$T\overline T$-deformed modular forms

TL;DR

This paper investigates how $T\overline T$ deformation affects modular and Jacobi forms, showing that the deformation acts simply on their Mellin transforms and that Maass forms are eigenfunctions of this deformation.

Contribution

It extends the understanding of $T\overline T$ deformation to modular and Jacobi forms, demonstrating their transformation properties and eigenfunction behavior under the deformation.

Findings

Deformation acts as multiplication by a universal entire function on Mellin transforms.

Partition functions and one-point functions on the torus preserve modular properties under deformation.

Maass forms are eigenfunctions of the $T\overline T$ deformation.

Abstract

Certain objects of conformal field theory, for example partition functions on the rectangle and the torus, and one-point functions on the torus, are either invariant or transform simply under the modular group, properties which should be preserved under the $T\overline T$ deformation. The formulation and proof of this statement in fact extents to more general functions such as $T\overline T$ deformed modular and Jacobi forms. We show that the deformation acts simply on their Mellin transform, multiplying it by a universal entire function. Finally we show that Maass forms on the torus are eigenfunctions of the $T\overline T$ deformation.