  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.
