Categorified symmetries

TL;DR

This paper explores categorified symmetries in quantum field theory, focusing on higher groupoids, noncommutative geometry, and their role in modeling higher principal bundles, gerbes, and related structures, highlighting the appearance of the Drinfeld double.

Contribution

It introduces new perspectives on categorified symmetries, connecting noncommutative geometry, higher groupoids, and Hopf algebras, expanding the understanding of gauge theories and quantum symmetries.

Findings

Categorified symmetries include higher groupoids and noncommutative geometric structures.

Higher cocycles generalize group cohomology to describe higher principal bundles and gerbes.

The Drinfeld double Hopf algebra naturally arises in the context of categorified symmetries.

Abstract

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.