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arXiv · 2610.09922

Algebraic Proofs of Quantum Period--TBA Relations

Min-xin Huang
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Abstract

We give algebraic proofs of two conjectural relations between quantum $A$-periods and thermodynamic Bethe ansatz (TBA)-like equations for quantum mirror curves. The quadratic relation follows from constant terms of shifted product expansions. For higher-degree relations, a cyclic quantum-torus trace and a finite matrix determinant identify the period with the free energy of nonoverlapping rods. Their density equation yields the TBA-like equation, with normalization fixed by the rod length. Both proofs hold as formal power series in the complex structure parameter, with exact dependence on $q=e^{i\hbar}$.

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Min-xin Huang. 2026-10-07. Algebraic Proofs of Quantum Period--TBA Relations. https://arxiv.org/abs/2610.09922

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