An invitation to higher gauge theory

Abstract

In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge '2-group'. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2-group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincaré 2-group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a 'tangent 2-group', which serves as a gauge 2-group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an 'inner automorphism 2-group', which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an 'automorphism 2-group', which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a 'string 2-group'. We also touch upon higher structures such as the 'gravity 3-group', and the Lie 3-superalgebra that governs 11-dimensional supergravity.

D-algebra structure of topological insulators

“…At this level, the even and odd-dimensions are fundamentally different -in even dimensions, the structure factors of the algebra are proportional to the D/2'th Chern number, while in odd dimensions they are not proportional to the expected Chern-Simons form. The D-commutator hints at a different group structure from the usual gauge theories, such as higher gauge theories [27,28]. In light of this, the recent proposal [29] to describe topological insulators by a BF theory [30] looks very promising.…”

“…In higher dimensions, the D-algebra involves the D-form F ∧ · · · ∧ F . The natural objects that can couple to a D-form are D − 2 dimensional membranes [28]. Interestingly, the classical limit of the D-commutator is the Nambu-Poisson bracket [33], which is a natural setup to describe the dynamics of classical membranes [32].…”

Higher gauge theories from Lie n-algebras and off-shell covariantization

“…Along this line, the authors of [15] have proposed a generalization of parallel transport of point-like objects to parallel transport of string-like objects. This higher parallel transport leads to a gauge theory of a 2-form gauge field.…”

“…Then, as is discussed in [15], an action with nonabelian gauge symmetry cannot be formulated except for a topological action of BF type. As a result, the theory becomes topological or essentially free.…”

“…The corresponding physical field strength is the degree |z| + 1 part of the super field strength F Z , 15) where | |z|+1 denotes taking the degree |z| + 1 part. We can define a degree-preserving map,ã : T [1]Σ → M n , such that, for a function of degree k on the target space, the pullbackã * chooses the component of k-th order in θ µ in the superfield expansion, i.e.,…”