# TTT in CFT: Trace Identities and the Conformal Anomaly Effective Action

## TL;DR

This paper develops a covariant framework for stress-energy correlators in 4D conformal field theories, revealing the role of anomaly-induced effective actions and massless poles in correlator structures.

## Contribution

It explicitly computes metric variations of the anomaly effective action for stress-energy correlators, linking anomaly-induced poles to solutions of conformal Ward identities in curved space.

## Key findings

- Massless propagator poles appear in stress-energy correlators due to conformal anomalies.
- The covariant approach fixes all Ward identities, including contact terms, in a curved background.
- The explicit variation of the anomaly effective action matches algebraic reconstructions of correlators.

## Abstract

Stress-energy correlation functions in a general Conformal Field Theory (CFT) in four dimensions are described in a fully covariant approach, as metric variations of the quantum effective action in an arbitrary curved space background field. All Conservation, Trace and Conformal Ward Identities (CWIs), including contact terms, are completely fixed in this covariant approach. The Trace and CWIs are anomalous. Their anomalous contributions may be computed unambiguously by metric variation of the exact 1PI quantum effective action determined by the conformal anomaly of $\left\langle T^{\mu\nu}\right\rangle$ in $d = 4$ curved space. This action implies the existence of massless propagator poles in three and higher point correlators of $T^{\mu\nu}$ . The metric variations of the anomaly effective action in its local form in terms of a scalar conformalon field are carried out explicitly for the case of the correlator of three CFT stress-energy tensors, and the result is shown to coincide with the algebraic reconstruction of $\left\langle TTT\right\rangle$ from its transverse, tracefree parts, determined independently by the solution of the CWIs in d dimensional flat space in the momentum representation. This demonstrates that the specific analytic structure and massless poles predicted by the general curved space anomaly effective action are in fact a necessary feature of the exact solution of the anomalous CWIs in any $d = 4$ CFT.