mathematical physics - Weyl anomaly in 2d CFT (string theory lectures by D.Tong)

In his lectures on String Theory (http://www.damtp.cam.ac.uk/user/tong/string.html), Tong gives a proof of the Weyl anomaly, using equation $(4.36)$. It seems wrong to me.

Here he uses the OPE between the stress-energy tensors $T_{zz}T_{ww}$ obtained when trace vanishes, i.e. $T_{z \bar{z}} = 0$: this implies that they are holomorphic functions $T_{zz} = T_{zz}(z)$. But in this proof he starts from the fact that $T_{z \bar{z}} \neq 0$ (we want to proof this thing after all) and so $T_{zz}$ is not a holomorphic function anymore! In the OPE $(4.36)$ I should have also terms with $(\bar{z}- \bar{w})$.

I can't also understand why he uses in the rest of the proof only the singular term $(z-w)^{-4}$, neglecting the other ones $(z-w)^{-2}$, $(z-w)^{-1}$.

(The same proof is given in these lectures https://arxiv.org/abs/1511.04074 on conformal field theory, equation $(6.9)$).

I'll be really thankful if someone could explain me this proof :)

Answer

TL;DR. The main point is that Tong only needs to identify the leading singularity in order to determine the Weyl anomaly $$\langle T^{\alpha}{}_{\alpha}\rangle~=~-\frac{c}{12} R^{(2)}. \tag{4.34}$$ It is indeed unclear how to properly account for subleading terms in Tong's approach.

Let us introduce a regulator $\varepsilon>0$ in the $XX$ OPE

$${\cal R} X(z,\bar{z})X(w,\bar{w})~=~-\frac{\alpha^{\prime}}{2} \ln(|z-w|^2+\varepsilon)~+~: X(z,\bar{z})X(w,\bar{w}): \tag{4.22}$$

to better identify the singular structure. The $\partial X\partial X$ OPE becomes:

$$\begin{align} {\cal R} \partial_zX(z,\bar{z})& \partial_wX(w,\bar{w}) \cr ~\stackrel{(4.22)}{=}&~-\frac{\alpha^{\prime}}{2}\frac{(\bar{z}-\bar{w})^2}{(|z-w|^2+\varepsilon)^2}~+~:\partial_zX(z,\bar{z}) \partial_wX(w,\bar{w}): ~.\end{align} \tag{4.23}$$

The stress-energy-momentum tensor is

$$T_{zz}~~=~ -\frac{1}{\alpha^{\prime}} :\partial_zX\partial_zX:~.\tag{4.25}$$

The $TT$ OPE becomes

$$\begin{align} {\cal R}T_{zz}(z,\bar{z}) &T_{ww}(w,\bar{w})\cr ~\stackrel{(4.23)+(4.25)}{=}&~\frac{c}{2}\frac{(\bar{z}-\bar{w})^4}{(|z-w|^2+\varepsilon)^4} \cr &-\frac{2}{\alpha^{\prime}} \frac{(\bar{z}-\bar{w})^2}{(|z-w|^2+\varepsilon)^2}:\partial_zX(z,\bar{z}) \partial_wX(w,\bar{w}):~+~\ldots. \end{align}\tag{4.28}$$

We next use the energy conservation

$$\partial_z T_{\bar{z}z} + \partial_{\bar{z}} T_{zz}~=~0 \tag{4.35z}$$

to calculate$^1$

$$\begin{align} {\cal R}\partial_z T_{z\bar{z}}(z,\bar{z}) &\partial_wT_{w\bar{w}}(w,\bar{w}) \cr ~\stackrel{(4.35z)}{=}&~{\cal R}\partial_{\bar{z}} T_{zz}(z,\bar{z})\partial_{\bar{w}}T_{ww}(w,\bar{w})\cr ~\stackrel{(4.28)}{=}&~ \partial_{\bar{z}}\partial_{\bar{w}} \left[ \frac{c}{2} \frac{(\bar{z}-\bar{w})^4}{(|z-w|^2+\varepsilon)^4} +\ldots \right] \cr ~=&~\partial_{\bar{w}} \left[ 2c \frac{\varepsilon(\bar{z}-\bar{w})^3}{(|z-w|^2+\varepsilon)^5} +\ldots \right] \cr ~=&~-10c\frac{\varepsilon^2(\bar{z}-\bar{w})^2}{(|z-w|^2+\varepsilon)^6}+4c\frac{\varepsilon(\bar{z}-\bar{w})^2}{(|z-w|^2+\varepsilon)^5}+\ldots \cr ~=&~\frac{c}{12}\partial_z\partial_w\left[\frac{6\varepsilon^2}{(|z-w|^2+\varepsilon)^4}-\frac{4\varepsilon}{(|z-w|^2+\varepsilon)^3}\right]+\ldots \cr ~=&~\frac{c}{12}\partial_z\partial_w\partial_z\partial_{\bar{w}}\frac{\varepsilon}{(|z-w|^2+\varepsilon)^2}+\ldots, \end{align}\tag{4.36}$$

