Strings, Symmetric Products, $T \bar{T}$ deformations and Hecke Operators

TL;DR

This paper derives a modular invariant formula for the torus partition sum of symmetric product $Tar T$ deformed conformal field theories, linking long string spectra, Hecke operators, and universality in $AdS_3$ contexts.

Contribution

It introduces a novel integral transform formula for the partition function involving Hecke operators, extending the understanding of $Tar T$ deformations in symmetric product CFTs.

Findings

Partition sum expressed as an integral transform with Hecke sums

Manifestly modular invariant formulation

Spectrum interpreted as a gas of multiply wound long strings

Abstract

We derive a formula for the torus partition sum of the symmetric product of $T\bar T$ deformed CFT's, using previous work on long strings in (deformed) $AdS_3$, and universality. The result is given by an integral transform of the partition function for the block of the symmetric product, summed over its Hecke transforms, and is manifestly modular invariant. The spectrum is interpretable as a gas of multiply wound long strings with a particular orientation.