Topological Quantum Field Theory on non-Abelian gerbes

Abstract

The infinitesimal symmetries of a fully decomposed non-Abelian gerbe can be generated in terms of a nilpotent BRST operator, which is here constructed. The appearing fields find a natural interpretation in terms of the universal gerbe, a generalisation of the universal bundle. We comment on the construction of observables in the arising Topological Quantum Field Theory. It is also shown how the BRST operator and the trace part of a suitably truncated set of fields on the non-Abelian gerbe reduce directly to the coboundary operator and the pertinent cochains of the underlyingČech-de Rham complex.• The generalisation of theČech coboundary operator, ∂ λ .

A new kind of geometric phases in open quantum systems and higher gauge theory

“…[29,30,30]) defined from (A α , B α , η αβ ). P is a non-abelian bundle gerbes [26,27,28,29,30,31])♯ with the structure Lie crossed module (G, J, t, Ad) where t : J → G is the canonical injection (J is a subgroup of G) and Ad : G → Aut(J) is the adjoint representation of G on itself restricted to J (in the homomorphisms domain). The replacement of the single gauge group of closed quantum systems by a gauge Lie crossed module for open quantum systems is explained by the need for a gauge group associated with the evolution of the quantum system (G) and also for a gauge group associated with the decoherence induced by the environment (J).…”

Non-geometric magnetic flux and crossed modules

“…In Refs. [2,18] these fields are really group valued differential forms [20], though for the present discussion they reduce to algebra valued forms. To write the BRST operator down, we need the following notation:…”

“…Twisting in N = 2 supersymmetric Yang-Mills was introduced in [14]. There, twisting means identifying the R-symmetry group SU(2) R with one of the factors of the Euclidean spin-group Spin(4) = SU(2) × SU (2). Generalisations to N = 4 were proposed in [3,15].…”

“…The BRST operator that generates infinitesimal symmetries of a gerbe was constructed in [2]. The local fields involve the local connection which is a Lie Aut G valued one-form m i ; a Lie G valued one-form γ ij on the double intersection of local charts U ij ; and a local Lie G valued two-form B i .…”

Non-abelian higher gauge theory and categorical bundle

“…Since H-transition functions are trivial with a spherical affine 2-space (h ij (y, x) = e H , ∀xRy ⇐⇒ x = y), these local data of the 2-bundles are absent from the theory of Baez etal. Because the non-abelian bundle gerbes [19,20,21] are weakly equivalent to 2-bundles [32] the same remarks can be applied in the comparison of our definition with the constructions of non-abelian bundle gerbes or twisted bundles. Nevertheless, the non-abelian bundle gerbes present a kind of H-transition functions obeying to a structure equation similar to equation 32 (see [22]).…”