$c_{\rm eff}$ from Resurgence at the Stokes Line

TL;DR

This paper explores how resurgence analysis on the Stokes line reveals the algebraic structure of resurgent cyclic orbits, determining the growth rate of dual q-series coefficients related to quantum field theories.

Contribution

It demonstrates that the algebraic structure of resurgent cyclic orbits and the leading term of q-series fully determine the asymptotic growth of coefficients, linking to effective central charge.

Findings

Resurgent cyclic orbits encode the large order growth of dual q-series coefficients.

The asymptotic growth exponent has a Cardy-like interpretation as an effective central charge.

Resurgence manifests as a deeper structure in quantum field theoretic path integrals.

Abstract

In recent papers [1,2], a new method to cross the natural boundary has been proposed, and applied to Mordell-Borel integrals arising in the study of Chern-Simons theory, based on decompositions into {\it resurgent cyclic orbits}. Resurgent analysis on the Stokes line leads to a unique transseries decomposition in terms of unary false theta functions, which can be continued across the natural boundary to produce dual $q$-series whose integer-valued coefficients enumerate BPS states. This constitutes a deeper new manifestation of resurgence in quantum field theoretic path integrals. In this paper we show that the algebraic structure of the {\it resurgent cyclic orbits}, combined with just the leading term of the $q$-series, completely determines the large order rate of growth of the dual $q$-series coefficients. The essential exponent of this asymptotic growth has a Cardy-like interpretation [10] of an effective central charge in a 3 dimensional quantum field theory with $\mathcal{N}=2$ supersymmetry related to the Chern-Simons theory through the $3d$-$3d$ correspondence.