# Expectation value of $\mathrm{T}\overline{\mathrm{T}}$ operator in curved spacetimes

## TL;DR

This paper investigates the expectation value of the $ ext{T}ar{ ext{T}}$ operator in curved spacetimes, revealing curvature-dependent behavior and extending previous flat spacetime results.

## Contribution

It introduces a diffeomorphism invariant biscalar to compute the expectation value of the $ ext{T}ar{ ext{T}}$ operator in curved backgrounds, generalizing Zamolodchikov's flat spacetime result.

## Key findings

- In flat spacetime, the biscalar is a constant, matching Zamolodchikov's result.
- In curved spacetimes, the expectation value depends on stress-energy tensor correlations.
- The approach links the expectation value to both one-point and two-point functions of the stress-energy tensor.

## Abstract

We study the expectation value of the $\mathrm{T}\overline{\mathrm{T}}$ operator in spacetimes with constant curvature. We define an diffeomorphism invariant biscalar whose coinciding limit gives the expectation value of the $\mathrm{T}\overline{\mathrm{T}}$ operator. We show that this biscalar is a constant in flat spacetime, which reproduces Zamolodchikov's result in 2004. For spacetimes with non-zero curvature, we show that this is no longer true and the expectation value of the $\mathrm{T}\overline{\mathrm{T}}$ operator depends on both the one-point and two-point functions of the stress-energy tensor.