Vortex counting and the quantum Hall effect

TL;DR

This paper explores conjectural dualities between nonrelativistic Chern-Simons-matter theories and quantum Hall fluids, analyzing vortex dynamics, geometric quantization, and vortex degeneracies to uncover novel quantum properties and dualities.

Contribution

It provides new evidence linking vortex moduli space quantization to quantum Hall states and proposes dualities involving nonrelativistic Chern-Simons theories and anyon composites.

Findings

Vortices behave as fermions in the lowest Landau level.

Euler characteristic computations relate to vortex Hilbert space dimension.

Degeneracy of Abelian vortices matches a q-analog of classical case.

Abstract

We provide evidence for conjectural dualities between nonrelativistic Chern-Simons-matter theories and theories of (fractional, nonAbelian) quantum Hall fluids in $2+1$ dimensions. At low temperatures, the dynamics of nonrelativistic Chern-Simons-matter theories can be described in terms of a nonrelativistic quantum mechanics of vortices. At critical coupling, this may be solved by geometric quantisation of the vortex moduli space. Using localisation techniques, we compute the Euler characteristic ${\chi}(\mathcal{L}^\lambda)$ of an arbitrary power $\lambda$ of a quantum line bundle $\mathcal{L}$ on the moduli space of vortices in $U(N_c)$ gauge theory with $N_f$ fundamental scalar flavours on an arbitrary closed Riemann surface. We conjecture that this is equal to the dimension of the Hilbert space of vortex states when the area of the metric on the spatial surface is sufficiently large. We find that the vortices in theories with $N_c = N_f = \lambda$ behave as fermions in the lowest nonAbelian Landau level, with strikingly simple quantum degeneracy. More generally, we find evidence that the quantum vortices may be regarded as composite objects, made of dual anyons. We comment on potential links between the dualities and three-dimensional mirror symmetry. We also compute the expected degeneracy of local Abelian vortices on the $\Omega$-deformed sphere, finding it to be a $q$-analog of the undeformed case.