Exponential map in DT theory

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.

Companion paper to arXiv:2409.16013. There is some overlap which will be removed in a new version. Comments welcome!