Resurgence of $T\bar{T}$-deformed Partition Function

TL;DR

This paper investigates non-perturbative effects in the $T\bar{T}$-deformed 2D CFT partition functions using resurgence, revealing asymptotic series behavior and proposing new complex saddle points as non-perturbative contributions.

Contribution

The study develops efficient methods to compute high-order perturbative coefficients and demonstrates the non-Borel resummability of the series, providing a resurgence-based framework for non-perturbative effects.

Findings

Perturbative series is asymptotic and not Borel resummable.

Non-perturbative contributions originate from complex saddle points.

Numerical data matches resurgence predictions.

Abstract

We study non-perturbative effects of torus partition function of the $T\bar{T}$-deformed 2d CFTs by resurgence. The deformed partition function can be written as an infinite series of the deformation parameter $\lambda$. We develop highly efficient methods to compute perturbative coefficients in the $\lambda$ expansion. To exemplify, the first 600 coefficients for the $T\bar{T}$-deformed free boson and free fermion are computed. Equipped with the large order perturbative data, we provide convincing numerical evidence that the $\lambda$ expansion series is asymptotic and not Borel resummable. We extract the non-perturbative contribution by resurgence and propose that they originate from new complex saddle points after analytically continuing the modular parameters in the integral representation of the partition function. The proposal is checked by comparing the predicted asymptotic behavior of the coefficients and large order perturbative data, which match nicely. The implications of these non-perturbative contributions for the Stokes phenomenon, which relates the positive and negative signs of $\lambda$, is also discussed.