Quantum Cohomology and Quantum Hydrodynamics from Supersymmetric Quiver Gauge Theories

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.

70 pages, 2 figures. Invited contribution to Journal of Geometry and Physics, special issue "Instanton Counting: Moduli Spaces, Representation Theory and Integrable Systems"

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• Defects and Quantum Seiberg-Witten Geometry
• Small Instantons, Little Strings and Free Fermions
• Bethe/Gauge correspondence on curved spaces
• Six-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N) Quantum Intermediate Long Wave Hydrodynamics
• On spectrum of ILW hierarchy in conformal field theory II: coset CFT's
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• Toric Calabi-Yau threefolds as quantum integrable systems. R-matrix and RTT relations
• Anomaly in RTT relation for DIM algebra and network matrix models
• A slow review of the AGT correspondence
• Supersymmetric localization in two dimensions
• Bands and gaps in Nekrasov partition function
• Elliptic non-Abelian Donaldson-Thomas invariants of ${\mathbb{C}}^3$
• On Elliptic Algebras and Large-n Supersymmetric Gauge Theories
• Nested coordinate Bethe wavefunctions from the Bethe/gauge correspondence
• The Condensate from Torus Knots
• Solution of quantum integrable systems from quiver gauge theories
• Bethe/Gauge Correspondence for $A_N$ Spin Chains with Integrable Boundaries