Wednesday, April 24, 2013

1304.6292 (Domenico Fiorenza et al.)

L-infinity algebras of local observables from higher prequantum bundles    [PDF]

Domenico Fiorenza, Christopher L. Rogers, Urs Schreiber
The second author has defined a class of L-infinity algebras that are naturally associated to manifolds equipped with a closed higher-degree differential form, which reduce to the underlying Lie algebra of the usual Poisson algebra in the case of symplectic manifolds. Here we show how these L-infinity algebras of local observables can be interpreted as the infinitesimal autoequivalences of higher prequantum bundles covering higher Hamiltonian symplectomorphisms. More precisely, we define a dg Lie algebra of such infinitesimal autoequivalences and exhibit an explicit homotopy equivalence between it and the L-infinity algebra of local observables. By truncation of the connection data for the prequantum bundle, this produces higher analogues of the Lie algebra of sections of the Atiyah Lie algebroid and the Lie 2-algebra of sections of the Courant Lie 2-algebroid. Finally we exhibit the L-infinity cocycle that realizes the local observables as a Kostant-Souriau-type L-infinity extension of the Hamiltonian vector fields. When restricted along a Lie algebra action, this yields L-infinity analogs of the Heisenberg Lie algebras and their classifying cocycles. In particular, this recovers the string Lie 2-algebra of a semisimple Lie algebra as well as the 3-cocycle that classifies it.
View original:

No comments:

Post a Comment