Cumrun Vafa proposed a new unifying model for the principal series of FQHE
which predicts non-Abelian statistics of the quasi-holes. The many-body
Hamiltonian supporting these topological phases of matter is invariant under
four supersymmetries. In the thesis we study the geometrical properties of this
Landau-Ginzburg theory. The emerging picture is in agreement with the Vafa's
predictions. The 4-SQM Vafa Hamiltonian is shown to capture the topological
order of FQHE and the tt∗ monodromy representation of the braid group
factors through a Temperley-Lieb/Hecke algebra with q=±exp(πi/ν).
In particular, the quasi-holes have the same non-Abelian braiding properties of
the degenerate field ϕ1,2 in Virasoro minimal models. Part of the
thesis is dedicated to minor results about the geometrical properties of the
Vafa model for the case of a single electron. In particular, we study a special
class of models which reveal a beautiful connection between the physics of
quantum Hall effect and the geometry of modular curves. Despite it is not
relevant for phenomenological purposes, this class of theories has remarkable
properties which enlarge further the rich mathematical structure of FQHE.Comment: Doctoral thesis, SISSA, Trieste, Italy (2019