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

TL;DR

This work addresses the non-perturbative completion of the topological string dual to 2d U(N) Yang-Mills on a torus by leveraging resurgence theory. It derives closed-form instanton amplitudes to all orders, constructs a real non-perturbative partition function incorporating real instantons through Bell-polynomial resummations, and extends the analysis to complex instantons interpreted as BPS D-brane states, including wall-crossing behavior. The approach aligns the non-perturbative sector with holomorphic anomaly equations and provides boundary conditions in multiple frames (large-radius, conifold) to fix ambiguities. The results advance a precise finite-N duality between 2d YM on T^2 and topological string theory, with implications for OSV-type relations and potential generalizations to other gauge groups and Riemann surfaces.

Abstract

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.

Figures (3)

• Figure 4.1: Borel singularities of perturbative free energy in the large radius frame. We use perturbative series truncated to 200 terms, and use Padé approximant to mimic the analytic continuation of the Borel transform. The singular points (red) of the approximation would condense to branch cuts if the truncation is pushed to infinity. At $t=4\pi$ (a), the branch points (black) have charges $\pm (1,0,0), \pm (1,2,2), \pm (1,-2,2)$. At $t=\pi$ (b), the branch points (black) have charges $\pm (1,0,0)$.
• Figure 4.2: Borel singularities of perturbative free energy in the large radius frame. We use perturbative series truncated to 200 terms, and use Padé approximant to mimic the analytic continuation of the Borel transform. The singular points (red) of the approximation would condense to branch cuts if the truncation is pushed to infinity. At $t = 16\pi/3$$(a)$, the branch points (black) have charges $\pm(1,0,0)$, $\pm(1,2,2)$, $\pm(1,-2,2)$, $\pm(1,4,8)$, $\pm(1,-4,8)$. At $t=4\pi/3\,{\rm e}^{\pi{\mathsf{i}}/6}$$(b)$, the branch points (black) in the 1st and 3rd quadrants have charges $\pm(1,0,0)$, $\pm(1,2,2)$, and the branch points in the 2nd and 4th quadrants have charges $\pm(2,2,1)$.
• Figure 4.3: The moduli space and the possible wall of marginal stability of type IIA string compactified on $X_E$.