Laughlin states on higher genus Riemann surfaces

Abstract

Considering quantum Hall states on geometric backgrounds has proved over the past few years to be a useful tool for uncovering their less evident properties, such as gravitational and electromagnetic responses, topological phases and novel geometric adiabatic transport coefficients. One of the transport coefficients, the central charge associated with the gravitational anomaly, appears as a Chern number for the adiabatic transport on the moduli spaces of higher genus Riemann surfaces. This calls for a better understanding of the QH states on these backgrounds. Here we present a rigorous definition and give a detailed account of the construction of Laughlin states on Riemann surfaces of genus $\mathrm{g}>1$. By the first principles construction we prove that the dimension of the vector space of Laughlin states is at least ${\beta }^{\mathrm{g}}$ for the filling fraction $\nu =1/\beta$. Then using the path integral for the 2d bosonic field compactified on a circle, we reproduce the conjectured ${\beta }^{\mathrm{g}}$-degeneracy as the number of independent holomorphic blocks. We also discuss the lowest Landau level, integer QH state and its relation to the bosonization formulas on higher genus Riemann surfaces.

33 pages, 2 figures, v2: Fay's conventions for the $\sigma$-differential and the Arakelov metric are adopted, resulting in slight modifications of the affected formulas. Several other cosmetic changes and fixed typos, v3: further corrections, version to appear in Commun. Math. Phys

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• Genons, twist defects, and projective non-Abelian braiding statistics
• Framing Anomaly in the Effective Theory of Fractional Quantum Hall Effect
• Low-energy effective theory in the bulk for transport in a topological phase
• Density-curvature response and gravitational anomaly
• Topological central charge from Berry curvature: gravitational anomalies in trial wavefunctions for topological phases
• FQHE on curved backgrounds, free fields and large N
• Emergent Conformal Symmetry of Quantum Hall States on Singular surfaces
• Geometric Defects in Quantum Hall States
• Lowest Landau level on a cone and zeta determinants
• Quantum Hall States and Conformal Field Theory on a Singular Surface
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Cited by in corpus (6)

• Geometric responses of the Pfaffian state
• Integrable sigma models on Riemann surfaces
• Geometric test for topological states of matter
• Fractional Quantum Hall States on CP2 Space
• Liouville perturbation theory for Laughlin state and Coulomb gas
• Laughlin states change under large geometry deformations and imaginary time Hamiltonian dynamics