2021 journal article
Exact-WKB, complete resurgent structure, and mixed anomaly in quantum mechanics on S-1
JOURNAL OF HIGH ENERGY PHYSICS, (7).
author keywords: Discrete Symmetries; Nonperturbative Effects; Resummation; Solitons Monopoles and Instantons
open access via link.springer.com (publisher PDF)
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Abstract We investigate the exact-WKB analysis for quantum mechanics in a periodic potential, with N minima on S1. We describe the Stokes graphs of a general potential problem as a network of Airy-type or degenerate Weber-type building blocks, and provide a dictionary between the two. The two formulations are equivalent, but with their own pros and cons. Exact-WKB produces the quantization condition consistent with the known conjectures and mixed anomaly. The quantization condition for the case of N-minima on the circle factorizes over the Hilbert sub-spaces labeled by discrete theta angle (or Bloch momenta), and is consistent with ’t Hooft anomaly for even N and global inconsistency for odd N. By using Delabaere-Dillinger-Pham formula, we prove that the resurgent structure is closed in these Hilbert subspaces, built on discrete theta vacua, and by a transformation, this implies that fixed topological sectors (columns of resurgence triangle) are also closed under resurgence.