Lie n-algebras of BPS charges

Abstract

We uncover higher algebraic structures on Noether currents and BPS charges. It is known that equivalence classes of conserved currents form a Lie algebra. We show that at least for target space symmetries of higher parameterized WZW-type sigma-models this naturally lifts to a Lie (p + 1)-algebra structure on the Noether currents themselves. Applied to the Green-Schwarz-type action functionals for super p-brane sigma-models this yields super Lie (p + 1)-algebra refinements of the traditional BPS brane charge extensions of supersymmetry algebras. We discuss this in the generality of higher differential geometry, where it applies also to branes with (higher) gauge fields on their worldvolume. Applied to the M5-brane sigma-model we recover and properly globalize the M-theory super Lie algebra extension of 11-dimensional superisometries by 2-brane and 5-brane charges. Passing beyond the infinitesimal Lie theory we find cohomological corrections to these charges in higher analogy to the familiar corrections for D-brane charges as they are lifted from ordinary cohomology to twisted K-theory. This supports the proposal that M-brane charges live in a twisted cohomology theory.

More on the hidden symmetries of 11D supergravity

“…The DF-algebra of [2] raised recently a certain interest in the mathematical-physicists community, due to the fact that it can be reformulated in terms of L n ⊂ L ∞ algebras, or strong homotopy Lie algebras. A comprehensive reference to this approach can be found in [25,26].…”

Supersymmetric minisuperspace models in self-dual loop quantum cosmology

“…This demonstrates that the existence of the graded analog of Ashtekar's connection in N = 1 supergravity is not just a mere coincidence. Since extended supersymmetry also involves higher gauge fields [23,24], this may, among other things, provide important insights for quantizing higher gauge fields in the framework of loop quantum gravity.…”

Brane current algebras and generalised geometry from QP manifolds. Or, “when they go high, we go low”

“…The relation must necessarily be somewhat indirect, because the current algebra of phase space currents with the Poisson brackets (1.12) is a Lie algebra, while the L ∞algebra of the tensor hierarchy produced by the QP-structure will be a Lie P -algebra, so it will in general have non-vanishing brackets of up to P + 1 arguments. A natural conjecture is that the relation is through the transgression (of Lie P -algebras to Lie algebras) of Sati and Schreiber [71]: roughly speaking, they produce a Lie algebra from a Lie P -algebra by considering the space of embeddings of a p-brane worldvolume onto a geometry. This is of course strongly reminiscent of our brane phase space construction.…”

“…This is of course strongly reminiscent of our brane phase space construction. A technical difference is that our n-form currents (1.10) are integrated against test (p − n)-forms to produce an integral over the whole worldspace Σ, whereas those of [71] are not. It is in the latter formulation that Poisson brackets in field theory and L ∞ -algebras were originally connected (by Barnich, Fulp, Lada, and Stasheff [72]; see also [73]).…”