## Primary operators

Noether's theorem is replaced quantum mechanically by the Ward identities:

\begin{eqnarray}\frac{i}{2\pi}\oint_{\partial\epsilon} dz J_z(z)O(w,\bar{w})=-{\rm Res}(J_zO)=\delta O(w,\bar{w}).\label{one}\end{eqnarray}

These identities are equivalent to the operator product expansions:

\begin{eqnarray}J_z(z)O(w,\bar{w})&=&...+\frac{{\rm Res}(J_z(z)O(w,\bar{w}))}{z-w}+...\nonumber\\&=&...-\frac{\delta O(w,\bar{w})}{z-w}+...~,~z\longrightarrow w.\label{two}\end{eqnarray}

In other words, the OPE between a given operator and Noether's currents (stress-energy-momentum tensor) determine the transformation law of the operator under conformal transformations.

Translations

In this case $\delta z=\epsilon={\rm constant}$, $O(z-\epsilon)=O(z)-\epsilon\partial O(z)$, i.e. $\delta O=-\partial O$  and $J_z(z)=T(z)$. Hence, we have the OPE

\begin{eqnarray}T(z)O(w,\bar{w})&=&...+\frac{\partial O(w,\bar{w})}{z-w}+...~,~z\longrightarrow w.\end{eqnarray}

Rotations and Scalings

In this case we have:

\begin{eqnarray}\delta z=\epsilon z~,~\delta \bar{z}=\bar{\epsilon}\bar{z}.\end{eqnarray}

\begin{eqnarray}O(z-\epsilon z,\bar{z}-\bar{\epsilon}\bar{z})=O(z,\bar{z})-\epsilon (h O+z\partial O)-\bar{\epsilon}(\tilde{h} O+\bar{z}\bar{\partial}O).\end{eqnarray}

The pair $(h,\tilde{h})$ is called the weight of the operator $O$. The spin $s$ (eigenvalue under rotations) and the scaling dimension $\Delta$ (eigenvalue under scaling transformations which plays the role of energy) are defined by:

\begin{eqnarray}s=h-\tilde{h}~,~\Delta=h+\tilde{h}.\end{eqnarray}

In the case of rotations and scaling transformations we have $\delta z=\epsilon z$, $\delta O=-h O-z\partial O$ while Noether's current is $J_z(z)=zT(z)$. The OPE between $J$ and $O$ will now determine the $1/z^2$ term in the OPE between $T$ and $O$, viz

\begin{eqnarray}T(z)O(w,\bar{w})&=&...+h\frac{O(w,\bar{w})}{(z-w)^2}+\frac{\partial O(w,\bar{w})}{z-w}+...~,~z\longrightarrow w.\end{eqnarray}

Similarly, we obtain for the anti-holomorphic transformations $\delta\bar{z}=\bar{\epsilon} \bar{z}$:

\begin{eqnarray}\bar{T}(\bar{z})O(w,\bar{w})&=&...+\tilde{h}\frac{O(w,\bar{w})}{(\bar{z}-\bar{w})^2}+\frac{\bar{\partial} O(w,\bar{w})}{\bar{z}-\bar{w}}+...~,~z\longrightarrow w.\end{eqnarray}

Thus, translations determine the $1/z$ term in the OPE while rotations and scaling transformations determine the $1/z^2$ term in the OPE.

The operators whose OPE's with the stress-energy-momentum tensor truncates at the $1/z^2$ order are called  primary operators. These operators transform covariantly under conformal transformations. Indeed, general holomorphic conformal transformations are given by

\begin{eqnarray}\delta z=\epsilon(z)=\epsilon(w)+\epsilon^{\prime}(w)(z-w)+...\end{eqnarray}

The corresponding transformation laws:

\begin{eqnarray}\delta O &=&-{\rm Res}\big[\epsilon(z)T(z)O(w,\bar{w})\big]\nonumber\\&=&-h\epsilon^{\prime}(w)O(w,\bar{w})-\epsilon(w)\partial O(w,\bar{w}).\end{eqnarray}

And similarly, for anti-holomorphic transformations. The corresponding finite transformation law of a primary operator under the conformal transformations $z\longrightarrow \tilde{z}(z)$ and $\bar{z}\longrightarrow \bar{\tilde{z}}(\bar{z})$ reads:

\begin{eqnarray}O(z,\bar{z})\longrightarrow \tilde{O}(\tilde{z},\bar{\tilde{z}})=(\frac{\partial \tilde{z}}{\partial z})^{-h}(\frac{\partial \bar{\tilde{z}}}{\partial \bar{z}})^{-\tilde{h}}O(z,\bar{z}).\end{eqnarray}

The goal is to compute the spectrum of weights $(h,\tilde{h})$.

