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L$_\infty$-algebras of observables from higher prequantum line bundles

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Domenico Fiorenza
Mit, 10/04/2013 - 10:30 - 12:00
MPIM Lecture Hall

Rogers has defined a class of L$_\infty$-algebras that are naturally associated with manifolds equipped with closed higher-degree forms, and that reduce to Poisson bracket Lie algebras in the case of symplectic manifolds. Here we show that these L$_\infty$-algebras can be naturally identified with the L$_\infty$-algebras of infinitesimal autoequivalences of higher prequantum bundles. In particular, they are a Kostant-Souriau-type L$_\infty$ extension of (higher) Hamiltonian symplectomorphisms. By truncation of the connection data for the prequantum bundle, this produces higher analogues of the Lie algebra of sections of the Atiyah algebroid and of the Lie 2-algebra of sections of the Courant Lie 2-algebroid. Restriction along higher moment maps yields L$_\infty$ analogs of the Heisenberg Lie algebra and of the string Lie 2-algebra. Joint work with Christopher Rogers and Urs Schreiber.

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