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L∞algebra connections and applications to String and ChernSimons ntransport
, 2008
"... We give a generalization of the notion of a CartanEhresmann connection from Lie algebras to L∞algebras and use it to study the obstruction theory of lifts through higher Stringlike extensions of Lie algebras. We find (generalized) ChernSimons and BFtheory functionals this way and describe aspect ..."
Abstract

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We give a generalization of the notion of a CartanEhresmann connection from Lie algebras to L∞algebras and use it to study the obstruction theory of lifts through higher Stringlike extensions of Lie algebras. We find (generalized) ChernSimons and BFtheory functionals this way and describe aspects of their parallel transport and quantization. It is known that over a Dbrane the KalbRamond background field of the string restricts to a 2bundle with connection (a gerbe) which can be seen as the obstruction to lifting the P U(H)bundle on the Dbrane to a U(H)bundle. We discuss how this phenomenon generalizes from the ordinary central extension U(1) → U(H) → P U(H) to higher categorical central extensions, like the Stringextension BU(1) → String(G) → G. Here the obstruction to the lift is a 3bundle with connection (a 2gerbe): the ChernSimons 3bundle classified by the first Pontrjagin class. For G = Spin(n) this obstructs the existence of a Stringstructure. We discuss how to describe this obstruction problem in terms of Lie nalgebras and their corresponding categorified CartanEhresmann connections. Generalizations even beyond Stringextensions are then straightforward. For G = Spin(n) the next step is “Fivebrane structures” whose existence is obstructed by certain generalized ChernSimons 7bundles classified by the second Pontrjagin class.