## Summary so far

Conformal systems have no preferred length scale.

Example: A two-dimensional massless free scalar field $\phi$ with action

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

This is invariant under the conformal transformations:

\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}

A conformally flat spacetime remains therefore conformal.

Euclidean flat spacetime is the complex plane with coordinates $z$ and $\bar{z}$.

Conformal transformations are conformal mapping in the complex plane:

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

These are global transformations which are symmetries of the action and thus they must be associated with conserved quantities called Noether's currents. This is Noether's theorem.

\begin{eqnarray}&&\delta z=\epsilon (z) ~,~J_z=\frac{1}{2}{T}({z}){\epsilon}({z})~,~\bar{\partial} {J}_{{z}}=0\nonumber\\&&\delta\bar{z}=\bar{\epsilon}(\bar{z})~,~\bar{J}_{\bar{z}}=\frac{1}{2}\bar{T}(\bar{z})\bar{\epsilon}(\bar{z})~,~\partial \bar{J}_{\bar{z}}=0.\end{eqnarray}

The stress-energy-momentum tensor, with components $T(z)$ and $\bar{T}(\bar{z})$, is conserved (under translations) and traceless (under Weyl transformations) and it is the Noether's current associated with translations.

In the case of rotations and scaling transformations we have $\delta z=\epsilon z$, while Noether's current is $J_z(z)=zT(z)$. And similarly, for the anti-holomorphic transformations.

Noether's theorem is replaced in the quantum theory by Ward identities with arbitrary number of operator insertions in the complex plane. This can be turned into the operator identities

\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}

The conformal transformation $\epsilon (x)$ is non-zero only around the operator insertion $O_1(x_1)$ and $\delta O_1(x_1)$ is the variation of the operator $O_1(x_1)$ under conformal transformations.

In two dimensions this result reduces to the contour integral:

\begin{eqnarray}\frac{i}{2\pi}\oint_{\partial\epsilon} dz J_z(z)O(w,\bar{w})=-{\rm Res}(J_zO)=\delta O(w,\bar{w}).\end{eqnarray}

These identities are equivalent to the operator product expansions:

\begin{eqnarray}J_z(z)O(w,\bar{w})&=&...+\frac{{\rm Res}(J_z(z)O(w,\bar{w}))}{z-w}+...\nonumber\\&=&...-\frac{\delta O(w,\bar{w})}{z-w}+...~,~z\longrightarrow w.\end{eqnarray}

In other words, the OPE between a given operator and Noether's currents (stress-energy-momentum tensor) determine the transformation law of the operator under conformal transformations.

The weights $h$ and $\tilde{h}$ of an operator $O$ are defined by

\begin{eqnarray}O(z-\epsilon z,\bar{z}-\bar{\epsilon}\bar{z})=O(z,\bar{z})-\epsilon (h O+z\partial O)-\bar{\epsilon}(\tilde{h} O+\bar{z}\bar{\partial}O).\end{eqnarray}

The spin $s$ (eigenvalue under rotations) and the scaling dimension $\Delta$ (eigenvalue under scaling transformations which plays the role of energy) are defined by:\begin{eqnarray}s=h-\tilde{h}~,~\Delta=h+\tilde{h}.\end{eqnarray}

Thus, translations determine the $1/z$ term in the OPE while rotations and scaling transformations determine the $1/z^2$ term in the OPE, viz

\begin{eqnarray}T(z)O(w,\bar{w})&=&...+h\frac{O(w,\bar{w})}{(z-w)^2}+\frac{\partial O(w,\bar{w})}{z-w}+...~,~z\longrightarrow w.\end{eqnarray}

\begin{eqnarray}\bar{T}(\bar{z})O(w,\bar{w})&=&...+\tilde{h}\frac{O(w,\bar{w})}{(\bar{z}-\bar{w})^2}+\frac{\bar{\partial} O(w,\bar{w})}{\bar{z}-\bar{w}}+...~,~z\longrightarrow w.\end{eqnarray}

The operators whose OPE's with the stress-energy-momentum tensor truncates at the $1/z^2$ order are called  primary operators. These operators transform covariantly under conformal transformations.

## The conformal anomaly

In two dimensions the propagator is given by

\begin{eqnarray}\langle 0|T(\phi(x)\phi(x^{'}))|0\rangle=-\frac{\alpha^{'}}{2}\ln(x-x^{'})^2.\end{eqnarray}

We can now include operator insertions $O_i(x_i)$ in the Feynman path integral defining this expectation value (assuming also that $x_i \neq x$ and $x_i\neq x$) to obtain the operator product expansion

\begin{eqnarray}\phi(x)\phi(x^{'})=-\frac{\alpha^{'}}{2}\ln(x-x^{'})^2+....\end{eqnarray}

By using the fact that the field $\phi$ splits into a holomorphic (left-moving) and  an anti-holomorphic (right-moving) parts which do not communicate we can rewrite the above OPE in the complex plane as

\begin{eqnarray}\phi(z)\phi(w)=-\frac{\alpha^{'}}{2}\ln(z-w)+....\end{eqnarray}

We conclude immediately that $\phi$ is not a primary operator. In fact $\phi$ is not even a good conformal field  theory operator since it has no good transformation properties under conformal transformations.

