Conformal field theory 1

Introduction

Conformal symmetry is a crucial ingredient in:

-Statistical mechanics (second order phase transitions, critical exponents)

-Quantum field theory (fixed points of the renormalization group equation).

-String theory (Polyakov path integral, AdS/CFT correspondence). 

A conformal field theory is a quantum field theory with conformal symmetry, i.e. there is no preferred length scale.

 

 

 

Conformal field theory

Conformal transformations are:

\begin{eqnarray}x^{\alpha}\longrightarrow \tilde{x}^{\alpha}(x)~\Rightarrow ~g_{\alpha\beta}(x)\longrightarrow\tilde{g}_{\alpha\beta}(\tilde{x})=\Omega^2(x)g_{\alpha\beta}(x).\end{eqnarray}

Here there are two possible interpretations:

-The metric is dynamical. In this case conformal transformations are diffeomorphisms, i.e. local gauge transformations which can be undone by Weyl transformations.

-The metric is fixed. In this case conformal transformations are global transformations which physically change the point $x$ to the point $\tilde{x}$. 

In here we are interested in the second interpretation.

A typical example is two-dimensional massless scalar field $\phi$. The Hamilton action is:

\begin{eqnarray}S=\frac{1}{4\pi\alpha^{'}}\int d^2x \sqrt{g}\partial_{\mu}\phi\partial^{\mu}\phi.\end{eqnarray}

The coordinates of the flat and Euclidean spacetime are:

\begin{eqnarray}z=x^1+ix^2~,~\bar{z}=x^¹-ix^2.\end{eqnarray}

We have then a complex plane. The holomorphic derivatives:

\begin{eqnarray}\partial_z=\partial=\frac{1}{2}(\partial_1-i\partial_2)~,~\partial_{\bar{z}}=\bar{\partial}=\frac{1}{2}(\partial_1+i\partial_2).\end{eqnarray}

Conformal transformations in a flat background (the metric is fixed)

\begin{eqnarray}x^{\mu}\longrightarrow \tilde{x}^{\mu}(x)=x^{\mu}+\epsilon^{\mu}(x) \label{conf}\end{eqnarray}

become in the complex plane the conformal mappings:

\begin{eqnarray}z\longrightarrow {z}^{'}=f(z)~,~\bar{z}\longrightarrow \bar{z}^{'}=\bar{f}(\bar{z}).\end{eqnarray}

The most important conformal transformations are:

\begin{eqnarray}&&z\longrightarrow z^{'}=z+a~,~{\rm Translations}\nonumber\\&&z\longrightarrow z^{'}=\zeta z~,~|\zeta|=1~,~{\rm Rotations}\nonumber\\&&z\longrightarrow z^{'}=\zeta z~,~|\zeta|\neq 1~,~{\rm Dilatations}.\end{eqnarray} 

The stress-energy-momentum tensor

Infinitesimal diffeomorphisms (used in the computation of the stress-energy-tensor):

\begin{eqnarray}x^{\mu}\longrightarrow x^{\mu}+\epsilon^{\mu}~,~g_{\mu\nu}\longrightarrow g_{\mu\nu}+\partial_{\mu}\epsilon_{\nu}+\partial_{\nu}\epsilon_{\mu}.\end{eqnarray}

The variation of the action under the conformal transformations  (\ref{conf}):

\begin{eqnarray}\delta S=\frac{1}{2\pi}\int d^2x T^{\alpha\beta}\partial_{\alpha}\epsilon_{\beta}.\end{eqnarray}

The stress-energy-momentum tensor:

\begin{eqnarray}T_{\alpha\beta}&=&-\frac{4\pi}{\sqrt{g}}\bigg(\frac{\delta S}{\delta g^{\alpha\beta}}\bigg)_{g=\delta}\nonumber\\&=&-\frac{1}{\alpha^{'}}(\partial_{\alpha}\phi\partial_{\beta}\phi-\frac{1}{2}\delta_{\alpha\beta}(\partial\phi)^2).\end{eqnarray}

