An analytic torsion for graded D-branes

TL;DR

This paper links the semiclassical partition function of graded Chern-Simons theories describing topological A-branes to a generalized analytic torsion, extending Ray-Singer invariants to graded superbundles in Calabi-Yau contexts.

Contribution

It introduces a new version of analytic torsion for flat graded superbundles and explores its metric independence, connecting topological invariants with D-brane physics.

Findings

Partition function expressed via a generalized analytic torsion.

Metric independence of the twisted Ray-Singer norm.

Reduction of torsion to classical Ray-Singer invariant in trivial connection case.

Abstract

I consider the semiclassical approximation of the graded Chern-Simons field theories describing certain systems of topological A type branes in the large radius limit of Calabi-Yau compactifications. I show that the semiclassical partition function can be expressed in terms of a certain (differential) numerical invariant which is a version of the analytic torsion of Ray and Singer, but associated with flat graded superbundles. I also discuss a `twisted' version of the Ray-Singer norm, and show its independence of metric data. As illustration, I consider graded D-brane pairs of unit relative grade with a scalar condensate in the boundary condition changing sector. For the particularly simple case when the reference flat connections are trivial, I show that the generalized torsion reduces to a power of the classical Ray-Singer invariant of the base 3-manifold.