Weyl anomaly in 2d CFT (string theory lectures by D.Tong)
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
$$\begin{align} {\cal R} X(z,\bar{z})X(w,\bar{w})~=~&-\frac{\alpha^{\prime}}{2} \ln(|z-w|^2+\varepsilon)\cr &~+~: X(z,\bar{z})X(w,\bar{w}): \end{align}\tag{4.22}$$
to better identify the singular structure. The $\partial X\partial X$ OPE becomes:
$$\begin{align} {\cal R} \partial_zX(z,\bar{z})& \partial_wX(w,\bar{w}) \cr ~\stackrel{(4.22)}{=}&~-\frac{\alpha^{\prime}}{2}\frac{(\bar{z}-\bar{w})^2}{(|z-w|^2+\varepsilon)^2}\cr &~+~:\partial_zX(z,\bar{z}) \partial_wX(w,\bar{w}): ~.\end{align} \tag{4.23}$$
The stress-energy-momentum tensor is
$$ T_{zz}~~=~ -\frac{1}{\alpha^{\prime}} :\partial_zX\partial_zX:~.\tag{4.25} $$
The $TT$ OPE becomes
$$ \begin{align} {\cal R}T_{zz}(z,\bar{z}) &T_{ww}(w,\bar{w})\cr ~\stackrel{(4.23)+(4.25)}{=}&~\frac{c}{2}\frac{(\bar{z}-\bar{w})^4}{(|z-w|^2+\varepsilon)^4} \cr &-\frac{2}{\alpha^{\prime}} \frac{(\bar{z}-\bar{w})^2}{(|z-w|^2+\varepsilon)^2}:\partial_zX(z,\bar{z}) \partial_wX(w,\bar{w}):\cr &~+~\ldots. \end{align}\tag{4.28}$$
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$^2$
$$\begin{align} \delta^2(z\!-\!w,\bar{z}\!-\!\bar{w})~:=~& \delta({\rm Re} (z\!-\!w))~\delta({\rm Im} (z\!-\!w))\cr ~=~&\lim_{\varepsilon\searrow 0^+} \frac{1}{\pi}\frac{\varepsilon}{(|z-w|^2+\varepsilon)^2}. \end{align} \tag{4.2d}$$
Now proceed as in Tong's notes. $\Box$
References:
- 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
$$\begin{align} {\cal R} \partial_zX(z,\bar{z}) &\partial_{\bar{w}}X(w,\bar{w})\cr ~=~&{\cal R} \partial_{\bar{z}}X(z,\bar{z}) \partial_wX(w,\bar{w})\cr ~=~&\frac{\alpha^{\prime}}{2}\frac{\varepsilon}{(|z-w|^2+\varepsilon)^2}+\ldots \cr ~\stackrel{(4.2d)}{=}&~\frac{\alpha^{\prime}\pi}{2}\delta^2(z\!-\!w,\bar{z}\!-\!\bar{w}) +\ldots. \end{align}$$
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
$$ \begin{align} {\cal R}T_{zz}(z,\bar{z})&T_{\bar{w}\bar{w}}(w,\bar{w})\cr ~=&~\frac{c}{2}\frac{\varepsilon^2}{(|z-w|^2+\varepsilon)^4}\cr &+\frac{2}{\alpha^{\prime}} \frac{\varepsilon}{(|z-w|^2+\varepsilon)^2}:\partial_zX(z,\bar{z})\partial_{\bar{w}}X(w,\bar{w}):\cr &~+~\ldots. \end{align}$$
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
$$ \begin{align} {\cal R}T_{z\bar{z}}(z,\bar{z}) &T_{w\bar{w}}(w,\bar{w})\cr ~=~&\frac{c}{2}\frac{\varepsilon^2}{(|z-w|^2+\varepsilon)^4}+\ldots\cr ~\stackrel{(4.2d)}{=}&~\frac{c\pi^2}{2}\delta^2(z\!-\!w,\bar{z}\!-\!\bar{w})^2+\ldots. \end{align}$$
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.
Ref. 1 has a different proof of the Weyl anomaly
$$ T^{\alpha}_{\alpha}~=~-\frac{c}{12} R^{(2)}, \tag{4.9.8} $$
which we outline in this answer, and which is possibly more convincing than the proofs in Refs. 2 & 3.
Sketched proof of eq. (4.9.8):
We start with a $(1,1)$ Hermitian metric $$\begin{align} \mathbb{g}~=~&2g_{z\bar{z}} \mathrm{d}z \odot \mathrm{d}\bar{z}, \cr g_{zz}~=~&0~=~g_{\bar{z}\bar{z}}. \end{align} \tag{A}$$ The Levi-Civita Christoffel symbols are $$\begin{align} \Gamma^{z}_{zz}~=~&g^{z\bar{z}}\partial_{z} g_{z\bar{z}}, \cr \Gamma^{\bar{z}}_{\bar{z}\bar{z}}~=~&g^{z\bar{z}}\partial_{\bar{z}} g_{z\bar{z}}, \cr \Gamma(\text{mixed indices})~=~&0, \end{align} \tag{B}$$ i.e. the Levi-Civita connection $\nabla$ is Hermitian.
