nLab gauge theory

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:

• Ordinary electromagnetism in the absence of magnetic charges is a gauge theory of $U(1)$-principal bundles with connection.

• Fields in Yang-Mills theory (such as appearing in the standard model of particle physics and in GUTs) are vector bundles with connection.

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

• The magnetic current in electromagnetism is a bundle gerbe with connection, a Deligne cocycle refining a cocycle in degree-$3$ Eilenberg-MacLane cohomology: the magnetic charge .

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

• The Kalb-Ramond field is a bundle gerbe with connection, a Deligne cocycle with curvature 3-form.

• The supergravity C-field is a Deligne cocycle with curvature 4-form.

• The RR-field is a cocycle in differential K-theory.

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.

• Yang-Mills field

  • cocycle in lowest degree nonabelian differential cohomology

    • originally realized in terms of differential Čech cocycles

      $$\hat F \in \mathbf{H}(X, \bar \mathbf{B} G)$$

      with coefficients in the groupoid of Lie-algebra valued forms,

      then traditionally in terms of vector bundles with connection

  • field strength depending on the group $G$ we have

    • $G = U(1)$ - electromagnetism (see below)

    • $G = SU(2)\times U(1)$ - electroweak force field strength

    • $G = SU(3)$ - strong nuclear force field

  • parallel transport: Wilson lines

• electromagnetic field

  • cocycle in degree-$2$ ordinary differential cohomology

    • naturally/historically realized in terms of Maxwell-Dirac presentation as a cocycle in Čech–Deligne cocycle
      $$\hat F \in \mathbf{H}(X,\bar \mathbf{B} U(1))$$

  • field strength: the electric field $E$ and magnetic field $B$, locally at a point $x \in X$

    $$F = E \wedge d t + \star_3 B$$

  • on $X = \mathbb{R}^3\backslash \{0\}$: underlying class in integral cohomology $cl(\hat F) \in H(X,\mathbf{B} U(1)) \simeq H^2(X,\mathbb{Z})$ is the magnetic charge

  • parallel transport: gauge interaction piece of action functional of the electrically charged quantum 1-particle

• Kalb-Ramond field

  • cocycle in degree-$3$ ordinary differential cohomology

    • naturally/historically realized in terms of

      • a cocycle in Čech–Deligne cocycle

        $$\hat H \in \mathbf{H}(X,\bar \mathbf{B}^2 U(1))$$

      • a bundle gerbe with connection

  • field strength: $H \in \Omega^3(X)$ the “$H$-field” – on a D-brane this is the magnetic current for the Yang-Mills field on the brane

  • parallel transport: gauge interaction piece of action functional of the electrically charged quantum 2-particle (the string).

• supergravity C-field

  • cocycle in degree-$4$ ordinary differential cohomology

    • naturally/historically realized in terms of as a cocycle in Čech–Deligne cocycle

      $$\hat H \in \mathbf{H}(X,\bar \mathbf{B}^3 U(1))$$

      using the D'Auria-Fre formulation of supergravity it may also be thought of as a nonabelian differential cocycle given by a Cartan-Ehresmann ∞-connection

  • field strength: $H \in \Omega^4(X)$ the “$G$-field” – in heterotic supergravity this is the 5-brane magnetic current for the twisted Kalb-Ramond field

  • parallel transport: gauge interaction piece of action functional of the electrically charged quantum 3-particle (the membrane).

• RR field

  • cocycle in differential K-theory

    • in presence of nontrivial Kalb-Ramond field: cocycle in differential twisted K-theory

  • field strength: RR-forms

Related concepts

• fiber bundles in physics

• gauge

• gauge group

• gauge transformation, higher gauge transformation

• BRST complex, BV-BRST formalism

• ghost field, ghost-of-ghost field

• gauge fixing, gauge fixing fermion, gauge invariant

• lattice gauge theory

• Gribov ambiguity

• quiver gauge theory

• arithmetic gauge theory

gauge field: models and components

physics	differential geometry	differential cohomology
gauge field	connection on a bundle	cocycle in differential cohomology
instanton/charge sector	principal bundle	cocycle in underlying cohomology
gauge potential	local connection differential form	local connection differential form
field strength	curvature	underlying cocycle in de Rham cohomology
gauge transformation	equivalence	coboundary
minimal coupling	covariant derivative	twisted cohomology
BRST complex	Lie algebroid of moduli stack	Lie algebroid of moduli stack
extended Lagrangian	universal Chern-Simons n-bundle	universal characteristic map

