The $T\overline T$ deformation of quantum field theory as random geometry

TL;DR

The paper shows that the two-dimensional $Tar T$ deformation, when viewed as coupling to a random metric via a Hubbard–Stratonovich transformation, reduces to a topological bulk action whose nontrivial contributions come from boundaries or topology. This leads to linear diffusion-type PDEs for partition functions on manifolds such as the torus, cylinder, and disk, effectively describing a stochastic evolution in the moduli space of domain shapes. By connecting two derivations (Gaussian integration and saddle-point analysis) and aligning with Zamolodchikov’s spectrum flow, the work derives explicit evolution equations and analyzes their solutions, including Burgers-type dynamics for energy levels and exact thermodynamic expressions in certain cases. The framework also extends to domains with boundaries and polygonal domains, interprets the results as stochastic boundary dynamics, and discusses possible generalizations to higher dimensions and nonlocal deformations, as well as implications for entanglement and curvature-driven processes.

Abstract

We revisit the results of Zamolodchikov and others on the deformation of two-dimensional quantum field theory by the determinant $\det T$ of the stress tensor, commonly referred to as $T\overline T$. Infinitesimally this is equivalent to a random coordinate transformation, with a local action which is, however, a total derivative and therefore gives a contribution only from boundaries or nontrivial topology. We discuss in detail the examples of a torus, a finite cylinder, a disk and a more general simply connected domain. In all cases the partition function evolves according to a linear diffusion-type equation, and the deformation may be viewed as a kind of random walk in moduli space. We also discuss possible generalizations to higher dimensions.