L-infinity algebras of local observables from higher prequantum bundles

TL;DR

This paper develops a framework linking L-infinity algebras of local observables on manifolds with higher prequantum bundles, providing explicit homotopy equivalences and extensions that generalize classical symplectic structures.

Contribution

It introduces a homotopy-theoretic interpretation of local observable L-infinity algebras via higher prequantum bundles and constructs new algebraic structures generalizing classical Lie algebroids.

Findings

Explicit homotopy equivalence between L-infinity algebras and autoequivalences of prequantum bundles

Construction of higher Lie algebra analogues from connection data truncation

Identification of L-infinity cocycles as Kirillov-Kostant-Souriau extensions

Abstract

To any manifold equipped with a higher degree closed form, one can associate an L-infinity algebra of local observables that generalizes the Poisson algebra of a symplectic manifold. Here, by means of an explicit homotopy equivalence, we interpret this L-infinity algebra in terms of infinitesimal autoequivalences of higher prequantum bundles. By truncating the connection data on the prequantum bundle, we produce analogues of the (higher) Lie algebras of sections of the Atiyah Lie algebroid and of the Courant Lie 2-algebroid. We also exhibit the L-infinity cocycle that realizes the L-infinity algebra of local observables as a Kirillov-Kostant-Souriau-type L-infinity extension of the Hamiltonian vector fields. When restricted along a Lie algebra action, this yields Heisenberg-like L-infinity algebras such as the string Lie 2-algebra of a semisimple Lie algebra.