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# Homology, Homotopy and Applications

## Volume 16 (2014)

### Number 2

### $L_{\infty}$-algebras of local observables from higher prequantum bundles

Pages: 107 – 142

DOI: http://dx.doi.org/10.4310/HHA.2014.v16.n2.a6

#### Authors

#### Abstract

To any manifold equipped with a higher degree closed form, one can associate an $L_\infty$-algebra of local observables that generalizes the Poisson algebra of a symplectic manifold. Here, by means of an explicit homotopy equivalence, we interpret this $L_\infty$-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 $L_\infty$-cocycle that realizes the $L_\infty$-algebra of local observables as a Kirillov-Kostant-Souriau-type $L_\infty$-extension of the Hamiltonian vector fields. When restricted along a Lie algebra action, this yields Heisenberg-like $L_\infty$-algebras such as the string Lie 2-algebra of a semisimple Lie algebra.

#### Keywords

geometric quantization, gerbes, homotopical algebra

#### 2010 Mathematics Subject Classification

18G55, 53C08, 53D50