Genus 2 Superstring Chiral Measure From The 3-Dimensional Gelca-Hamilton TQFT

In the path integral formulation of the superstring, the chiral measure acquires a phase under the modular transformation of a Riemann surface. This motivated the use of anomaly inflow to define the superstring chiral measure by a path integral formalism of a modular invariant $3$-dimensional theory. A Gelca-Hamilton topological field theory (TQFT) is one of the Atiyah's TQFT on a $3$-dimensional extended manifold with the boundary Jacobi variety of a Riemann surface, whose Hilbert space is spanned by the theta series. We show that genus $g\leq 2$ superstring chiral measure in the path integral can be obtained by the path integral of the Gelca-Hamilton TQFT on some $3$-dimensional bulk extended manifolds. The modular transformation of the superstring chiral measure can be understood as the action of the extended mapping class group on the bulk $3$-dimensional extended manifolds.