# nLab gauge theory

Contents

## 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 notably 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 whole tower of higher and generalized gauge theories became visible with the study of higher supergravity theories,

### Gravity as a (non-)gauge theory

In the first order formulation of gravity also the theory of gravity looks a little like a gauge theory. However, there is a crucial difference. What really happens here is Cartan geometry: the field of gravity may be encoded in a vielbein field, namelely an orthogonal structure on the tangent bundle, hence as an example of a G-structure, and the torsion freedom of this G-structure may be encoded by an auxiliary connection, namely a Cartan connection, often called the “spin connection” in this context. Hence while in the formulation of Cartan geometry gravity is described by many of the ingredients from differential geometry that also govern pure gauge theory, it’s not quite the same. In particular there is a constraint on a Cartan connection, which in terms of vielbein fields is the constraint that the vielbein (which is part of the Cartan connection) is non-degenerate, and hence really a “soldering form”. Such a constraint is absent in a “genuine” gauge theory such as Yang-Mills theory or Chern-Simons theory.

## 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.

## References

### General

Historical origins:

Textbook accounts:

Basics on fiber bundles in physics are recalled in

• Adam Marsh, Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results (arXiv:1607.03089)

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

Lecture notes on this:

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

### In AQFT

Standard discussion of gauge theory in the context of algebraic quantum field theory (AQFT) includes

For AQFT on curved spacetimes the axioms of AQFT need to be promoted to a context of higher geometry unless locality is broken, see the expositions at

This was established in

and the program of improving the axioms of AQFT on curved spacetimes to the stacky context in order to accomodate gauge theory includes the following articles:

### 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: Vorlesungen über die Allgemeine Relativitätstheorie, Springer Berlin Heidelberg 1923

The manuscript of Weyl’s first book on mathematical physics, Space – Time – Matter (STM) (Raum – Zeit – Materie), delivered to the publishing house (Springer) Easter 1918, did not contain Weyl’s new geometry and proposal for a UFT. It was prepared from the lecture notes of a course given in the Summer semester of 1917 at the Polytechnical Institute (ETH) Zürich. Weyl included his recent findings only in the 3rd edition (1919) of the book. The English and French versions (Weyl 1922b, Weyl 1922a), translated from the fourth revised edition (1921), contained a short exposition of Weyl’s generalized metric and the idea for a scale gauge theory of electromagnetism. (Scholz)

See

• Erhard Scholz, H. Weyl’s and E. Cartan’s proposals for infinitesimal geometry in the early 1920s (pdf)

Early surveys include

Quick reviews include

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

• Afriat, Weyl’s gauge argument (pdf)

More comprehensive historical accounts include

Last revised on July 29, 2024 at 11:49:10. See the history of this page for a list of all contributions to it.