Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence

TL;DR

This paper introduces the concept of quantum spectral curves, establishing their role in quantum integrable systems and the geometric Langlands correspondence, with applications to quantum Hamiltonians, affine algebras, and the KZ-equation.

Contribution

It develops a construction of quantum spectral curves and demonstrates their unifying role in quantum integrable systems and the geometric Langlands program, providing explicit formulas and conjectures.

Findings

Explicit description of maximal commutative subalgebras in quantum groups

Connection between quantum spectral curves and the KZ-equation

Construction of the Langlands correspondence via quantum spectral curves

Abstract

The spectral curve is the key ingredient in the modern theory of classical integrable systems. We develop a construction of the ``quantum spectral curve'' and argue that it takes the analogous structural and unifying role on the quantum level also. In the simplest, but essential case the ``quantum spectral curve'' is given by the formula "det"(L(z)-dz) [Talalaev04] (hep-th/0404153). As an easy application of our constructions we obtain the following: quite a universal receipt to define quantum commuting hamiltonians from the classical ones, in particular an explicit description of a maximal commutative subalgebra in U(gl(n)[t])/t^N and in U(\g[t^{-1}])\otimes U(t\g[t]); its relation with the center on the of the affine algebra; an explicit formula for the center generators and a conjecture on W-algebra generators; a receipt to obtain the q-deformation of these results; the simple and explicit construction of the Langlands correspondence; the relation between the ``quantum spectral curve'' and the Knizhnik-Zamolodchikov equation; new generalizations of the KZ-equation; the conjecture on rationality of the solutions of the KZ-equation for special values of level. In the simplest cases we observe the coincidence of the ``quantum spectral curve'' and the so-called Baxter equation. Connection with the KZ-equation offers a new powerful way to construct the Baxter's Q-operator.