On W-algebras and ODE/IM correspondence

TL;DR

This paper develops a systematic method to compute eigenvalues of local integrals of motion in conformal field theories with W-algebra symmetry using WKB expansions, geometric structures, and applications to minimal models and topological strings.

Contribution

It introduces a new algorithm for calculating eigenvalues via WKB expansions and explores the geometric interpretation through mirror curves, connecting algebraic, geometric, and physical aspects.

Findings

Explicit formulas for eigenvalues of quantum KdV Hamiltonians for Virasoro, W3, and W4.

Identification of mirror curves as geometric structures underlying the algebraic properties.

Agreement between analytical and numerical results in Argyres-Douglas models and connection to topological string theory.

Abstract

We study the ODE/IM correspondence for two-dimensional conformal field theories with Virasoro and $\mathcal{W}_N$ symmetry. Building on earlier work establishing the correspondence, we develop a systematic algorithm for calculating the eigenvalues of local integrals of motion in terms of the Bethe roots using formal WKB expansions of wave functions associated to the differential operators. The method is demonstrated explicitly for Virasoro, $\mathcal{W}_3$, and $\mathcal{W}_4$ algebras, yielding closed expressions for the eigenvalues of the first few local quantum KdV Hamiltonians. A key geometric structure emerging from our analysis is the mirror curve, a three-punctured sphere that is naturally covered by the WKB curve. We show how the algebraic properties of the $\mathcal{W}$-symmetry algebras are reflected in the geometry of these curves, and how period integrals on these curves reproduce the spectral data of the integrable system. Applications to Argyres-Douglas minimal models allow us to test the prescription both analytically and numerically and we find complete agreement between the calculations in different triality frames. Finally, we examine large rank limits of ground state eigenvalues and show that they match the genus expansion of the topological string partition function on $\mathbb{C}^3$.