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

Abstract

To any manifold equipped with a higher degree closed form, one can associate an L ∞ -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 ∞ -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 ∞ -cocycle that realizes the L ∞ -algebra of local observables as a KirillovKostant-Souriau-type L ∞ -extension of the Hamiltonian vector fields. When restricted along a Lie algebra action, this yields Heisenberg-like L ∞ -algebras such as the string Lie 2-algebra of a semisimple Lie algebra.

Lie n-algebras of BPS charges

“…We identify the correct higher current algebra for higher WZW-type Lagrangians as the higher Poisson bracket L ∞ -algebra of local observables considered in [32], for the pre-(p + 1)-plectic form being the higher WZW curvature form on target spacetime. In fact we find two different but quasi-isomorphic L ∞ -algebras of higher WZW-type Noether currents, following [17], corresponding to the two different incarnations of the Dickey bracket observed in [8, (3.10)-(3.11)].…”

“…We start by briefly reviewing the pertinent concepts and results from [17] in a way that makes manifest their relation to conserved currents. In order to see the origin of the higher symmetry in current algebras fully transparently, it is useful to adopt the following general perspective.…”

“…This is in the general spirit of refining field theories to what are called "extended" or "multi-tiered" field theories; we refer the reader to the introductions of [17,20] for more on this general background. The urge to refine Lie 1-algebras of infinitesimal symmetries to L ∞ -algebras becomes more pronounced as one aims to pass from infinitesimal to finite symmetries.…”

“…The algebras we discuss in this letter form a hierarchy of the following form (see [17,36] for discussion and references).…”

“…Whenever one is in such a situation where an algebraic structure exists on the homology of a chain complex, it is natural to ask whether a homotopy-theoretic refinement of the algebraic structure may exist on the chain complex itself, where algebraic conditions such as Jacobi identities may hold only up to higher coherent homotopies. For the case of Lie algebras this means lifting to L ∞ -algebras (see [17,36] for review and references), called Lie (p + 1)-algebras when the underlying chain complex is concentrated in degrees 0 up to p.…”