A Lie 2-algebra is a ‘categorified ’ version of a Lie algebra: that is, a category equipped with structures analogous to those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poisson bracket of functions on this space gives a Lie algebra of observables. Multisymplectic geometry describes an n-dimensional field theory using a phase space that is an ‘n-plectic manifold’: a finite-dimensional manifold equipped with a closed nondegenerate (n + 1)-form. Here we consider the case n = 2. For any 2-plectic manifold, we construct a Lie 2-algebra of observables. We then explain how this Lie 2-algebra can be used to describe the dynamics of a classical bosonic string. Just as the presence of an electromagnetic field affects the symplectic structure for a charged point particle, the presence of a B field affects the 2-plectic structure for the string.

This paper contains some basic results on 2-groupoids, with special emphasis on computing derived mapping 2-groupoids between 2-groupoids and proving their invariance under strictification. Some of the results proven here are presumably folklore (but do not appear in the literature to the author’s knowledge) and some of the results seem to be new. The main technical tool used throughout the paper is the Quillen model structure on the category of 2-groupoids introduced by Moerdijk and Svensson.

Abstract not found

Abstract. This paper contains some basic results on 2-groupoids, with special emphasis on computing derived mapping 2-groupoids between 2-groupoids and proving their invariance under strictification. Some of the results proven here are presumably folklore (but do not appear in the literature to the author’s knowledge) and some of the results seem to be new. The main technical tool used throughout the paper is the Quillen model structure on the category of 2-groupoids introduced by Moerdijk and Svensson. 1.

We introduce nonabelian differential cohomology classifying ∞-bundles with smooth connection and their higher gerbes of sections, generalizing [138]. We construct classes of examples of these from lifts, twisted lifts and obstructions to lifts through shifted central extensions of groups by the shifted abelian n-group B n−1 U(1). Notable examples are String 2-bundles [9] and Fivebrane 6-bundles [133]. The obstructions to lifting ordinary principal bundles to these, hence in particular the obstructions to lifting Spin-structures to String-structures [13] and further to Fivebrane-structures [133, 52], are abelian Chern-Simons 3- and 7-bundles with characteristic class the first and second fractional Pontryagin class, whose abelian cocycles have been constructed explicitly by Brylinski and McLaughlin [35, 36]. We realize their construction as an abelian component of obstruction theory in nonabelian cohomology by ∞-Lieintegrating the L∞-algebraic data in [132]. As a result, even if the lift fails, we obtain twisted String 2- and twisted Fivebrane 6-bundles classified in twisted nonabelian (differential) cohomology and generalizing the twisted bundles appearing in twisted K-theory. We explain the Green-Schwarz mechanism in heterotic string theory in terms of twisted String 2-bundles and its magnetic dual version – according to [133] – in terms of twisted Fivebrane 6-bundles. We close by transgressing differential cocycles to mapping

The string group String(n) is the 3-connected cover of Spin(n). Given and compact simply connected group G, we will let String G be its 3-connected cover. The group String G is only defined up to homotopy, and various models have appeared in the literature. Stephan Stolz and Peter Teichner [7], [6] have a couple of models of String G, one of which, inspired by Anthony Wassermann, is an extension of G by the group of projective unitary operators in a particular Von-Neuman algebra. Jean-Luc Brylinski [4] has a model which is a U(1)-gerbe with connection over the group G. More recently, John Baez et al [2] came up with a model of String G in their quest for a 2-Lie group integrating a given 2-Lie algebra. We show how to produce their model by applying a certain canonical procedure to their 2-Lie algebra. A 2-Lie algebra is a two step L∞-algebra. It consists of two vector spaces V0 and V1, and three brackets [], [,], [ , , ] acting on V: = V0 ⊕ V1. They are of degree-1, 0, and 1 respectively and satisfy various axioms, see [1] for more details.

Every finite strict 2-group has a canonical 2-representation on Vectmodule

Abelian differential generalized cohomology as developed by Hopkins and Singer has been shown by Freed to formalize the global description of anomaly cancellation problems in String theory, such as notably the Green-Schwarz mechanism. On the other hand, this mechanism, as well as the Freed-Witten anomaly cancellation, are fundamentally governed by the cohomology classes represented by the relevant nonabelian O(n)- and U(n)-principal bundles underlying the tangent and the gauge bundle on target space. In this article we unify the picture by describing nonabelian differential cohomology and twisted nonabelian differential cohomology and apply it to these situations. We demonstrate that the Freed-Witten mechanism for the B-field, the Green-Schwarz mechanism for the H3-field, as well as its magnetic dual version for the H7-field define cocycles in twisted nonabelian differential cohomology that may be addressed, respectively, as twisted Spin(n)-, twisted String(n)- and twisted Fivebrane(n)structures on target space, where the twist in each case is provided by the obstruction to lifting the gauge bundle through a higher connected cover of U(n). We work out the (nonabelian) L∞-algebra valued connection data provided by the differential refinements of these twisted cocycles and demonstrate that this reproduces locally the differential form data with the twisted Bianchi identities as known from the

Quantum field theory allows more general symmetries than groups and Lie algebras. For instance quantum groups, that is Hopf algebras, have been familiar to theoretical physicists for a while now. Nowdays many examples of symmetries of categorical flavor – categorical groups, groupoids, Lie algebroids and their higher analogues – appear in physically motivated constructions and faciliate constructions of geometrically sound models and quantization of field theories. Here we consider two flavours of categorified symmetries: one coming from noncommutative algebraic geometry where varieties themselves are replaced by suitable categories of sheaves; another in which the gauge groups are categorified to higher groupoids. Together with their gauge groups, also the fiber bundles themselves become categorified, and their gluing (or descent data) is given by nonabelian cocycles, generalizing group cohomology, where ∞-groupoids appear in the role both of the domain and the coefficient object. Such cocycles in particular represent higher principal bundles, gerbes, – possibly equivariant, possibly with connection – as well as the corresponding associated higher vector bundles. We show how the Hopf algebra known as the Drinfeld double arises in this context. 1.

Recent work applying higher gauge theory to the superstring has indicated the presence of ‘higher symmetry’. Infinitesimally, this is realized by a ‘Lie 2-superalgebra ’ extending the Poincaré superalgebra in precisely the dimensions where the classical superstring makes sense: 3, 4, 6 and 10. In the previous paper in this series, we constructed this Lie 2-superalgebra using the normed division algebras. In this paper, we use an elegant geometric technique to integrate this Lie 2-superalgebra to a ‘Lie 2-supergroup ’ extending the Poincaré supergroup in the same dimensions. Briefly, a ‘Lie 2-superalgebra ’ is a two-term chain complex with a bracket like a Lie superalgebra, but satisfying the Jacobi identity only up to chain homotopy. Simple examples of Lie 2-superalgebras arise from 3-cocycles on Lie superalgebras, and it is in this way that we constructed the Lie 2-superalgebra above. Because this 3-cocycle is supported on a nilpotent subalgebra, our geometric technique applies, and we obtain a Lie 2-supergroup integrating the Lie 2-superalgebra in the guise of a smooth 3-cocycle on the Poincaré supergroup. 1