FQHE and $tt^{*}$ geometry

TL;DR

This paper explores the monodromy representation in Vafa's supersymmetric model of the FQHE, revealing its algebraic structure and developing new geometric concepts in $tt^*$ geometry with broad applications.

Contribution

It provides a detailed analysis of the monodromy representation in Vafa's model, showing it factors through a Temperley-Lieb/Hecke algebra, and introduces novel concepts in $tt^*$ geometry.

Findings

Monodromy representation factors through Temperley-Lieb/Hecke algebra

Consistent with Vafa's predictions on non-Abelian statistics

Develops new $tt^*$ geometric techniques

Abstract

Cumrun Vafa has proposed a microscopic description of the Fractional Quantum Hall Effect (FQHE) in terms of a many-body Hamiltonian $H$ invariant under four supersymmetries. The non-Abelian statistics of the defects (quasi-holes and quasi-particles) is then determined by the monodromy representation of the associated $tt^*$ geometry. In this paper we study the monodromy representation of the Vafa 4-susy model. Modulo some plausible assumption, we find that the monodromy representation factors through a Temperley-Lieb/Hecke algebra with $q=\pm\exp(\pi i/\nu)$. The emerging picture agrees with the other Vafa's predictions as well. The bulk of the paper is dedicated to the development of new concepts, ideas, and techniques in $tt^*$ geometry which are of independent interest. We present several examples of these geometric structures in various contexts.