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Vortex counting and the quantum Hall effect

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Peer-reviewed

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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 χ(Lλ) of an arbitrary power λ of a quantum line bundle L on the moduli space of vortices in U(Nc) gauge theory with Nf 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 Nc=Nf=λ 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 Ω-deformed sphere, finding it to be a q-analog of the undeformed case.

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Keywords

Regular Article - Theoretical Physics, Chern-Simons Theories, Duality in Gauge Field Theories, Solitons Monopoles and Instantons, Differential and Algebraic Geometry

Journal Title

Journal of High Energy Physics

Conference Name

Journal ISSN

1126-6708
1029-8479

Volume Title

2022

Publisher

Springer Science and Business Media LLC