Quantum curves

Abstract

One says that a pair (P,Q) of ordinary differential operators specify a quantum curve if [P,Q]=const. If a pair of difference operators (K,L) obey the relation KL=const LK we say that they specify a discrete quantum curve. This terminology is prompted by well known results about commuting differential and difference operators, relating pairs of such operators with pairs of meromorphic functions on algebraic curves obeying some conditions. The goal of this paper is to study the moduli spaces of quantum curves. We will show how to quantize a pair of commuting differential or difference operators (i.e. to construct the corresponding quantum curve or discrete quantum curve). The KP-hierarchy acts on the moduli space of quantum curves; we prove that similarly the discrete KP-hierarchy acts on the moduli space of discrete quantum curves.

New results, some corrections

References in corpus (2)

• Remodeling the B-model
• Quantum curves for Hitchin fibrations and the Eynard-Orantin theory

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• Quantization of classical curves
• $q$-Difference Kac-Schwarz Operators in Topological String Theory
• Spectral curve duality beyond the two-matrix model