Types of quantum field thories
the objects are the (field-)configurations of the physical system;
the k-morphisms are “gauge-transformations of gauge transformation”:
(these higher order gauge transformations are in the traditional physics literature mainly known in their infinitesimal approximation where the configuration Lie ∞-groupoid is approximated by a Lie-∞-algebroid whose Chevalley-Eilenberg algebra is the BRST complex: here they correspond to ghosts of ghosts ).
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 on Minkowski space, and a gauge transformation is a 0-form, i.e. a function, such that .
Here are more details on how one may think of gauge fixing from the nPOV.
where is some -groupoid called the space of fields. The space of fields comes equipped with a projection to an -groupoid called the moduli space of the quantum theory. Then the (path integral) quantization is, if it exists, the Kan extension of along . The functor is customary called the partition function of the theory.
A gauge fixing is a choice of a subgroupoid of such that the inclusion is an equivalence. The basic idea of gauge fixing is that the gauge fixed partition functions , 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 (since is equivalent to ).
A classical instance of gauge fixing is when is an action groupoid, for the action of some group (the gauge group) on a manifold . In this case a classical gauge fixing is the choice of a slice in intersecting each orbit of exactly once. If the action of on is not free, there still will be nontrivial automorphisms in the groupoid ; these residual internal symmetries are sometimes called ghost symmetries