Resurgent structure of 2d Yang-Mills theory on a torus

We study the resurgent structure of the topological string dual to 2d $U(N)$ Yang-Mills on torus. We find closed form formulas for instanton amplitudes up to arbitrarily high instanton orders, based on which we propose the non-perturbative partition function including contributions from all the real instantons, which is real for positive modulus and string coupling. We also explore complex instantons and find two infinite towers of them. We expect them to correspond to BPS states in type II string.

💡 Research Summary

This paper investigates the resurgence structure of the topological‑string theory that is dual to two‑dimensional U(N) Yang–Mills theory on a torus. Starting from the well‑known factorisation of the Yang–Mills partition function into chiral and anti‑chiral pieces, the authors recall that the chiral sector coincides with the perturbative topological‑string free energy on the elliptic Calabi–Yau threefold given by the total space of O(1)⊕O(−1) over the torus. They review the modular properties of the perturbative free energies, the BCOV holomorphic anomaly equations and the existing non‑perturbative proposals (notably the Okuyama‑Sakai construction).

The core of the work is the systematic construction of instanton amplitudes using resurgence theory. By analysing the Stokes phenomenon of the asymptotic perturbative series, the authors derive closed‑form expressions for single‑instanton contributions that agree with previous results, and, more importantly, they obtain recursive closed formulas for multi‑instanton amplitudes valid to arbitrary instanton order. These amplitudes are functionals of the perturbative partition function with discrete shifts of the Kähler modulus, precisely the effect expected from D‑brane insertions. The associated Stokes constants turn out to be integers, interpreted as BPS multiplicities.

With these ingredients the authors propose a non‑perturbative partition function (eq. 4.93) that sums over all real instantons. Although an infinite family of real parameters appears, they argue that a canonical choice (eq. 4.96) yields a unique, real‑valued partition function for positive modulus and string coupling. This construction resolves the inconsistencies of earlier proposals, especially in the presence of a non‑zero θ‑angle.

In addition to real instantons, the paper explores complex instanton sectors. Two infinite towers of complex instantons are identified; they do not affect the real‑instanton non‑perturbative completion but are expected to correspond to bound states of D‑branes with complex central charges in type IIA string theory. The authors discuss the wall‑crossing behaviour of these sectors and how the BPS spectrum reorganises under modular transformations.

Overall, the study demonstrates that resurgence theory provides a powerful framework to obtain the full non‑perturbative completion of the topological‑string dual of 2d Yang–Mills on a torus, clarifies the role of instantons as BPS D‑brane states, and sets the stage for extending these methods to more general backgrounds and higher‑genus surfaces.