By taking the derivatives with respect to $z$ and $w$ we obtain the good OPE:

\begin{eqnarray}\partial\phi(z)\partial\phi(w)=-\frac{\alpha^{'}}{2}\frac{1}{(z-w)^2}+....\end{eqnarray}

Indeed, $\partial\phi$ is a primary operator of weights $h=1$ and $\tilde{h}=0$.

The stress-energy-momentum tensor in the classical theory is given by

\begin{eqnarray}T=-\frac{1}{\alpha^{'}}\partial\phi\partial\phi.\end{eqnarray}

In the quantum theory this is defined using the normal-ordering prescription:

\begin{eqnarray}T=-\frac{1}{\alpha^{'}}:\partial\phi\partial\phi:=-\frac{1}{\alpha^{'}}\bigg(\partial\phi(z)\partial\phi(w)-\langle 0|T(\partial\phi(z)\partial\phi(w))|0\rangle\bigg)~,~z\longrightarrow w.\label{Tno}\end{eqnarray}

Physically this means that the vacuum energy is identically zero, i.e. $\langle T\rangle=0$.

Let us recall that the OPE's involve time-ordered products of operators. Here, we want to compute the time-ordered product $T(z)\partial\phi(w)$. To this end, we use Wick's theorem which allows us to convert the normal-ordered product to a time-ordered product minus all contractions. We write this schematically as

\begin{eqnarray}({\rm normal~ordered})=({\rm time~ordered})-\sum {\rm contractions}.\end{eqnarray}

In our case, this reads explicitly

\begin{eqnarray}:T(z)\partial\phi(w):=T(z)\partial\phi(z)+\frac{1}{\alpha^{'}}\partial\phi(z)\langle\partial\phi(z)\partial\phi(w)\rangle+\frac{1}{\alpha^{'}}\langle\partial\phi(z)\partial\phi(w)\rangle\partial\phi(z).\end{eqnarray}

We get immediately the OPE:

\begin{eqnarray}T(z)\partial\phi(w)&=&-\frac{2}{\alpha^{'}}\big(-\frac{\alpha^{'}}{2}\frac{1}{(z-w)^2}\big)\partial\phi(z)\nonumber\\&=&\frac{\partial\phi(z)}{(z-w)^2}+..\nonumber\\&=&\frac{\partial \phi(w)}{(z-w)^2}+\frac{\partial^2\phi(w)}{z-w}+...\end{eqnarray}

Thus, $\partial\phi$ is a primary operator of weights $h=1$ and $\tilde{h}=0$.

As a second example, we compute the OPE between the stress-energy-momentum tensor and itself. We have

\begin{eqnarray}:T(z)T(w):=T(z)T(w)-\frac{2}{\alpha^{'2}}\langle \partial\phi(z)\partial\phi(w)\rangle \langle \partial\phi(z)\partial\phi(w)\rangle-\frac{4}{\alpha^{'2}}\langle\partial\phi(z)\partial\phi(w)\rangle :\partial\phi(z)\partial\phi(w):.\end{eqnarray}

The second term contains the two ways in which we can perform two contractions. The third term contains the four ways in which we can perform a single contraction. (In both cases we exclud those contractions involving operators at the same point which were already taken into account in the definition (\ref{Tno}) of the stress-energy-momentum tensor $T$.)

We get then the OPE:

\begin{eqnarray}T(z)T(w)&=&\frac{1/2}{(z-w)^4}-\frac{1}{\alpha^{'}}\frac{2}{(z-w)^2} :\partial\phi(z)\partial\phi(w):\nonumber\\&=&\frac{1/2}{(z-w)^4}+\frac{2T(w)}{(z-w)^2}-\frac{1}{\alpha^{'}}\frac{2}{z-w} :\partial^2\phi(w)\partial\phi(w):\nonumber\\&=&\frac{1/2}{(z-w)^4}+\frac{2T(w)}{(z-w)^2}+\frac{\partial T(w)}{z-w}.\end{eqnarray}

We conclude immediately that the stress-energy-momentum tensor $T(z)$ is a conformal operator of weights $h=2$ and $\tilde{h}=0$ and hence it is of spin $s=2$ (it is a symmetric $2-$tensor which couples to the metric) and scaling dimension $\Delta =2$ (it involves two derivatives each giving a mass dimension $+1$ whereas the inverse of the string tension $\alpha^{'}$ is of mass dimension $-2$ and the filed $\phi$ is of mass dimension $-1$).

