gauge fixing


\infty-Chern-Weil theory


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Configuration spaces in physics are typically not just plain spaces, but are groupoids and generally ∞-groupoids. This is traditionally most familiar for configuration spaces of gauge theories:

The operation called gauge fixing can essentially be understood as the passage to a more skelatized ∞-groupoid: in the simplest case, it amounts to picking one representative configuration from each equivalence class.

Generally, since the configuration ∞-groupoid typically carries extra structure in that it is a topological ∞-groupoid or a Lie ∞-groupoid or the like, the choice of gauge slice as it is called is similarly required to preserve that structure.


Standard textbook examples of gauge fixings include the following:

  • in the gauge theory of the electromagnetic field, a field configuration is, on a given pseudo-Riemannian manifold a line bundle with connection. Often the special case is considered where the underlying manifold is just Minkowski space and the bundle is assumed to be trivial, in which case a configuration of the gauge field configuration is just a 1-form AA on Minkowski space, and a gauge transformation λ:AA\lambda: A \to A' is a 0-form, i.e. a function, such that A=A+dλA' = A + d \lambda.

    • in the Lorenz gauge? the gauge field AA is taken to be a harmonic 1-form ddA=0d \star d \star A = 0.

Category-theoretic description

Here are more details on how one may think of gauge fixing from the nPOV.

In the Freed and Alm-Schreiber approach to quantization, the action functional is a functor

e iS:XnVect, e^{\mathrm{i}S}:X \to nVect,

where XX is some (,n)(\infty,n)-groupoid called the space of fields. The space of fields comes equipped with a projection π:XM\pi:X\to M to an (,n)(\infty,n)-groupoid MM called the moduli space of the quantum theory. Then the (path integral) quantization is, if it exists, the Kan extension Z:MnVectZ:M\to nVect of e iSe^{\mathrm{i}S} along π\pi. The functor ZZ is customary called the partition function of the theory.

A gauge fixing is a choice of a subgroupoid X gfX_{gf} of XX such that the inclusion X gfXX_{gf}\hookrightarrow X is an equivalence. The basic idea of gauge fixing is that the gauge fixed partition functions Z gfZ_{gf}, when they exist, are independent of the particular gauge fixing (since gauge fixing are all equivalent each other) and are equivalent to the original partition function ZZ (since X gfX_{gf} is equivalent to XX).

A classical instance of gauge fixing is when X=X˜//GX=\tilde{X}//G is an action groupoid, for the action of some group GG (the gauge group) on a manifold X˜\tilde{X}. In this case a classical gauge fixing is the choice of a slice SS in X˜\tilde{X} intersecting each orbit of GG exactly once. If the action of GG on X˜\tilde{X} is not free, there still will be nontrivial automorphisms in the groupoid S//GS//G; these residual internal symmetries are sometimes called ghost symmetries

Classical gauge fixings


Revised on December 28, 2013 14:30:06 by Urs Schreiber (