x^1 &= \frac{1}{2}(z+\bar{z})
& \partial_z &= \frac{1}{2}(\partial_1-i\partial_2)
& \partial_1 = \frac{\partial}{\partial x^1}
 &= \partial_z+\partial_{\bar z} \\
\bar{z} &= x^1-ix^2 & x^2 &= \frac{i}{2}(\bar{z}-z)
& \partial_{\bar z} &= \frac{1}{2}(\partial_1+i\partial_2)
& \partial_2 = \frac{\partial}{\partial x^2}
 &= i(\partial_z-\partial_{\bar z})
\end{align}
These and similar formulas stem from
\begin{equation}
\label{}
\tilde{A}_{\nu'}^{\mu'}(\tilde{x})=
\frac{\partial\tilde{x}^{\mu'}}{\partial x^\mu}
\frac{\partial x^\nu}{\partial\tilde{x}^{\nu'}}A_{\nu}^{\mu}(x)\,, \ \ \
\frac{\partial(z,\bar{z})}{\partial(x^1,x^2)}
 = \left(\begin{array}{rr} 1 & i \\ 1 & -i \end{array}\right)\,, \ \ \
\frac{\partial(x^1,x^2)}{\partial(z,\bar{z})} = \frac{1}{2}
 \left(\begin{array}{rr} 1 & 1 \\ -i & i \end{array}\right) \ \ \
\end{equation}
and formal complex conjugation:
\begin{align}
A^z &= A^1+iA^2 & A^1 &= \frac{1}{2}\,(A^z+A^{\bar z})
& A_z &= \frac{1}{2}\,(A_1-iA_2)
& A_1 &= A_z+A_{\bar z} \\
A^{\bar z} &= A^1-iA^2 & A^2 &= \frac{i}{2}\,(A^{\bar z}-A^z)
& A_{\bar z} &= \frac{1}{2}\,(A_1+iA_2)
& A_2 &= i(A_z-A_{\bar z})
\end{align}
\begin{align}
A_{zz} &= \frac{1}{4}\,[A_{11}-A_{22}-i(A_{12}+A_{21})]
& A_{\bar z\bar z} &= A_{zz}^* \\
A_{z\bar z} &= \frac{1}{4}\,[A_{11}+A_{22}+i(A_{12}-A_{21})]
& A_{\bar zz} &= A_{z\bar z}^*
\end{align}
For a symmetric tensor $T_{\mu\nu}=T_{\nu\mu}$ we have
\begin{equation}
\label{}
T_{zz} = \frac{1}{4}\,(T_{11}-T_{22}-2iT_{12})\,, \ \ \
T_{z\bar z} = T_{\bar zz} = \frac{1}{4}\,(T_{11}+T_{22})
 = \frac{1}{4}\,\text{tr}T\,.
\end{equation}
From
\begin{equation}
\label{}
h_{zz}=h_{\bar z\bar z}=0\,, \ \ \
h_{z\bar z}=h_{\bar zz}=\rho/2\,, \ \ \
h^{zz}=h^{\bar z\bar z}=0\,, \ \ \ h^{z\bar z}=h^{\bar zz}=2\rho^{-1}
\end{equation}
one finds
\begin{gather}
A_z=\frac{\rho}{2}A^{\bar z}\,, \ \ \ A_{\bar z}=\frac{\rho}{2}A^z\,,
\ \ \ A^z=2\rho^{-1}A_{\bar z}\,, \ \ \ A^{\bar z}=2\rho^{-1}A_z\,, \\
A^\mu B_\mu = A^zB_z + A^{\bar z}B_{\bar z}\,, \ \ \ \ \
\partial_\mu A^\mu = \partial_z A^z + \partial_{\bar z}A^{\bar z}\,.
\end{gather}
At last
\begin{gather}
ds^2 = \rho\,[(dx^1)^2 + (dx^2)^2]
 = \rho\,dz\,d\bar z = \rho\,|dz|^2\,, \\
d^2x = dx^1\,dx^2 = \Bigl|\frac{i}{2}\Bigr|dz\,d\bar z
 = \frac{1}{2}\,dz\,d\bar z\,.
\end{gather}
An infinite dimensional conformal transformation takes the form
$(\varepsilon = \varepsilon_1 +i\varepsilon_2)$
\begin{equation}
\label{}
\,dz\,d\bar z \rightarrow (1+\partial_z\varepsilon)
(1+\partial_{\bar z}\bar\varepsilon)\,dz\,d\bar z\,, \ \ \ \ \
\partial_{\bar z}\varepsilon = \partial_z\bar\varepsilon = 0\,.
\end{equation}

