On the Duflo formula for $L_{\infty }$-algebras and Q-manifolds

Abstract

We prove a direct analogue of the classical Duflo formula in the case of $L_{\infty }$-algebras. We conjecture an analogous formula in the case of an arbitrary Q-manifold. When $G$ is a compact connected Lie group, the Duflo theorem for the Q-manifold $(\Pi TG,d_{DR})$ is exactly the Duflo theorem for the Lie algebra $g=LieG$. The corresponding theorem for the Q-manifold $(\Pi TM,d_{DR})$, where $M$ is an arbitrary smooth manifold, is a generalization of the Duflo theorem for the case of smooth manifolds. On the other hand, the Duflo theorem for the Q-manifold $(\Pi {\bar{T}}_{hol}M,\bar{\partial })$, where $M$ is a complex manifold, is a generalization of the M. Kontsevich's ``theorem on complex manifold'' [K1], Sect. 8.4.

11 pages, LaTeX2e