We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group String(n). A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the 'Jacobiator.' Similarly, a Lie 2-group is a categorified version of a Lie group. If G is a simply-connected compact simple Lie group, there is a 1-parameter family of Lie 2-algebras g k each having g as its Lie algebra of objects, but with a Jacobiator built from the canonical 3-form on G. There appears to be no Lie 2-group having g k as its Lie 2-algebra, except when k = 0. Here, however, we construct for integral k an infinite-dimensional Lie 2-group P k G whose Lie 2-algebra is equivalent to g k . The objects of P k G are based paths in G, while the automorphisms of any object form the level-k Kac-Moody central extension of the loop group ΩG. This 2-group is closely related to the kth power of the canonical gerbe over G. Its nerve gives a topological group |P k G| that is an extension of G by K(Z, 2). When k = ±1, |P k G| can also be obtained by killing the third homotopy group of G. Thus, when G = Spin(n), |P k G| is none other than String(n).

Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1-dimensional objects (e.g. strings) using 2-connections on 2-bundles. A 2-bundle is a categorified version of a bundle: that is, one where the fiber is not a manifold but a category with a suitable smooth structure. Where gauge theory uses Lie groups and Lie algebras, higher gauge theory uses their categorified analogues: Lie 2-groups and Lie 2algebras. We describe a theory of 2-connections on principal 2-bundles and explain how this is related to Breen and Messing's theory of connections on nonabelian gerbes. The distinctive feature of our theory is that a 2-connection allows parallel transport along paths and surfaces in a parametrization-independent way. In terms of Breen and Messing's framework, this requires that the 'fake curvature' must vanish. In this paper we summarize the main results of our theory without proofs.

Abstract. In the background effective field theory of heterotic string theory, the Green-Schwarz anomaly cancellation mechanism plays a key role. Here we reinterpret it and its magnetic dual version in terms of, differential twisted String-and differential twisted Fivebrane-structures that generalize the notion of Spinstructures and Spin-lifting gerbes and their differential refinement to smooth Spin-connections. We show that all these structures can be encoded in terms of nonabelian cohomology, twisted nonabelian cohomology, and differential twisted nonabelian cohomology, extending the differential generalized abelian cohomology as developed by Hopkins and Singer and shown by Freed to formalize the global description of anomaly cancellation problems in higher gauge theories arising in string theory. We demonstrate that the GreenSchwarz mechanism for the H 3 -field, as well as its magnetic dual version for the H 7 -field define cocycles in differential twisted nonabelian cohomology that may be called, respectively, differential twisted Spin(n)-, String(n)-and Fivebrane(n)-structures on target space, where the twist in each case is provided by the obstruction to lifting the classifying map of the gauge bundle through a higher connected cover of U (n) or O(n). We show that the twisted Bianchi identities in string theory can be captured by the (nonabelian) L∞-algebra valued differential form data provided by the differential refinements of these twisted cocycles.

We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L∞algebras and use it to study the obstruction theory of lifts through higher String-like extensions of Lie algebras. We find (generalized) Chern-Simons and BF-theory functionals this way and describe aspects of their parallel transport and quantization.It is known that over a D-brane the Kalb-Ramond 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 D-brane 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 String-extension BU (1) → String(G) → G. Here the obstruction to the lift is a 3-bundle with connection (a 2-gerbe): the Chern-Simons 3-bundle classified by the first Pontrjagin class. For G = Spin(n) this obstructs the existence of a String-structure. We discuss how to describe this obstruction problem in terms of Lie n-algebras and their corresponding categorified Cartan-Ehresmann connections. Generalizations even beyond String-extensions are then straightforward. For G = Spin(n) the next step is "Fivebrane structures" whose existence is obstructed by certain generalized Chern-Simons 7-bundles classified by the second Pontrjagin class.