Super-Lie n-algebra extensions, higher WZW models and super-p-branes with tensor multiplet fields

Abstract

We formalize higher-dimensional and higher gauge WZW-type sigma-model local prequantum field theory, and discuss its rationalized/perturbative description in (super-)Lie n-algebra homotopy theory (the true home of the “FDA”-language used in the\ud supergravity literature). We show generally how the intersection laws for such higher WZW-type σ-model branes (open brane ending on background brane) are encoded precisely in (super-)L∞-extension theory and how the resulting “extended (super-)spacetimes”\ud formalize spacetimes containing σ-model brane condensates. As an application we prove in Lie n-algebra homotopy theory that the complete super-p-brane spectrum of superstring/M-theory is realized this way, including the pure σ-model branes (the “old brane scan”) but also the branes with tensor multiplet worldvolume fields, notably\ud the D-branes and the M5-brane. For instance the degree-0 piece of the higher symmetry algebra of 11-dimensional (11D) spacetime with an M2-brane condensate turns out to be the “M-theory super-Lie algebra”. We also observe that in this formulation there is a\ud simple formal proof of the fact that type IIA spacetime with a D0-brane condensate is the 11D sugra/M-theory spacetime, and of (prequantum) S-duality for type IIB string theory. Finally we give the non-perturbative description of all this by higher WZWtype\ud σ-models on higher super-orbispaces with higher WZW terms in stacky differential cohomology

Lie n-algebras of BPS charges

“…Rather, in general it is only locally so, but is globally a (p + 1)-form connection on a "(p + 1)-bundle" or "p-gerbe". This is notably the case for field theories of Wess-Zumino-Witten type; see [21] for review of this fact and for pointers to the literature.…”

“…Now the proper global analysis of [2, p. 8] and of proposition 5.2 only ever produces extensions by charges of a single brane species with no (higher) gauge fields on its worldvolume. But in [21,22,39] we explained that the WZW-type sigma models for super p 2 -branes with (higher) gauge fields on the their worldvolume and on which super p 1 -branes may end, are globally defined not on target superspacetime X itself, but on the total space X of a super p 1 -stack extension X → X of superspacetime, which itself is a differential refinement of the p 1 -gerbeX that underlies the WZW term of the p 1 -branes. Moreover, in [16] we showed that the higher Heisenberg-Kostant-Souriau extensions of remark 2.11 generalizes to such higher stacky base spaces.…”

“…We note that all the previous discussion generalizes directly to the situation of superalgebra and supergeometry, where all manifolds are generalized to supermanifolds, differential forms to super-differential forms, and where Lie n-algebras are generalized to super Lie n-algebras (see [21] for details and pointers to the literature). This means that we may apply the previous results to supermanifolds X as they appear in higher supergravity theory, where they carry certain special super (p + 2)-forms which serve as the curvatures of WZW terms for the super p-brane sigma-models of Green-Schwarz type, with target space X.…”

“…is a super Lie algebra cocycle of the given form (see [21] for review and details). This is then the WZW-curvature term for the Green-Schwarz sigma model for super p-branes propagating on R d−1,1|N .…”

“…Key examples of higher-dimensional WZW models are the Green-Schwarz-type sigma models describing the worldvolume theory of all super p-branes [13,21,23]. Their extended current algebras have been argued in [2] to yield extensions of superisometry algebras of curved superspacetimes by brane charges [24,41].…”