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.
Highlights
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 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
Summary
Our goal is to describe the dynamics of Yang-Mills instantons in a background magnetic field, and determine the resulting quantum Hall wavefunctions governing their ground state. We’re not interested in including any fermions so supersymmetry provides little motivation After an integration by parts, this can be written as a 3d Chern-Simons form, coupled to a constant background magnetic field ω = Bη3, Smag =. Such a term arises in the “Kahler Chern-Simons theory” of Nair and Schiff [40, 41]. As we will see, there is an elegant description of the dynamics using the ADHM construction
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