• higher U(1)-gauge theory

  • higher electric background charge coupling

• self-dual higher gauge theory

• higher spin gauge theory

• quantum anomaly

  • Green-Schwarz mechanism

• infinity-Chern-Simons theory

• free field theory

References

General

Historical origins:

• A. C. T. Wu, Chen Ning Yang, Evolution of the concept of vector potential in the description of the fundamental interactions, International Journal of Modern Physics A 21 16 (2006) 3235-3277 [doi:10.1142/S0217751X06033143]

• Chen Ning Yang, The conceptual origins of Maxwell’s equations and gauge theory, Phyics Today 67 11 (2014) [doi:10.1063/PT.3.2585, pdf]

Textbook accounts:

• Yuri Makeenko, Methods of contemporary gauge theory, Cambridge Monographs on Math. Physics, Cambridge University Press (2002) [doi:10.1017/CBO9780511535147, gBooks]

• Mikio Nakahara, Section 10.5 of: Geometry, Topology and Physics, IOP 2003 (doi:10.1201/9781315275826, pdf)

• Mike Guidry, Gauge Field Theories: An Introduction with Applications, Wiley 2008 (ISBN:978-3-527-61736-4)

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

• Nicolai Reshitikhin, Lectures on quantization of gauge systems (pdf)

See also:

• Garth Warner: Quantum Gauge Theory, seminar notes, University of Washington [pdf, pdf]

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

• Marc Henneaux, Claudio Teitelboim, Quantization of Gauge Systems, Princeton University Press (1992) [doi:10.2307/j.ctv10crg0r]

Lecture notes on this:

• Urs Schreiber: geometry of physics – perturbative quantum field theory (2017)

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

• Dan Freed, Dirac charge quantization and generalized differential cohomology

• Alessandro Valentino, Differential cohomology and quantum gauge fields (pdf)

• José Figueroa-O'Farrill, Gauge theory (web page)

• Tohru Eguchi, Peter Gilkey, Andrew Hanson, Gravitation, gauge theories and differential geometry, Physics Reports 66:6 (1980) 213—393 (pdf)

For discussion in the context of gravity see also

• Edward Witten, Symmetry and Emergence (arXiv:1710.01791)

In AQFT

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

• Franco Strocchi, section 4 of Relativistic Quantum Mechanics and Field Theory, Found.Phys. 34 (2004) 501-527 (arXiv:hep-th/0401143)

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

• Higher field bundles for gauge fields

• Alexander Schenkel, On the problem of gauge theories

  in locally covariant QFT_, talk at Operator and Geometric Analysis on Quantum Theory Trento, 2014 (web)

This was established in

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

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:

• Marco Benini, Alexander Schenkel, Richard Szabo, Homotopy colimits and global observables in Abelian gauge theory (arXiv:1503.08839)

• Marco Benini, Alexander Schenkel, Quantum field theories on categories fibered in groupoids (arXiv:1610.06071)

• Marco Benini, Alexander Schenkel, Urs Schreiber, The stack of Yang-Mills fields on Lorentzian manifolds (arXiv:1704.01378)

Dualities

An exposition of the relation to geometric Langlands duality is in

• Edward Frenkel, Gauge theory and Langlands duality (pdf)

History

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

• Georg Hegel, §714 of Science of Logic, 1812

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

• John Iliopoulos, Progress in Gauge Theories, 1975 (spire:3000)

Quick reviews include

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

• Afriat, Weyl’s gauge argument (pdf)

More comprehensive historical accounts include

• Lochlainn O'Raifeartaigh, The Dawning of Gauge Theory, Princeton University Press (1997)

• Lochlainn O'Raifeartaigh, Norbert Straumann, Gauge Theory: Historical Origins and Some Modern Developments Rev. Mod. Phys. 72, 1-23 (2000).

• Norbert Straumann, Early History of Gauge Theories and Weak Interactions (arXiv:hep-ph/9609230)

• Norbert Straumann, Gauge principle and QED, talk at PHOTON2005, Warsaw (2005) (arXiv:hep-ph/0509116)