Unimodular homotopy algebras and Chern-Simons theory

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.

37 pages, expanded introduction and made minor corrections

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• Combinatorics and formal geometry of the master equation
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• Unimodular L-infinity algebras
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Cited by in corpus (8)

• Quantum $L_{\infty }$ Algebras and the Homological Perturbation Lemma
• Minimal models of quantum homotopy Lie algebras via the BV-formalism
• On homotopy Lie bialgebroids
• Higher current algebras, homotopy Manin triples, and a rectilinear adelic complex
• Modular classes of Q-manifolds: a review and some applications
• Homotopy relative Rota-Baxter Lie algebras, triangular $L_{\infty }$-bialgebras and higher derived brackets
• Full S-matrices and Witten diagrams with (relative) L-infinity algebras
• The $L_{\infty }$-algebra of the S-matrix