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

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.