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

TL;DR

This work extends the Zamolodchikov factorization of the $T\overline{T}$ operator to curved spacetimes with constant curvature by defining a diffeomorphism-invariant biscalar $\mathsf{C}(x,y)$ constructed from the stress-energy two-point function and the parallel propagator. In flat space (or at large central charge) the biscalar yields the familiar factorization $\langle T\overline{T}\rangle=\langle T\rangle\langle\overline{T}\rangle$, but in curved backgrounds the expectation value acquires a finite-$c$ correction given by a curvature-weighted integral of the connected two-point function projection $\mathsf{\Delta}_{\text{con}}(x,y)$. The authors derive a 2D differential equation for $\mathsf{C}(x,y)$ in maximally symmetric spacetimes, present an explicit integral formula for $\langle T\overline{T}\rangle$ in terms of geodesic distance and curvature, and illustrate the formalism with concrete sphere and Poincaré-disc coordinates. Overall, the paper reveals that curvature generally obstructs the exact solvability of the $T\overline{T}$ deformation at finite $c$, while providing a tractable curvature-dependent framework for computing the deformation’s finite-$c$ corrections and guiding further explorations in curved holography and related deformations.

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.