Torsion volume forms

TL;DR

This paper introduces a new approach to defining volume forms on mapping stacks in derived algebraic geometry, connecting torsion invariants, Todd classes, and symplectic forms, with applications to 3-manifold invariants.

Contribution

It develops a novel construction of volume forms on derived mapping stacks using parametrized Reidemeister-Turaev torsion and relates them to Todd classes and symplectic forms.

Findings

Volume forms on derived loop stacks expressed via Todd class.

Comparison of volume forms on mapping stacks from surfaces to symplectic forms.

Construction of canonical orientation data for cohomological DT invariants.

Abstract

We introduce volume forms on mapping stacks in derived algebraic geometry using a parametrized version of the Reidemeister-Turaev torsion. In the case of derived loop stacks we describe this volume form in terms of the Todd class. In the case of mapping stacks from surfaces, we compare it to the symplectic volume form. As an application of these ideas, we construct canonical orientation data for cohomological DT invariants of closed oriented 3-manifolds.