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A gauge transformation of categorical principal bundles arises from a functorial isomorphism between such bundles. We determine the geometric nature of such gauge transformations. For a twisted-product categorical principal bundle whose structure group is given by a pair of Lie groups G and H we show that a pair consisting of a traditional gauge transformation θ , given by a G -valued function, and an L ( H ) -valued 1-form Λ H determine a categorical gauge transformation. More general gauge transformations are also studied.

For a principal bundle $P\to M$ equipped with a connection ${\bar A}$, we study an infinite dimensional bundle ${\mathcal P}^{\rm dec}_{\bar A}P$ over the space of paths on $M$, with the points of ${\mathcal P}^{\rm dec}_{\bar A}P$ being horizontal paths on $P$ decorated with elements of a second structure group. We construct parallel transport processes on such bundles and study holonomy bundles in this setting. We explain the relationship with categorical geometry and explore the notion of categorical connections on categorical principal bundles in a concrete differential geometric way., Comment: 53pp, 3 figs

Categorical bundles provide a natural framework for gauge theories involving multiple gauge groups. Unlike the case of traditional bundles there are distinct notions of triviality, and hence also of local triviality, for categorical bundles. We study categorical principal bundles that are product bundles in the categorical sense, developing the relationship between functorial sections of such bundles and trivializations. We construct functorial cocycles with values in categorical groups using a suitable family of locally defined functions on the object space of the base category. Categorical product bundles being too rigid to give a widely applicable model for local triviality, we introduce the notion of a twisted-product categorical bundle. We relate such bundles to decorated categorical bundles that contain more information, specifically parallel transport data., Comment: v2:36 pages, 4 figures added to clarify statements about functorial cocycles and twisted product bundles

We study a type of connection forms, given by Chen integrals, over pathspaces by placing such forms within a category-theoretic framework of principal bundles and connections. We introduce a notion of 'decorated' principal bundles, develop parallel transport on such bundles, and explore specific examples in the context of pathspaces., Comment: 50 pages. Updated and revised on 16th August 2012

We prove that integration over the moduli space of flat connections can be obtained as a limit of integration with respect to the Yang–Mills measure defined in terms of the heat-kernel for the gauge group. In doing this we also give a rigorous proof of Witten’s formula for the symplectic volume of the moduli space of flat connections. Our proof uses an elementary identity connecting determinants of matrices along with a careful accounting of certain dense subsets of full measure in the moduli space.

A formula is derived which expresses the symplectic volume measure on the moduli space of flat connections over a closed Riemann surface Σ in terms of the corresponding volume measures for the surfaces Σ 1 and Σ 2 , when the latter have one boundary component each.

All the connected components of the moduli space of flat connections on SU (2) and SO (3) (trivial and non-trivial) bundles over closed oriented surfaces are determined. The symplectic structure and volumes of the non-maximal strata of the moduli space are also determined.