# Ward identity Ward identities are relations satisfied by *correlation functions* due to symmetries of a theory. It takes the form$-\frac{1}{2 \pi} \int_{\epsilon} \partial_{\alpha}\left\langle J^{\alpha}(\sigma) \mathcal{O}_{1}\left(\sigma_{1}\right) \ldots\right\rangle=\left\langle\delta \mathcal{O}_{1}\left(\sigma_{1}\right) \ldots\right\rangle,$where the integral is over a region where the transformation $\epsilon$ has support. ## Derivation - quantum Noether's theorem: $\left\langle\partial_{\alpha} J^{\alpha}\right\rangle=0$ - upon $\phi^{\prime}=\phi+\epsilon \delta \phi$: $\begin{aligned} Z & \longrightarrow \int \mathcal{D} \phi^{\prime} \exp \left(-S\left[\phi^{\prime}\right]\right) \\ &=\int \mathcal{D} \phi \exp \left(-S[\phi]-\frac{1}{2 \pi} \int J^{\alpha} \partial_{\alpha} \epsilon+O(\epsilon^2)\right) \\ &=\int \mathcal{D} \phi\, e^{-S[\phi]}\left(1-\frac{1}{2 \pi} \int J^{\alpha} \partial_{\alpha} \epsilon\right) \end{aligned}$where in the second line, we wrote down $\partial_\alpha \epsilon$ because we know a constant $\epsilon$ is a symmetry - $\left\langle\partial_{\alpha} J^{\alpha}(\sigma) \mathcal{O}_{1}\left(\sigma_{1}\right) \ldots \mathcal{O}_{n}\left(\sigma_{n}\right)\right\rangle=0 \quad$ for $\sigma \neq \sigma_{i}$ - same proof with insertions - **Ward identity** - $\frac{1}{Z} \int \mathcal{D} \phi e^{-S[\phi]}\left(1-\frac{1}{2 \pi} \int J^{\alpha} \partial_{\alpha} \epsilon\right)\left(\mathcal{O}_{1}+\epsilon \delta \mathcal{O}_{1}\right) \mathcal{O}_{2} \ldots \mathcal{O}_{n}$ - working to leading order leads to the Ward identity: - $-\frac{1}{2 \pi} \int_{\epsilon} \partial_{\alpha}\left\langle J^{\alpha}(\sigma) \mathcal{O}_{1}\left(\sigma_{1}\right) \ldots\right\rangle=\left\langle\delta \mathcal{O}_{1}\left(\sigma_{1}\right) \ldots\right\rangle$ ## Relation to [[0030 Operator product expansion|OPE]] - knowing OPE between *conformal current* and an operator => knowing how a field transforms under *conformal symmetry*; - knowing how an operator transforms => know something about the OPE ## Conformal transformations in 2d - first use Stokes' theorem: - $\int_\epsilon \partial_\alpha J^\alpha=\oint_{\partial \epsilon} J_\alpha \hat{n}^\alpha=\oint_{\partial \epsilon}\left(J_1 d \sigma^2-J_2 d \sigma^1\right)=-i \oint_{\partial \epsilon}\left(J_z d z-J_{\bar{z}} d \bar{z}\right)$ - then Ward identity says - $\frac{i}{2 \pi} \oint_{\partial \epsilon} d z\left\langle J_z(z, \bar{z}) \mathcal{O}_1\left(\sigma_1\right) \ldots\right\rangle-\frac{i}{2 \pi} \oint_{\partial \epsilon} d \bar{z}\left\langle J_{\bar{z}}(z, \bar{z}) \mathcal{O}_1\left(\sigma_1\right) \ldots\right\rangle=\left\langle\delta \mathcal{O}_1\left(\sigma_1\right) \ldots\right\rangle$ - in the case of 2d conformal currents - $J_z$ is holomorphic; $J_{\bar{z}}$ is anti-holomorphic - so $\frac{i}{2 \pi} \oint_{\partial \epsilon} d z J_z(z) \mathcal{O}_1\left(\sigma_1\right)=-\operatorname{Res}\left[J_z \mathcal{O}_1\right]$ - result: - for $\delta z=\epsilon(z)$: $\delta \mathcal{O}_1\left(\sigma_1\right)=-\operatorname{Res}\left[J_z(z) \mathcal{O}_1\left(\sigma_1\right)\right]=-\operatorname{Res}\left[\epsilon(z) T(z) \mathcal{O}_1\left(\sigma_1\right)\right]$ - for $\delta \bar{z}=\epsilon(\bar{z})$: $\delta \mathcal{O}_1\left(\sigma_1\right)=-\operatorname{Res}\left[\bar{J}_{\bar{z}}(\bar{z}) \mathcal{O}_1\left(\sigma_1\right)\right]=-\operatorname{Res}\left[\bar{\epsilon}(\bar{z}) \bar{T}(\bar{z}) \mathcal{O}_1\left(\sigma_1\right)\right]$ - i.e., how operator transforms under conformal transformation is related to the OPE with the stress tensor components ## Relation to [[0010 Celestial holography|CCFT]] Recently, as an important part of the development of [[0010 Celestial holography|celestial holography]], [[0009 Soft theorems|soft theorems]] are understood to be Ward identifies of [[0060 Asymptotic symmetry|asymptotic symmetries]]. ## Refs - [[Rsc0002 Alday Notes on CFT]] - [[Rsc0006 Tong String theory]]