\newpage

\begin{center}
\large\textbf{Stress tensor and conformal symmetry}
\end{center}

\vspace{.1cm}

Let $T_{\mu\nu}$ be a (symmetric) stress tensor of some field theory
in a flat 2-dimensional Euclidean space. Then
\begin{equation}
\label{}
\delta S = \int\!d^2\!x\,T_{\mu\nu}\,\partial_\mu\varepsilon_\nu(x)
 = \frac{1}{2}\int\!d^2\!x\,T_{\mu\nu}\,(\partial_\mu\varepsilon_\nu
 + \partial_\nu\varepsilon_\mu)
\end{equation}
and we see from (\ref{cond}) that $T_{\mu\mu}=0$ is an (off-shell) condition
of conformal symmetry. In conformal theories stress tensor is traceless.
In complex coordinates all this looks like
\begin{equation}
\label{vars}
T_{z\bar z} = T_{\bar zz} = 0\,, \ \ \ \
\delta S = \int\!dzd\bar z\,(T_{zz}\partial_{\bar z}\varepsilon^z
 + T_{\bar z\bar z}\partial_z\varepsilon^{\bar z})\,.
\end{equation}
Conservation of $T_{\mu\nu}$ (on-shell) implies
\begin{equation}
\label{cons}
\partial_{\bar z}\,T_{zz} = \partial_z\,T_{\bar z\bar z} = 0\,.
\end{equation}
From (\ref{cons}) one obtains
$\partial_{\bar z}(T_{zz}\varepsilon^z)=0$ for analytic
$\varepsilon^z = \varepsilon_1 + i\varepsilon_2$\,,
which corresponds to infinite number of conserved quantities
in 2-dimensional conformally invariant theory.

In what follows, we will consider only `holomorphic' components
$T_{zz}\doteq T(z)$ and $\varepsilon^z \doteq \varepsilon$\,.
Integrating (\ref{vars}) by parts and renormalizing
$T$ by the factor \,$-2\pi$\,, we obtain
\begin{equation}
\label{quant}
\delta_{\varepsilon}S=\frac{-1}{2\pi i}\!\oint\!dz\,T(z)\,\varepsilon\ .
\end{equation}
Now, to constitute a stress tensor as a quantum generator of conformal
symmetry, we
consider the $\varepsilon$-transformation as a change of variables in
a continual integral (with conformally invariant functional measure) of
the type
\begin{equation}
\label{}
\int\!\mathcal{D}\varphi\,e^{-S}\mathcal{O}_1\ldots\mathcal{O}_n \
\equiv \ <\mathcal{O}_1\ldots\mathcal{O}_n>\,,
\end{equation}
observe that $\delta_{\varepsilon}S$ should compensate for explicit
variations of the operators $\mathcal{O}_k$\,, and,
relying on (\ref{quant}), postulate
\begin{equation}
\label{ward}
\frac{1}{2\pi i}\!\oint\!dz\,\varepsilon(z)
<T(z)\,\mathcal{O}_1(z_1)\ldots\mathcal{O}_n(z_n)> \
= -\sum_{k=1}^{n}<\ldots\delta_\varepsilon\mathcal{O}_k\ldots>
\end{equation}
to be a quantum conformal action principle. Here $\varepsilon(z)$
is assumed to be analytic inside the contour, while $T(z)$ may exhibit
singular (pole) behaviour near the locations (surrounded by the same
contour) of other field operators.
With $\mathcal{O}_k$ transforming as chiral primary fields of conformal
weight $\Delta_k$\,,
\begin{equation}
\label{}
\delta_{\varepsilon}\mathcal{O}_k(z)
 = -(\Delta_k\varepsilon'+\varepsilon\partial)\mathcal{O}_k(z)\,,
\end{equation}
we arrive at the main formula of CFT${}_2$\,:
\begin{equation}
\label{main}
<T(z)\,\mathcal{O}_1(z_1)\ldots\mathcal{O}_n(z_n)> \
= \sum_{k=1}^{n}\left(\frac{\Delta_k}{(z-z_k)^2}
 +\frac{1}{z-z_k}\,\frac{\partial}{\partial z_k}\right)
<\ldots\mathcal{O}_k(z_k)\ldots>\,.
\end{equation}
The Ward identity (\ref{ward}) is recovered from here by the contour
integration with analytic function $\varepsilon(z)$\,. Both (\ref{ward})
and (\ref{main}) are equivalent to (and may be viewed as substantiation
of) the OPE relations. This is shown, for example, by transforming the
contour in the l.h.s. of (\ref{ward}) into several small contours
around each $z_k$\,.

\end{document}