Motivic Joyce–Song formulas for Behrend function identities

Let $E_1$, $E_2$, and $E_1\oplus E_2$ be semi-Schur objects in the derived category of coherent sheaves over $Y$. Their extension space decomposes as

$$\operatorname{Ext}^1(E,E)=\operatorname{Ext}^1(E_1,E_1)\oplus\operatorname{Ext}^1(E_2,E_2)\oplus\operatorname{Ext}^1(E_1,E_2)\oplus\operatorname{Ext}^1(E_2,E_1).$$

Here ${\mathcal S}_0(E)$ denotes the motivic Milnor fiber, and $\int$ denotes pushforward of motives to ${\mathcal M}^{\hat{\mu}}_{\kappa}$. The motivic Joyce–Song conjecture. The two identities in the displayed statement hold: the motivic Milnor fiber of $E_1\oplus E_2$ satisfies the product formula, and the difference of the two projective extension-space integrals equals the corresponding difference of projective-space motives multiplied by the motivic Milnor fiber restricted to the direct sum of the self-extension spaces. These formulas are intended as the motivic refinement of the Joyce–Song Behrend-function identities and are used to establish that the integration map is a Poisson algebra homomorphism.