The Thom-Sebastiani theorem for the Euler characteristic of cyclic L-infinity algebras

Let $L$ be a cyclic $L_{\mathrm{\infty }}$-algebra of dimension $3$ with finite dimensional cohomology only in dimension one and two. By transfer theorem there exists a cyclic $L_{\mathrm{\infty }}$-algebra structure on the cohomology $H^{\ast }(L)$. The inner product plus the higher products of the cyclic $L_{\mathrm{\infty }}$-algebra defines a superpotential function $f$ on $H^1(L)$. We associate with an analytic Milnor fiber for the formal function $f$ and define the Euler characteristic of $L$ is to be the Euler characteristic of the 'etale cohomology of the analytic Milnor fiber. In this paper we prove a Thom-Sebastiani type formula for the Euler characteristic of cyclic $L_{\mathrm{\infty }}$-algebras. As applications we prove the Joyce-Song formulas about the Behrend function identities for semi-Schur objects in the derived category of coherent sheaves over Calabi-Yau threefolds. A motivic Thom-Sebastiani type formula and a conjectural motivic Joyce-Song formulas for the motivic Milnor fiber of cyclic $L_{\mathrm{\infty }}$-algebras are also discussed.