Narrow Results

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 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-)space-times" 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 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 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 WZW-type σ-models on higher super-orbispaces with higher WZW terms in stacky differential cohomology.

We give a review of truncated L_∞ algebras, as used in the study of higher gauge theory. These structures are believed to hold the correct properties to adequately describe gauge theory of extended objects. We discuss how to construct topological higher-gauge-invariant theories and how their solutions relate to multisymplectic geometries. We also show how Courant algebroids fit into this formalism, so as to be able to study higher gauge theory on generalized geometric bundles, i.e. on $T\Sigma \oplus T^{*}\Sigma$, for some space-time Σ. We will see that via this formalism we can match and explain a recently proposed M5-brane model, arrived at in a more heuristic way, whose field content seemed difficult to interpret but finds a natural motivation in this framework.