gauge theory



physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics

Differential cohomology



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.

The group GG 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-11 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 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.


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


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

physicsdifferential geometrydifferential cohomology
gauge fieldconnection on a bundlecocycle in differential cohomology
instanton/charge sectorprincipal bundlecocycle in underlying cohomology
gauge potentiallocal connection differential formlocal connection differential form
field strengthcurvatureunderlying cocycle in de Rham cohomology
gauge transformationequivalencecoboundary
minimal couplingcovariant derivativetwisted cohomology
BRST complexLie algebroid of moduli stackLie algebroid of moduli stack
extended Lagrangianuniversal Chern-Simons n-bundleuniversal characteristic map



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


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:


An exposition of the relation to geometric Langlands duality is in


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)


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

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 6, 2017 02:29:11 by Urs Schreiber (