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arXiv · 1910.05022

FQHE and $tt^{*}$ geometry

Riccardo Bergamin·Sergio Cecotti
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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.

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Riccardo Bergamin, Sergio Cecotti. 2019-10-11. FQHE and $tt^{*}$ geometry. https://doi.org/10.1007/jhep12(2019)172

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