Quantum Cohomology and Quantum Hydrodynamics from Supersymmetric Quiver Gauge Theories

TL;DR

This paper explores the deep connections between supersymmetric gauge theories, quantum cohomology, and integrable systems, revealing how gauge theory computations can determine spectra of quantum hydrodynamic models.

Contribution

It establishes a novel link between N=2 supersymmetric gauge theories on ALE spaces and the quantum cohomology of Nakajima quiver varieties, providing explicit Bethe Ansatz equations for integrable systems.

Findings

Gauge theories describe quantum cohomology of Nakajima's quiver varieties.

Exact BPS observable calculations yield spectra of quantum Hamiltonians.

Bethe Ansatz equations derived from gauge theories characterize integrable systems.

Abstract

We study the connection between N = 2 supersymmetric gauge theories, quantum cohomology and quantum integrable systems of hydrodynamic type. We consider gauge theories on ALE spaces of A and D-type and discuss how they describe the quantum cohomology of the corresponding Nakajima's quiver varieties. We also discuss how the exact evaluation of local BPS observables in the gauge theory can be used to calculate the spectrum of quantum Hamiltonians of spin Calogero integrable systems and spin Intermediate Long Wave hydrodynamics. This is explicitly obtained by a Bethe Ansatz Equation provided by the quiver gauge theory in terms of its adjacency matrix.