Exponential map in DT theory

TL;DR

This paper explores the exponential map in derived algebraic geometry, comparing additive and multiplicative structures in shifted symplectic stacks, and applies this to moduli spaces and nonabelian Hodge theory.

Contribution

It introduces an exponential map linking tangent and loop stacks in shifted symplectic geometry and applies it to moduli of 2-Calabi-Yau categories and nonabelian Hodge theory.

Findings

Established an exponential map preserving shifted symplectic structures.

Proved a loop dimensional reduction theorem for certain moduli spaces.

Developed a loop version of nonabelian Hodge theory for $ ext{GL}_n$ stacks.

Abstract

This paper studies the Cohomological Donaldson-Thomas theory of loop stacks of $0$-shifted symplectic stacks. In particular, we compare $(-1)$-shifted tangent stacks of these moduli problems, which we view as additive, to loop stacks, which we view as multiplicative, via an exponential map that preserves induced $(-1)$-shifted symplectic structures. As an application, we prove for certain moduli of objects of $2$-Calabi-Yau categories a loop dimensional reduction theorem for the loop stacks of these moduli spaces. Finally, we prove a loop version of nonabelian Hodge theory for stacks in the $\mathrm{GL}_n$ case.