Mathematics – Quantum Algebra
Scientific paper
2010-04-14
5th Summer School of Modern Mathematical Physics, SFIN, XXII Series A: Conferences, No A1, (2009), 397-424 (Editors: Branko Dr
Mathematics
Quantum Algebra
about 55 pages, expansion of 28-page published proceedings
Scientific paper
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 infinity-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.
Schreiber Urs
Škoda Zoran
No associations
LandOfFree
Categorified symmetries does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Categorified symmetries, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Categorified symmetries will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-528384