Communications in Number Theory and Physics

Volume 16 (2022)

Number 3

$T \overline{T}$-deformed modular forms

Pages: 435 – 457



John Cardy (Department of Physics, University of California, Berkeley, Calif., U.S.A.; and All Souls College, Oxford, United Kingdom)


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.


modular forms, conformal field theory

2010 Mathematics Subject Classification

Primary 11F11, 11F50, 11M36. Secondary 81T40.

The author was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958, and by the Quantum Science Center (QSC) at the University of California, Berkeley, a National Quantum Information Science Research Center of the U.S. Department of Energy (DOE).

Received 15 January 2022

Accepted 20 February 2022

Published 4 October 2022