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

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.

Keywords

modular forms, conformal field theory

2010 Mathematics Subject Classification

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

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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