Schreiber L-∞ algebras of local observables from higher prequantum bundles

An article that we have written:

• Domenico Fiorenza, Chris Rogers, Urs Schreiber

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

  Homology, Homotopy and Applications,

  Volume 16 (2014) Number 2, p. 107-142

  (arXiv:1304.6292, HHA.2014.v16.n2.a6)

  Lecture notes:

  Domenico Fiorenza, Higher geometric prequantum theory II: $L_\infty$-algebras of local observables, talk at Bayrischzell workshop 2013 (pdf slides)

on L-∞ algebraic aspects of the generalization of prequantum field theory to higher geometry.

The companion article

• Higher geometric prequantum theory

provides more background and context. The followup

• Classical field theory via Cohesive homotopy types

shows how classical field theory in its local de Donder-Weyl theory-formulation is naturally formulated in terms of these higher $L_\infty$-algebras of observables.

Abstract

The second author has defined a class of L-∞ algebras that are naturally associated with manifolds equipped with closed higher-degree differential forms, and that reduce to Poisson bracket Lie algebras in the case of symplectic manifolds. Here we demonstrate that these L-∞ algebras of local observables are the infinitesimal autoequivalences of higher prequantum bundles covering Hamiltonian symplectomorphisms. Hence, these algebras fit into a robust theory of higher geometric prequantization. We exhibit an explicit dg-Lie algebra model of such infinitesimal autoequivalences and a homotopy equivalence of this with the L-∞ algebra of local observables. By truncation of the connection data for the higher prequantum bundle, this produces higher analogues of the Lie algebra of sections of the Atiyah Lie algebroid and of the Lie 2-algebra of sections of the Courant Lie 2-algebroid. Finally we exhibit the L-∞ cocycle that realizes the L-∞ algebras of local observables as Kostant-Souriau-type $L_\infty$-extension of the Hamiltonian vector fields. When restricted along a Hamiltonian action this yields $L_\infty$-analogs of the Heisenberg Lie algebras and of their classifying cocycle, in particular recovering the string Lie 2-algebra of a semisimple Lie algebra as well as the 3-cocycle that classifies it.

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