Stress-tensor Ward identity for primary insertions

Let $n\ge 1$, let $U\subseteq \mathbb{C}$ be the open coordinate image of the plane chart carrying holomorphic stress-tensor data, and let $w_1,\dots ,w_n\in U$ be pairwise distinct insertion labels for primary field data ${\mathcal{O}}_i$ of weights $(h_i,{\bar{h}}_i)$ on their common invariant domain. Write ${\mathrm{Conf}}_n(U)=\{(w_1,\dots ,w_n)\in U^n:w_i\mathrm{\ne }{w}_j\text{ for }i\mathrm{\ne }j\}$. The model's correlation prescription supplies a scalar function $G:{\mathrm{Conf}}_n(U)\to \mathbb{C}$ and, for each fixed insertion configuration, a single-valued scalar function $H_{\mathbf{w}}:\mathbb{C}\setminus \{w_1,\dots ,w_n\}\to \mathbb{C}$ that agrees on $U\setminus \{w_i\}$ with the one-stress-insertion correlation representative and is its specified analytic continuation. In this theorem, point-coordinate notation denotes these scalar representatives; it does not assert point evaluation of the weak fields as operators. Write $w_i=x_i+i{y}_i$ and use the Wirtinger derivatives ${\mathrm{\partial }}_{w_i}=\frac{1}{2}({\mathrm{\partial }}_{x_i}-i{\mathrm{\partial }}_{y_i})$ and ${\mathrm{\partial }}_{{\bar{w}}_i}=\frac{1}{2}({\mathrm{\partial }}_{x_i}+i{\mathrm{\partial }}_{y_i})$. Assume that $G$ is real $C^1$ on ${\mathrm{Conf}}_n(U)$. The contour integrals below use positive orientation in their displayed complex variables. Assume that $H_{\mathbf{w}}$ is holomorphic on $\mathbb{C}\setminus \{w_1,\dots ,w_n\}$ and meromorphic at each $w_i$ with local expansion

where $R_i$ is holomorphic near $w_i$. Assume also that no contact term is added to the prescribed scalar contour during deformation and that

for each $z\mathrm{\notin }\{w_1,\dots ,w_n\}$. Then, as an identity of scalar correlation representatives,

If the stress data also include a barred sector, $G$ is differentiable in each ${\bar{w}}_i$, and the prescribed single-valued barred scalar representative ${\bar{H}}_{\mathbf{w}}:\mathbb{C}\setminus \{{\bar{w}}_1,\dots ,{\bar{w}}_n\}\to \mathbb{C}$ agrees on $\bar{U}\setminus \{{\bar{w}}_i\}$, where $\bar{U}=\{\bar{z}:z\in U\}$, with the barred one-stress-insertion correlation representative and is its specified analytic continuation; it is holomorphic in its independent barred argument on $\mathbb{C}\setminus \{{\bar{w}}_1,\dots ,{\bar{w}}_n\}$ and meromorphic at the insertions with local expansions

where ${\bar{R}}_i$ is holomorphic in the independent coordinate $\bar{\zeta }$ near ${\bar{w}}_i$. Here and throughout the barred conclusion, $\xi =\bar{\zeta }$ is an independent complex integration variable: $\xi \mapsto {\bar{H}}_{\mathbf{w}}(\xi )$ is holomorphic away from ${\bar{w}}_i$ and meromorphic at them. The term “antiholomorphic” refers to the function $\zeta \mapsto {\bar{H}}_{\mathbf{w}}(\bar{\zeta })$ on the original plane, not to $\xi \mapsto {\bar{H}}_{\mathbf{w}}(\xi )$. Assume there are no barred contact terms during deformation and

for every $\bar{z}\mathrm{\notin }\{{\bar{w}}_1,\dots ,{\bar{w}}_n\}$, then, for every such $\bar{z}$, the separate antiholomorphic identity holds with $H_{\mathbf{w}},T,h_i,z,w_i$ replaced by ${\bar{H}}_{\mathbf{w}},\bar{T},{\bar{h}}_i,\bar{z},{\bar{w}}_i$ and ${\mathrm{\partial }}_{w_i}$ replaced by ${\mathrm{\partial }}_{{\bar{w}}_i}$. These are conditional model deductions; they assert neither existence of stress data in every CFT nor an empirical fact. Outside $U$, the identity concerns only the stipulated analytic continuation.

The picture is schematic: it shows an outer circle enclosing the possible poles and small positively oriented circles around those poles. The circle residue theorem and the stated boundary limit carry the proof independently of the drawing.

The picture is schematic and assumes all marked points are distinct and inside ${\mathrm{\Gamma }}_R$. Every arrow indicates the positive (counterclockwise) orientation used for the residue contours.