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

TL;DR

This paper explores the $T\overline T$ deformation in 2D quantum field theories, showing it acts like a random geometry transformation affecting partition functions, with potential extensions to higher dimensions.

Contribution

It provides a detailed analysis of the $T\overline T$ deformation as a random coordinate transformation and derives evolution equations for partition functions on various geometries.

Findings

Partition functions satisfy a linear diffusion-type equation.

Deformation acts as a random walk in moduli space.

Analysis extends to different topologies like torus, cylinder, and disk.

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.