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.

中文翻译：

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$T \overline{T}$-变形模形式

共形场论的某些对象，例如矩形和圆环上的分区函数，以及圆环上的单点函数，要么是不变的，要么只是在模群下变换，这些属性应该保留在 $T \overline{ T}$变形。这个陈述的制定和证明实际上扩展到了更一般的函数，例如 $T \overline{T}$ 变形模和 Jacobi 形式。我们证明了变形只作用于他们的梅林变换，将它乘以一个通用的完整函数。最后我们证明了环面上的 Maass 形式是 $T \overline{T}$ 变形的特征函数。