Orientation Reversal and the Chern-Simons Natural Boundary

TL;DR

This paper introduces a resurgent analysis perspective on crossing natural boundaries in quantum field theory, revealing deeper rigidity and providing a numerical algorithm for generating dual $q$-series related to complex Chern-Simons invariants.

Contribution

It develops a new resurgent framework to analyze orientation reversal in Chern-Simons theory, enabling the computation of dual $q$-series beyond known mock theta functions.

Findings

Resurgence identifies Mordell integrals as primary objects.

A numerical algorithm generates dual $q$-series from Mordell integrals.

The approach extends known mock modular identities and reveals new structures.

Abstract

We show that the fundamental property of preservation of relations, underlying resurgent analysis, provides a new perspective on crossing a natural boundary, an important general problem in theoretical and mathematical physics. This reveals a deeper rigidity of resurgence in a quantum field theory. We study the non-perturbative completion of complex Chern-Simons theory that associates to a 3-manifold a collection of $q$-series invariants labeled by Spin$^c$ structures, for which crossing the natural boundary corresponds to orientation reversal of the 3-manifold. Our new resurgent perspective leads to a practical numerical algorithm that generates $q$-series which are dual to unary $q$-series composed of false theta functions. Until recently, these duals were only known in a limited number of cases, essentially based on Ramanujan's mock theta functions, and the common belief was that the more general duals might not even exist. Resurgence analysis identifies as primary objects Mordell integrals: transforms of resurgent functions. Their unique Borel summed transseries decomposition on either side of the Stokes line is the unique decomposition into real and imaginary parts. The latter are combinations of unary $q$-series in terms of $q$ and its modular counterpart $\tilde{q}$, and are resurgent by construction. The Mordell integral is analytic across the natural boundary of the $q$ and $\tilde{q}$ series, and uniqueness of a similar decomposition which preserves algebraic relations on the other side of the boundary defines the unique boundary crossing of the $q$ series. This continuation can be efficiently implemented numerically. This identifies known unique mock modular identities, and extends well beyond. The resurgent approach reveals new aspects, and is very different from other approaches based on indefinite theta series, Appell-Lerch sums, and logarithmic vertex operator algebras.