nLab
gauge theory

Idea

A gauge theory is a quantum field theory whose field configurations are cocycles in differential cohomology.

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?:

• 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).

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 whoe 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 gauge theory

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

• Chern-Simons Actions for (Super)-Gravities

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

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

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

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

    • $G=\mathrm{SU}(3)$ - strong nuclear force field

  • parallel transport: Wilson line?s

• 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

      $$\hat{F}\in H(X,\overline{B}U(1))$$\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 $x\in X$

    $$F=E\wedge dt+{\star }_{3}B$$F = E \wedge d t + \star_3 B

  • on $X={ℝ}^3\backslash \{0\}$: underlying class in integral cohomology $\mathrm{cl}(\hat{F})\in H(X,BU(1))\simeq H^2(X,\mathbb{Z})$ 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

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

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