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

TL;DR

This paper extends the Duflo formula to $L_ ablafty$-algebras and Q-manifolds, connecting classical Lie theory with modern geometric structures and conjecturing broader generalizations.

Contribution

It proves a direct analogue of the Duflo formula for $L_ ablafty$-algebras and proposes a conjecture for arbitrary Q-manifolds, linking classical and modern geometric frameworks.

Findings

Duflo formula analogue for $L_ ablafty$-algebras proved

Duflo theorem for specific Q-manifolds connected to classical Lie theory

Generalizations to smooth and complex manifolds discussed

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 $g = Lie G$. 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.