Abstract
Cumrun Vafa [1] 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 = ± exp (πi/ν) as predicted in [1]. The emerging picture agrees with the other predictions of [1] 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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Bergamin, R., Cecotti, S. FQHE and tt* geometry. J. High Energ. Phys. 2019, 172 (2019). https://doi.org/10.1007/JHEP12(2019)172
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DOI: https://doi.org/10.1007/JHEP12(2019)172