A motivic integral identity for $(-1)$-shifted symplectic stacks

TL;DR

This paper establishes a motivic integral identity for $(-1)$-shifted symplectic stacks, extending known identities and aiming to facilitate the development of motivic Donaldson-Thomas theory in this broader context.

Contribution

It proves a new motivic integral identity for $(-1)$-shifted symplectic stacks, generalizing previous identities for moduli stacks in Calabi-Yau categories.

Findings

Relates motivic Behrend functions of $(-1)$-shifted symplectic stacks and their graded points

Generalizes identities used in wall-crossing formulas for Donaldson-Thomas invariants

Provides a foundation for extending motivic Donaldson-Thomas theory

Abstract

We prove a motivic integral identity relating the motivic Behrend function of a $(-1)$-shifted symplectic stack to that of its stack of graded points. This generalizes analogous identities for moduli stacks of objects in $3$-Calabi$\unicode{x2013}$Yau abelian categories obtained by Kontsevich$\unicode{x2013}$Soibelman and Joyce$\unicode{x2013}$Song, which are crucial in proving wall-crossing formulae for Donaldson$\unicode{x2013}$Thomas invariants. We expect our identity to be useful in extending motivic Donaldson$\unicode{x2013}$Thomas theory to general $(-1)$-shifted symplectic stacks.