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 subleading terms $$(z-w)^{-2}$$, $$(z-w)^{-1}$$.

(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 :)

TL;DR. The main point is that Tong only needs to identify the leading singularity in order to determine the Weyl anomaly

$$\langle T^{\alpha}{}_{\alpha}\rangle~=~-\frac{c}{12} R^{(2)}. \tag{4.34}$$

Concerning OP's question it is indeed unclear how to properly account for subleading terms in Tong's approach. They are presumably either contact terms or vanish on-shell.

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

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

which holds on-shell up to contact terms. We 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)}{\approx}&~{\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 ~=~&\frac{c}{12}\partial_{\bar{w}}\partial_z\partial_w\partial_z\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:

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} \left. \begin{array}{c} {\cal R} \partial_zX(z,\bar{z}) \partial_{\bar{w}}X(w,\bar{w})\cr\cr {\cal R} \partial_{\bar{z}}X(z,\bar{z}) \partial_wX(w,\bar{w})\end{array}\right\} ~=~&\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

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

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}{2}\pi^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 instead given by the second derivative of the 2D Dirac delta distribution,

\begin{align} \frac{c}{12}\partial_z\partial_{\bar{w}}\underbrace{\frac{\varepsilon}{(|z-w|^2+\varepsilon)^2}}_{=\pi\delta^2(z-w,\bar{z}-\bar{w})} ~=~&\frac{c}{2}\underbrace{\frac{\varepsilon^2}{(|z-w|^2+\varepsilon)^4}}_{=\pi^2\delta^2(z-w,\bar{z}-\bar{w})^2}\cr ~-~&\frac{c}{3}\underbrace{\frac{\varepsilon}{(|z-w|^2+\varepsilon)^3}}_{=4\pi^2\delta^2(z-w,\bar{z}-\bar{w})^2},\end{align}

as in eq. (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.

• I understand that, but what I am wondering is that in the OPE $T_{zz}(z, \bar{z})T_{ww}(w, \bar{w})$ there are other singularities, like $(z-w)^{-2}$ ,and anti-holomorphic pieces (with some singularities I think) that he isn't considering. – Weyl Nov 24 '18 at 15:50
• I mean, I see that the OPE $T_{z \bar{z}}(z, \bar{z})T_{w \bar{w}}(w, \bar{w})$ gets modified by contact terms at singularities, but I don't understand the OPE $T_{zz}(z, \bar{z})T_{ww}(w, \bar{w})$ that he uses for the proof. – Weyl Nov 24 '18 at 16:06
• Thank you for your answer, but what I don't understand is precisely why he can neglect these subleading terms (i.e. why they don't contribute to equations $(4.38)$, $(4.39)$). – Weyl Nov 25 '18 at 16:51
• They do contribute to eq. (4.38) unlike what Tong is writing. – Qmechanic Nov 26 '18 at 13:38
• Oh ok, I think the same. I was asking because I like the way he tries to give the proof and so I was hoping that it was right. On the other hand, other important references give a totally different proof (e.g. Polchinski). Thank you for your help :) – Weyl Nov 26 '18 at 19:43

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):

1. 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.

2. 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 $$\hat{T}_{zz}~=~(f^{\prime})^2 \hat{T}_{z^{\prime}z^{\prime}}. \tag{4.9.4}$$

3. 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}

4. 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:

1. E. Kiritsis, String Theory in a Nutshell, 2007; Section 4.9. NB: The minus sign in eq. (K) is mentioned in the Errata.

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

3. J. Polchinski, String Theory Vol. 1, 1998; Section 3.4.