Note: This page represents a failed attempt to convert one of Urs and Zoran’s papers from LaTeX to itex.
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.
This article is an expansion of a talk that the second author gave at the 5th Summer School of Modern Mathematical Physics in 2008.
URS13April: I expanded the following a bit, to more accurately reflect what the article is about.
The first part of this article is an overview for a general audience of mathematical physicists of (some appearances of) categorified symmetries of geometrical spaces and symmetries of constructions related to physical theories on spaces. Our main emphasis is on geometric and physical motivation, and the kind of mathematical structures involved. Sections 2-4 treat examples in noncommutative geometry, while 5-6 introduce nonabelian cocycles motivated in physics.
In sections 6-9 we discuss some technical details concerning differential cocycles and their quantization; part of these sections can be understood as a research unaouncement.