nLab
gauge group

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

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

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 potentiall 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 GG of the GG-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 GG, the moduli space is the delooping/classifying space BG\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 GG-principal bundles and locally on a piece of space UU these are all equivalent to the trivial principal bundle U×GU \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 BG conn:=[P 1(),BG]\mathbf{B}G_{conn} := [P_1(-),\mathbf{B}G] of BG\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 BG conn\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 encode information about the quantum 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)G^GU(1) \to \hat G \to G is a central extension of the gauge group GG, then there is a 2-group denoted (U(1)G^)(U(1) \to \hat G) (see crossed module) which is equivalent to GG. Therefore 2-gauge theory (such as, say, the Courant sigma-model) with gauge 2-group (U(1)G^)(U(1) \to \hat G) is in fact equivalent to ordinary GG-gauge theory: there is indeed a lot of redundancy in (U(1)G^)(U(1) \to \hat G). One can detect this by looking at the invariant information: the higher homotopy groups of (U(1)G^)(U(1) \to \hat G) or else of its moduli stack (now a 2-stack) B(U(1)G^)\mathbf{B}(U(1) \to \hat G) are the same as that of BG\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)G^)(U(1)1)=BU(1)(U(1) \to \hat G) \to (U(1) \to 1) = \mathbf{B}U(1). Together with the equivalence to GG this constitutes an 2-anafunctor

B(U(1)G^) B 2U(1) BG. \array{ \mathbf{B}(U(1) \to \hat G) &\to& \mathbf{B}^2 U(1) \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}G } \,.

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

A famous example of this in string theory comes from the case where G^G\hat G \to G is the projection U(n)PU(n)U(n) \to PU(n) from the unitary group to the projective unitary group. In this case a configuration of PU(n)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 BU(1)\mathbf{B}U(1)-gauge theory condiguration 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)U(n)-gauge theory.

Examples

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

References

Global gauge group

Chapter IV of

Local gauge group

Revised on May 16, 2014 22:02:28 by Tim Porter (2.26.24.117)