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

TL;DR

This paper constructs the genus 2 superstring chiral measure using a 3D Gelca-Hamilton TQFT, linking modular transformations to the action of the extended mapping class group on 3D manifolds.

Contribution

It introduces a novel approach to derive the superstring chiral measure for genus 2 using a 3D TQFT framework, connecting path integrals with topological field theories.

Findings

Genus 2 superstring chiral measure obtained from 3D TQFT.

Modular transformations correspond to extended mapping class group actions.

Path integral formalism ensures modular invariance of the measure.

Abstract

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.