  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 sigma-model branes (open brane ending on
background brane) are encoded precisely in (super-) L-infinity-extension theory
and how the resulting "extended (super-)spacetimes" formalize spacetimes
containing sigma 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 sigma-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 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 11-dimensional
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 sigma-models on higher super-orbispaces with higher WZW terms in
stacky differential cohomology.
