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Contents

Idea

In physics there are (at least) two different concepts that go by the name gauge group:

• a local gauge group $G$ is a structure group of $G$-principal bundles in the configuration space of a classical gauge theory: it acts by gauge transformations on the space of field configurations.

• a global gauge group is a group of automorphisms that acts on the (local net of) observables of a quantum field theory.

Notably after quantization the gauge group in the first sense does not become a gauge group in the second sense. On the contrary, observables in quantum field theory are required not to depend on gauge transformations in the first sense, and part of what makes quantization of gauge theory nontrivial is to find among all potential candidate observables those that actually are invariant under gauge transformations, i.e. under isomorphisms of principal bundles with connection in the configuration space of the gauge theory.

Global gauge group

The concept of gauge groups is most prominent in quantum field theory, where the gauge group of a physical system is the group of transformations of the mathematical model of the system that do not correspond to any measurable physical effects. In this sense, nontrivial gauge groups arise from redundancies of the mathematical description. Gauge groups are a central ingredient of gauge theories.

In AQFT gauge groups are introduced via a net of C-star-systems.

Local gauge groups

In Yang-Mills theory and other gauge theories the gauge groups is the structure group $G$ of the $G$-principal bundle on which the Yang-Mills field is a connection.

Local gauge groups are visible in the Lagrangian approach to quantum field theory, where they act on the configuration space on which the action functional is a function by gauge transformations. A large machinery has been developed to handle the (path integral) quantization of action functionals on such configuration spaces. See for instance BV-BRST formalism.

Properties

Not a redundancy

Sometimes one sees the statement being made that gauge symmetry in theories of physics is just a sign of a redundancy in the theory, since, after all, gauge equivalent configurations are equivalent.

But, instead, there is genuine information contained in the gauge group of a physical theory: it encodes the homotopy type of the moduli space of configurations; or in other words: the higher homotopy groups in the ∞-groupoid of configurations of the system. Infinitesimally this is given by the BRST complex of the system, and the nature of the gauge group controls its higher cohomology groups. For instance the degree-1 cohomology of the BRST complex (meaning: “ghost degree-1”) of a system contains the possible gauge quantum anomalies (as discussed there) of the system. This is clearly not redundant information.

Abstractly speaking, the idea that symmetry is just a redundancy is a mistake of decategorification: passing from a groupoid of configurations – where different configurations are related by morphisms called gauge transformations – to the quotient space of configurations modulo gauge transformations is the decategorification of the groupoid. More technically speaking, it is the 0-truncation . It computes the 0-th homotopy group and forgets all the higher homotopy groups.

While there is indeed some information extracted by this process, that about the gauge equivalence classes of configurations, other information is lost.

Physically speaking, notably the locality of field theory would break down if one insisted on always passing to gauge equivalence classes of configurations.

For instance in a finite QFT such as Dijkgraaf-Witten theory with gauge group $G$, the moduli space is the delooping/classifying space $\mathbf{B}G$. This space is connected, reflecting the fact that locally every configuration of DW-theory is gauge equivalent to every other: namely the field configurations are $G$-principal bundles and locally on a piece of space $U$ these are all equivalent to the trivial principal bundle $U \times G$. If one insisted that only gauge equivalence classes of configuration contain non-redundant information, then one would find that either DW theory is not a local QFT or else that it contains the trivial configuration.

Entirely analogous comments apply to more interesting theories such as Yang-Mills theory, only that there the moduli space is richer: it is the differential refinement $\mathbf{B}G_{conn} := [P_1(-),\mathbf{B}G]$ of $\mathbf{B}G$ (see connection on a bundle for details). Again, this cannot be reconstructed from its decategorification and if one insisted on passing to gauge equivalence classes of configurations then either Yang-Mills theory would no longer appear as a local QFT, or else it would have just the the globally trivial configurations.

To say this more precisely, here instead of “moduli space” we should be saying “moduli stack”. The mathematical theory of stacks is the theory that deals with systems that are both local (in that global configurations are glued together from local data) as well as equipped with (gauge) symmetries. In that more refined language of higher topos theory one would find and say that the 0-truncation of $\mathbf{B}G_{conn}$ is the sheaf of gauge equivalence classes of Lie algebra valued differential forms. Insisting that this is the only relevant information amounts to insisting that either Yang-Mills theory is not a local theory, or else that all configurations of Yang-Mills theory are given by globally defined differential form data. This would exclude from the theory all configurations with nontrivial Chern classes, hence in particular it would exclude the instanton solutions from the theory, which are known to crucuially crucially encode information about thequantum theory.

