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

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 H1(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 étale 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.