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

TL;DR

This paper establishes a Thom-Sebastiani type formula for the Euler characteristic of cyclic L-infinity algebras, linking algebraic structures to geometric invariants and applying it to derived categories of Calabi-Yau threefolds.

Contribution

It proves a new Thom-Sebastiani formula for the Euler characteristic of cyclic L-infinity algebras and applies it to Joyce-Song formulas in Calabi-Yau geometry.

Findings

Proved a Thom-Sebastiani type formula for Euler characteristics.

Applied the formula to Joyce-Song identities for semi-Schur objects.

Discussed motivic versions and conjectures related to the formula.

Abstract

Let $L$ be a cyclic $L_\infty$-algebra of dimension $3$ with finite dimensional cohomology only in dimension one and two. By transfer theorem there exists a cyclic $L_\infty$-algebra structure on the cohomology $H^*(L)$. The inner product plus the higher products of the cyclic $L_\infty$-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 $L_\infty$-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 $L_\infty$-algebras are also discussed.