engleski

Categorified symmetries

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 $\infty$-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.

categorification ; cocycle ; categorical group ; nonabelian cohomology ; quantum field theory ; Drinfeld double ; sigma model ; bibrane ; noncommutative geometry

Publicirana verzija je 28 stranica. Verzija arxiv/1004.2472v1 je verzija extended post publication i prosirena novim materijalom na 55 stranica. AMS MSC2010 Classification: 81T, 14A22, 18, 55N30