Under a holomorphic coordinate transformation $$z^{\prime}~=~f(z),\tag{C}$$ the Christoffel symbol does not transform as a tensor $$ \Gamma^{z}_{zz}~\stackrel{(B)+(C)}{=}~f^{\prime}\Gamma^{z^{\prime}}_{z^{\prime}z^{\prime}}+\frac{f^{\prime\prime}}{f^{\prime}}. \tag{D}$$ We construct for later an object $$r_{zz}~:=~ \partial_{z}\Gamma^{z}_{zz} -\frac{1}{2}(\Gamma^{z}_{zz})^2, \tag{E}$$ which transforms with the Schwarzian derivative: $$\begin{align} r_{zz}~\stackrel{(D)+(E)}{=}&~(f^{\prime})^2 r_{z^{\prime}z^{\prime}} +\{f,z\}, \cr \{f,z\}~:=~~&\frac{f^{\prime\prime\prime}}{f^{\prime}}-\frac{3}{2} \left(\frac{f^{\prime\prime}}{f^{\prime}}\right)^2.\end{align} \tag{F}$$ The holomorphic SEM tensor component also transforms with a Schwarzian derivative $$ T_{zz}~=~(f^{\prime})^2 T_{z^{\prime}z^{\prime}} +\frac{c}{12}\{f,z\},\tag{4.9.2} $$ where $$ \partial_{\bar{z}}T_{zz}~=~0. \tag{G} $$ We can therefore define the difference $$\hat{T}_{zz}~:=~T_{zz} - \frac{c}{12}r_{zz},\tag{4.9.3} $$ which transforms as a tensor $$ \hat{T}_{zz}~=~(f^{\prime})^2 \hat{T}_{z^{\prime}z^{\prime}}. \tag{4.9.4}$$
The Ricci curvature tensor is $$\begin{align} -R_{z\bar{z}}~=~&\partial_{\bar{z}}\Gamma^{z}_{zz}\cr ~=~&\partial_{z}\Gamma^{\bar{z}}_{\bar{z}\bar{z}}, \cr R_{zz}~=~&0~=~R_{\bar{z}\bar{z}}.\end{align} \tag{H}$$ The Ricci scalar curvature is $$ R^{(2)}~=~2g^{z\bar{z}}R_{z\bar{z}}.\tag{I} $$ Ignoring a possible cosmological constant $\Lambda$, the trace of the SEM tensor must be proportional to the Ricci scalar $$ T^{\alpha}_{\alpha}~=~A R^{(2)},\tag{4.9.5} $$ or equivalently, $$\begin{align} \hat{T}_{z\bar{z}}~:=~&T_{z\bar{z}}\cr ~=~&AR_{z\bar{z}}\cr ~=~&\frac{A}{2}g_{z\bar{z}}R^{(2)}.\end{align} \tag{4.9.6} $$
From diffeomorphism invariance we have the continuum equation $$ \nabla_{\alpha} \hat{T}^{\alpha\beta} ~=~0.\tag{J}$$ We calculate $$\begin{align} -\frac{c}{12}g^{z\bar{z}}&(\partial_{\bar{z}}\partial_{z}\Gamma^{z}_{zz} -\Gamma^{z}_{zz}\partial_{\bar{z}}\Gamma^{z}_{zz})\cr ~\stackrel{(E)}{=}~~~&-\frac{c}{12}g^{z\bar{z}}\partial_{\bar{z}}r_{zz}\cr ~\stackrel{(G)+(4.9.3)}{=}&g^{z\bar{z}}\partial_{\bar{z}}\hat{T}_{zz}\cr ~=~~~~&\nabla^{z}\hat{T}_{zz}\cr ~\stackrel{(J)}{=}~~~~&-\nabla^{\bar{z}}\hat{T}_{\bar{z}z}\cr ~\stackrel{(4.9.6)}{=}~~& -\nabla^{\bar{z}}T_{z\bar{z}}\cr ~\stackrel{(4.9.6)}{=}~~& -\frac{A}{2}\nabla^{\bar{z}}(g_{z\bar{z}}R^{(2)})\cr ~=~~~~& -\frac{A}{2}\partial_{z}R^{(2)}\cr ~\stackrel{(B)+(I)}{=}~& -Ag^{z\bar{z}}(\partial_{z} - \Gamma^{z}_{zz}) R_{z\bar{z}}\cr ~\stackrel{(H)}{=}~~~& Ag^{z\bar{z}}(\partial_{z}\partial_{\bar{z}}\Gamma^{z}_{zz} -\Gamma^{z}_{zz}\partial_{\bar{z}}\Gamma^{z}_{zz}) .\end{align} \tag{4.9.7} $$ We therefore deduce the Weyl anomaly $$ A~\stackrel{(4.9.7)}{=}~-\frac{c}{12}. \tag{K}$$ $\Box$
References:
E. Kiritsis, String Theory in a Nutshell, 2007; Section 4.9. NB: The minus sign in eq. (K) is mentioned in the Errata.
D. Tong, Lectures on String Theory; Subsection 4.4.2.
J. Polchinski, String Theory Vol. 1, 1998; Section 3.4.