### 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 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.

Original language | English (US) |
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Article number | 1550018 |

Journal | International Journal of Geometric Methods in Modern Physics |

Volume | 12 |

Issue number | 2 |

DOIs | |

State | Published - Feb 16 2015 |

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### Keywords

- Lie n-algebras
- Lie superalgebras
- WZW model
- p-branes

### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)

### Cite this

*International Journal of Geometric Methods in Modern Physics*,

*12*(2), [1550018]. https://doi.org/10.1142/S0219887815500188