The stress-energy-momentum tensor is conserved (Noether's theorem):

\begin{eqnarray}\partial^{\alpha}T_{\alpha\beta}=0.\end{eqnarray}

The stress-energy-tensor is traceless (which is due to the invariance under scale transformations):

\begin{eqnarray}g_{\mu\nu}\longrightarrow \epsilon g_{\mu\nu}~,~S\longrightarrow S+\delta S~:~\delta S=0\Rightarrow T_{\alpha}^{\alpha}=0.\end{eqnarray}

In the complex plane:

\begin{eqnarray}&&T(z)=T_{zz}(z)=-\frac{1}{\alpha^{'}}\partial \phi\partial\phi~,~\bar{\partial}T=0\nonumber\\&&\bar{T}(\bar{z})=T_{\bar{z}\bar{z}}(\bar{z})=-\frac{1}{\alpha^{'}}\bar{\partial} \phi\bar{\partial}\phi~,~{\partial}\bar{T}=0\nonumber\\&&T_{z\bar{z}}=0.\end{eqnarray}

 

Noether's currents (For generic conformal transformations $z\longrightarrow z+\epsilon(z)$, $\bar{z}\longrightarrow \bar{z}+\bar{\epsilon}(\bar{z})$):

 \begin{eqnarray}&&2J_z={J}^{\bar{z}}={T}({z}){\epsilon}({z})~,~\bar{\partial} {J}^{\bar{z}}=0\nonumber\\&&2\bar{J}_{\bar{z}}=\bar{J}^z=\bar{T}(\bar{z})\bar{\epsilon}(\bar{z})~,~\partial \bar{J}^z=0.\end{eqnarray}

 

Ward Identities: The operator product expansion (OPE)

After quantization the conservation of Noether's  currents $J^{\mu}$ is replaced as:

\begin{eqnarray}&&\partial_{\mu}J^{\mu}=0~,~{\rm Classical}\nonumber\\&& \Rightarrow-\frac{1}{2\pi}\int_{\epsilon\neq 0}\sqrt{g}d^2x \partial_{\mu}\langle 0| T\bigg( J^{\mu}(x)O_1(x_1)...O_N(x_N)\bigg)|0\rangle=\langle 0|T\big(\delta O_1(x_1)...O_N(x_N)\big)|0\rangle~,~x\longrightarrow x_1~,~{\rm Quantum}.\end{eqnarray} 

These are Ward identities.

In deriving this equation we have assumed that the support of the conformal transformation $\epsilon(x)$ includes only the operator insertion $O_1(x_1)$. 

The variation of the field $\phi$ and the operator insertions $O_i(x)$ under conformal transformations are  defined by $\phi\longrightarrow \phi+\epsilon \delta \phi$ and $O_i(x)\longrightarrow O_i(x)+\epsilon\delta O_i(x)$ respectively.

The above result is true for any $\epsilon(x)$. Thus we have actually the operator identity 

\begin{eqnarray}-\frac{1}{2\pi}\int_{\epsilon\neq 0}\sqrt{g}d^2x \partial_{\mu} \bigg(J^{\mu}(x)O_1(x_1)...O_N(x_N)\bigg)=\delta O_1(x_1)...O_N(x_N).\end{eqnarray}  

However, the expectation values of time-ordered products of operators computed using the Feynman path integral are implicitly understood if and when needed (for example the operator insertions all compute).

In two dimensions we can use Stokes' theorem (we can reduce to a contour integral) and for conformal transformations the Noether's currents are holomorphic or anti-holomorphic functions (we can use the residue theorem). We get then the result (for $z\longrightarrow w$)

\begin{eqnarray}\frac{i}{2\pi}\oint_{\partial\epsilon} dz J_z(z)O_1(w,\bar{w})=-{\rm Res}(J_zO_1)\Rightarrow \delta O_1(w,\bar{w})=-\frac{1}{2}{\rm Res}(\epsilon(z)T(z)O_1(w,\bar{w})).\end{eqnarray}

We have then the behavior 

\begin{eqnarray}J_z(z)O_1(w,\bar{w})=...+\frac{{\rm Res}(J_z(z)O_1(w,\bar{w}))}{z-w}+...\end{eqnarray}

This is an example of an operator product expansion (OPE). 

We conclude that if we know the OPE of an operator with the stress-energy-momentum tensor we can determine the transformation law of this operator under conformal transformations. 

References:

David Tong, String Theory.