Unimodular homotopy algebras and Chern-Simons theory

TL;DR

This paper explores the homological algebra framework of quantum Chern-Simons invariants, establishing conditions for L-infinity algebra structures to admit quantum lifts and introducing a doubling construction connecting unimodular and cyclic L-infinity algebras.

Contribution

It formulates criteria for quantum lifts of L-infinity algebras associated with manifolds and introduces a novel doubling construction linking unimodular and cyclic structures.

Findings

Conditions for L-infinity algebra quantum lifts

Structural results on unimodular L-infinity algebras

A doubling construction linking unimodular and cyclic L-infinity algebras

Abstract

Quantum Chern-Simons invariants of differentiable manifolds are analyzed from the point of view of homological algebra. Given a manifold M and a Lie (or, more generally, an L-infinity) algebra g, the vector space H^*(M) \otimes g has the structure of an L-infinity algebra whose homotopy type is a homotopy invariant of M. We formulate necessary and sufficient conditions for this L-infinity algebra to have a quantum lift. We also obtain structural results on unimodular L-infinity algebras and introduce a doubling construction which links unimodular and cyclic L-infinity algebras.