That all said, one should note that in higher gauge theory redundancies may appear after all. This is related to the fact that in higher category theory/homotopy theory the notion of resolutions becomes relevant: there are higher moduli stacks that look very rich on first sight but turn out to be equivalent to simpler moduli stacks.

For instance if $U(1) \to \hat G \to G$ is a central extension of the gauge group $G$, then there is a 2-group denoted $(U(1) \to \hat G)$ (see crossed module) which is equivalent to $G$. Therefore 2-gauge theory (such as, say, the Courant sigma-model) with gauge 2-group $(U(1) \to \hat G)$ is in fact equivalent to ordinary $G$-gauge theory: there is indeed a lot of redundancy in $(U(1) \to \hat G)$. One can detect this by looking at the invariant information: the higher homotopy groups of $(U(1) \to \hat G)$ or else of its moduli stack (now a 2-stack) $\mathbf{B}(U(1) \to \hat G)$ are the same as that of $\mathbf{B}G$.

But even in such a situation, these redundant resolutions have a good use, in general as well as in applications to physics. In the above example the resolution serves to support an evident morphism $(U(1) \to \hat G) \to (U(1) \to 1) = \mathbf{B}U(1)$. Together with the equivalence to $G$ this constitutes an 2-anafunctor

This structure exhibits a characteristic class of $G$-gauge theory (namely the class that classifies the group extension). The nontriviality of this class measures the failure of $G$-gauge theory to lift to $\hat G$-gauge theory.

A famous example of this in string theory comes from the case where $\hat G \to G$ is the projection $U(n) \to PU(n)$ from the unitary group to the projective unitary group. In this case a configuration of $PU(n)$-gauge theory is a twisted bundle representing a class in twisted K-theory as it appears on the worldvolume of D-branes. The corresponding higher $\mathbf{B}U(1)$-gauge theory configuration is the Kalb-Ramond field(B-field) restricted to the brane, which is the twist that twists the bundle and prevents it from being a configuration in genuine $U(n)$-gauge theory.

Examples

We list examples of local gauge groups and ∞-groups for various higher gauge theories.

• the gauge group of $G$-Yang-Mills theory is the given Lie group $G$; for the Yang-Mills theory appearing in the standard model of particle physics this is the unitary group $U(3) \times SU(2) \times U(1)$;

• the local gauge group of gravity on a manifold $X$ is the Poincare group;

• the gauge 2-group of the Kalb-Ramond field is the circle 2-group $\mathbf{B} U(1) = (U(1) \to 1)$;

• the gauge 3-group of the supergravity C-field is the circle 3-group $\mathbf{B}^2 U(1) = (U(1) \to 1 \to 1)$;

• the gauge group of abelian higher dimensional Chern-Simons theory in dimension $4 k+3$ is the circle (2k+1)-group $\mathbf{B}^{2k} U(1)$;

• the 7-dimensional “fivebrane Chern-Simons theory” has as gauge 2-group the string 2-group;

• the symmetry group of string field theory is some ∞-group that is not an $n$-group for any finite $n$;

• an ∞-Chern-Simons theory has in general not only a gauge ∞-group but an ∞-groupoid of symmetries:

  • the Poisson sigma-model has as gauge groupoid the symplectic groupoid that is the Lie integration of the given Poisson Lie algebroid;

  • the Courant sigma-model has as gauge 2-groupoid the symplectic 2-groupoid that integrates the given Courant Lie 2-algebroid;

  • generally, the AKSZ sigma-model in grade $n$ has as gauge $\infty$-groupoid a symplectic Lie n-groupoid.

Related concepts

• gauge

• structure group, equivariance group

• gauge symmetry

• gauge fixing

• spontaneously broken symmetry

• enhanced gauge symmetry

References

• Wikipedia: gauge group

Global gauge group

Chapter IV of

• Rudolf Haag, Local Quantum Physics

Local gauge group

(…)

Local gauging of global symmetries in quantum gravity

It is being argued that after embedding into consistent quantum gravity, all global symmetries must become local symmetries.

A substantiation of this argument via AdS/CFT is given in:

• Daniel Harlow, Hirosi Ooguri, Symmetries in quantum field theory and quantum gravity (arXiv:1810.05338)

• Daniel Harlow, Hirosi Ooguri, Constraints on symmetry from holography, Phys. Rev. Lett. 122, 191601 (2019) (arXiv:1810.05337)

See also:

• Sylvain Fichet, Prashant Saraswat, Approximate Symmetries and Gravity (arXiv:1909.02002)