LAUSR

Abstract Let L be a cyclic L ∞ -algebra of dimension 3 with finite dimensional cohomology only in dimension one and two. By transfer theorem there exists a cyclic L ∞ -algebra structure on the cohomology H ⁎ ( L ) . The inner product plus the higher products of the cyclic L ∞ -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 ∞ -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 formula for the motivic Milnor fiber of cyclic L ∞ -algebras are also discussed.