# nLab gauge theory

## Surveys, textbooks and lecture notes

#### Differential cohomology

differential cohomology

# Contents

## Idea

A gauge theory may denote either a classical field theory or a quantum field theory whose field configurations are cocycles in differential cohomology (abelian or nonabelian).

### Ordinary gauge theories

An ordinary gauge theory is a quantum field theory whose field configurations are vector bundles with connection.

This includes notable the fields that carry the three fundamental forces of the standard model of particle physics:

Other examples include formal physical models.

• Dijkgraaf-Witten theory is a gauge theory whose field configurations are $G$-principal bundles for $G$ a finite group (these come with a unique connection, so that in this simple case the connection is no extra datum).

The group $G$ in these examples is called the gauge group of the theory.

### Higher and generalized gauge theories

The above examples of gauge fields consisted of cocycles in degree-$1$ differential cohomology.

More generally, a higher gauge theory is a quantum field theory whose field configurations are cocycles in more general differential cohomology, for instance higher degree Deligne cocycles or more generally cocycles in other differential refinements, such as in differential K-theory.

This generalization does contain experimentally visible physics such as

But a whoe tower of higher and generalized gauge theories became visible with the study of higher supergravity theories,

### Gravity as a gauge theory

There are various models that realize gravity also as a gauge theory.

In particular supergravity theories have interpretations as higher gauge theories as described at D'Auria-Fre formulation of supergravity.

## Properties

### Non-redundancy and locality

Sometimes one see the view expressed that gauge symmetry is “just a redundancy” in the description of a theory of physics, for instance in that among observables it is only the gauge invariant ones which are physically meaningful.

This statement however

### Anomalies

In the presence of magnetic charge (and then even in the absence of chiral fermion anomalies?) the standard would-be action functional for higher gauge theories may be ill-defined. The Green-Schwarz mechanism is a famous phenomenon in differential cohomology by which such a quantum anomaly cancels against that given by chiral fermions.

## List of gauge fields and their models

The following tries to give an overview of some collection of gauge fields in physics, their models by differential cohomology and further details.

gauge field: models and components

## References

### General

An introduction to concepts in the quantization of gauge theories is in

A standard textbook on the BV-BRST formalism for the quantization of gauge systems is in

Discussion of abelian higher gauge theory in terms of differential cohomology is in

### In AQFT

Discussion in the context of AQFT (or at least with aiming for such a formualtion) includes the following

and more specifically on the problem of locality.

• Marco Benini, Claudio Dappiaggi, Alexander Schenkel, Quantized Abelian principal connections on Lorentzian manifolds, Communications in Mathematical Physics 2013 (arXiv:1303.2515)

### Dualities

An exposition of the relation to geometric Langlands duality is in

### History

A discussion of “gauge” and gauge transformation in metaphysics is in

Hermann Weyl’s historical argument motivating gauge theory in physics from rescaling of units of length was given in 1918 in

• Hermann Weyl, Raum, Zeit, Materie: Voerlesungen über die Allgemeine Relativitätstheorie, Springer Berlin Heidelberg 1923

Quick reviews include

• Quigley, On the origins of gauge theory (pdf)

• Afriat, Weyl’s gauge argument (pdf)

More comprehensive historical accounts include

Revised on April 15, 2014 06:03:07 by Urs Schreiber (88.128.80.29)