Schreiber Minicourse on nonabelian differential cohomology

This page provides background information on the

• Minicourse on Nonabelian Differential Cohomology

  Part: Categorical Apporach (Urs Schreiber)

  in the context of the

  ESI Program on Higher Structures in Mathematics and Physics

  Vienna, September 1 - November 7, 2010

Abstract

Connections on bundles describe gauge fields such as the electromagnetic field and more generally Yang-Mills fields. Higher gauge fields in string theory, such as the Kalb-Ramond B-field and the supergravity C-field, are described by higher analogs of this, known as gerbes or higher bundles with connection.

In this minicourse I give an introduction to the description of higher bundles with connection using the tools of higher category theory.

We start with some basics on $\infty$-Lie groupoids and the classification of principal $\infty$-bundles in terms of nonabelian cohomology. Then we describe connections on these in terms of parallel transport. As an example we describe Chern-Simons circle 3-bundles and indicate their role in string theory.

We follow the exposition at differential cohomology in a cohesive topos – full content.

Lectures

In three lectures we shall try to describe the general context of nonabelian differential cohomology and spell out aspects of some central constructions and applications. The keyword lists below indicate the rough plan. Following the links leads to pages with detailed and comprehensive treatments.

I – $\infty$-Lie groupoids

1. Motivation

   Some applications in mathematics and in fundamental physics that naturally can or have to be treated with tools from nonabelian differential cohomology.

2. Principal n-bundles in low dimension

   A review of some classical concepts, such as that of principal bundles, of Cech cohomology and of gerbes, in a functorial language that will lend itself nicely to generalization.

3. Principal ∞-Bundles in an ∞-topos

   The tools for taking the theory to infinite categorical degree.

II – Differential cohomology

1. Parallel transport in low dimensions

   A review of some classical concepts, such as the parallel transport of a connection, and the surface transport of a gerbe, in a functorial language that lends itself to generalization.

2. path ∞-groupoid

   The generalization of parallel transport to infinite categorical degree.

3. circle n-bundle with connection

   The ordinary differential cohomology of circle $n$-bundles with connection.

III – $\infty$-Chern-Weil theory

1. Connections on an ∞-bundle

   The local data of an connection on an $\infty$-bundle with values in an $\infty$-Lie algebra.

2. Chern-Simons circle 3-bundle

   The refined Chern-Weil homomorphism applied to the first Pontryagin class.

3. Differential string structures

   Twisted String-2-bundles with connection and the Green-Schwarz mechanism.

References

An exposition along the above lines is in the first chapter of

• Urs Schreiber, differential cohomology in a cohesive topos

Further relevant references are collected at

• differential cohomology in an (∞,1)-topos – references.