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–Yau abelian categories obtained by Kontsevich and Soibelman, and Joyce and Song, which are crucial in proving wall-crossing formulae for Donaldson–Thomas invariants. We expect our identity to be useful in extending motivic Donaldson–Thomas theory to general $(-1)$ -shifted symplectic stacks.