which leads to the sought-for OPE

$$\begin{align} {\cal R}T_{z\bar{z}}(z,\bar{z}) &T_{w\bar{w}}(w,\bar{w}) \cr ~\stackrel{(4.36)}{=}&~\frac{c}{12}\partial_z\partial_{\bar{w}}\frac{\varepsilon}{(|z-w|^2+\varepsilon)^2}+\ldots \cr ~\stackrel{(4.2d)}{=}&~\frac{c\pi}{12}\partial_z\partial_{\bar{w}}\delta^2(z\!-\!w,\bar{z}\!-\!\bar{w}) +\ldots . \end{align} \tag{4.38}$$

Here we use the following representation of the 2D Dirac delta distribution$^2$

$$\delta^2(z\!-\!w,\bar{z}\!-\!\bar{w})~:=~\delta({\rm Re} (z\!-\!w))~\delta({\rm Im} (z\!-\!w))~=~\lim_{\varepsilon\searrow 0^+} \frac{1}{\pi}\frac{\varepsilon}{(|z-w|^2+\varepsilon)^2}. \tag{4.2d}$$

Now proceed as in Tong's notes. $\Box$

References:

1. D. Tong, Lectures on String Theory; Subsection 4.4.2.

--

$^1$ Tong's trick (4.36) suggests another route: Let us instead consider the $\partial X \bar{\partial}X$ OPE

$$\begin{align} {\cal R} \partial_zX(z,\bar{z}) &\partial_{\bar{w}}X(w,\bar{w})\cr ~=&~{\cal R} \partial_{\bar{z}}X(z,\bar{z}) \partial_wX(w,\bar{w})\cr ~=&~\frac{\alpha^{\prime}}{2}\frac{\varepsilon}{(|z-w|^2+\varepsilon)^2}+\ldots ~\stackrel{(4.2d)}{=}~\frac{\alpha^{\prime}\pi}{2}\delta^2(z\!-\!w,\bar{z}\!-\!\bar{w}) +\ldots. \end{align}$$

It is comforting that the regularization $\varepsilon>0$ correctly predicts that the leading singularity is a 2D Dirac delta distribution. Then the $T\bar{T}$ OPE becomes

$$\begin{align} {\cal R}T_{zz}(z,\bar{z})&T_{\bar{w}\bar{w}}(w,\bar{w})\cr ~=&~\frac{c}{2}\frac{\varepsilon^2}{(|z-w|^2+\varepsilon)^4} +\frac{2}{\alpha^{\prime}} \frac{\varepsilon}{(|z-w|^2+\varepsilon)^2}:\partial_zX(z,\bar{z})\partial_{\bar{w}}X(w,\bar{w}):~+~\ldots. \end{align}$$

The leading singularity is given by double contractions, which are proportional to the square of the 2D Dirac delta distribution. This is ill-defined, cf. e.g. this Phys.SE post.

Nevertheless, let us now formally apply Tong's trick: Using the energy conservation (4.35z) leads to

$${\cal R}\partial_z T_{z\bar{z}}(z,\bar{z}) \partial_{\bar{w}}T_{w\bar{w}}(w,\bar{w}) ~\stackrel{(4.35z)}{=}~{\cal R}\partial_{\bar{z}} T_{zz}(z,\bar{z})\partial_w T_{\bar{w}\bar{w}}(w,\bar{w}),$$

so that the sought-for OPE leads to the square of the 2D Dirac delta distribution as well

$$\begin{align} {\cal R}T_{z\bar{z}}(z,\bar{z}) &T_{w\bar{w}}(w,\bar{w})\cr ~=&~\frac{c}{2}\frac{\varepsilon^2}{(|z-w|^2+\varepsilon)^4}+\ldots ~\stackrel{(4.2d)}{=}~\frac{c\pi^2}{2}\delta^2(z\!-\!w,\bar{z}\!-\!\bar{w})^2+\ldots. \end{align}$$

There might be a way to resolve the square of the 2D Dirac delta distribution, and argue that the leading singularity is given by (4.38), although we shall not pursue the matter here. $\Box$

$^2$ Note that there is a factor of 2 in Tong's definition of the 2D Dirac delta distribution, cf. the end of Section 4.0.1.