However, the stress-energy-momentum tensor $T(z)$ is not a conformal operator because of the first term which corresponds to the so-called conformal anomaly and is proportional to the so-called central charge.

For a general conformal field theory the above equation changes to

\begin{eqnarray}T(z)T(w)&=&\frac{c/2}{(z-w)^4}+\frac{2T(w)}{(z-w)^2}+\frac{\partial T(w)}{z-w}+....\end{eqnarray}

The number $c$ is called the central charge and it is the most important number characterizing a conformal field theory. For example, in the case of $N$ non-interacting free massless scalar fields in two dimensions we have $c=N$. In general, the central charge is a measure of the number of the degrees of freedom (Cardy’s formula and  Zamalodchikov's c-theorem).

We are ready now to compute the transformation law of the stress-energy-momentum tensor under infinitesimal conformal transformations $z\longrightarrow \tilde{z}+\epsilon(z)$ with Noether's current $J_z(z)=\epsilon(z)T(z)$. We get

\begin{eqnarray}\delta T(w)&=&-{\rm Res}\big[\epsilon (z)T(z).T(w)\big]\nonumber\\&=&-{\rm Res}\bigg[\bigg(\epsilon(w)+\epsilon^{\prime}(w)(z-w)+\frac{1}{2}\epsilon^{\prime\prime}(w)(z-w)^2+\frac{1}{6}\epsilon^{\prime\prime\prime}(w)(z-w)^3+...\bigg)\nonumber\\&\times&\bigg(\frac{c/2}{(z-w)^4}+\frac{2T(w)}{(z-w)^2}+\frac{\partial T(w)}{z-w}+....\bigg)\bigg]\nonumber\\&=&-\epsilon(w)\partial T(w)-2\epsilon^{'}(w)T(w)-\frac{c}{12}\epsilon^{\prime\prime\prime}(w).\end{eqnarray}

This can be integrated to obtain the transformation law:

\begin{eqnarray}\big(\frac{\partial\tilde{z}}{\partial z}\big)^2\tilde{T}(\tilde{z})=T(z)-\frac{c}{12}\{\tilde{z},z\}.\end{eqnarray}

The bracket $\{\tilde{z},z\}$ is called the Schwarzian derivative of $\tilde{z}$ with respect to $z$,  denoted also by $S(\tilde{z},z)$,  and it is given by

\begin{eqnarray}\{\tilde{z},z\}=S(\tilde{z},z)&=&\frac{\tilde{z}^{\prime\prime\prime}}{z^{\prime}}-\frac{3}{2}\frac{\tilde{z}^{\prime\prime}}{z^{\prime 2}}\nonumber\\&=&\big(\frac{\tilde{z}^{\prime\prime}}{z^{\prime}}\big)^{\prime}-\frac{1}{2}\big(\frac{\tilde{z}^{\prime\prime}}{z^{\prime}}\big)^{2}.\end{eqnarray}

This derivative is invariant under $SL(2,R)$ or Mobius transformations:

Thus, if $\langle T(z)\rangle =0$ then $\langle \tilde{T}(\tilde{z})\rangle\neq 0$ and it is completely detremined by the conformal anomaly and the central charge. This is the conformal anomaly.

The central charge $c$ determines therefore the Casimir energy.

The conformal anomaly is intimately related to the Weyl anomaly and to the gravitational anomaly.

The Weyl anomaly is the statement that  the tracelessness property $T_{\alpha}^{\alpha}=0$ of the stress-energy-momentum tensor is lost in the quantum theory. Indeed, we compute for a conformally flat spacetime the expectation value

\begin{eqnarray}\langle T_{\alpha}^{\alpha}\rangle&=&-\frac{c}{12}R.\label{R}\end{eqnarray}

$R$ is the Ricci scalar. Thus, the symmetry under Weyl transformations is lost since $R$ takes different values for the different metric related by Weyl transformations.

Of course $c$ is the central charge associated with the holomorphic (left-moving) secor. There is also a central charge $\tilde{c}$ associated with the anti-holomorphic (right-moving) sector. For the anti-holomorphic (right-moving) sector we obtain a similar formula to (\ref{R}), viz

\begin{eqnarray}\langle T_{\alpha}^{\alpha}\rangle&=&-\frac{\tilde{c}}{12}R.\end{eqnarray}

Thus, in curved background we must have $c=\tilde{c}$. This is a gravitational anomlay.

### References:

David Tong, String Theory.