ADHM and the 4d Quantum Hall Effect

TL;DR

This paper explores 4d quantum Hall states derived from Yang-Mills instantons, linking their excitations to supersymmetric gauge theory partition functions and extending mirror symmetry techniques to include Chern-Simons effects.

Contribution

It constructs quantum Hall states from Yang-Mills instantons and relates their excitations to Nekrasov partition functions, extending mirror symmetry methods to five-dimensional Chern-Simons theories.

Findings

Ground state wavefunctions for Abelian and non-Abelian states derived.

Partition functions interpreted as counting holomorphic sections of instanton moduli space.

Coulomb and Higgs branch formulas shown to coincide, reproducing Nekrasov partition functions.

Abstract

Yang-Mills instantons are solitonic particles in d=4+1 dimensional gauge theories. We construct and analyse the quantum Hall states that arise when these particles are restricted to the lowest Landau level. We describe the ground state wavefunctions for both Abelian and non-Abelian quantum Hall states. Although our model is purely bosonic, we show that the excitations of this 4d quantum Hall state are governed by the Nekrasov partition function of a certain five dimensional supersymmetric gauge theory with Chern-Simons term. The partition function can also be interpreted as a variant of the Hilbert series of the instanton moduli space, counting holomorphic sections rather than holomorphic functions. It is known that the Hilbert series of the instanton moduli space can be rewritten using mirror symmetry of 3d gauge theories in terms of Coulomb branch variables. We generalise this approach to include the effect of a five dimensional Chern-Simons term. We demonstrate that the resulting Coulomb branch formula coincides with the corresponding Higgs branch Molien integral which, in turn, reproduces the standard formula for the Nekrasov partition function.