Contents

Idea

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 objects are the (field-)configurations of the physical system;

• the morphisms between objects are gauge transformation?s between different but equivalent field configurations;

• 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.

Examples

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 $A$ on Minkowski space, and a gauge transformation $\lambda :A\to A\prime$ is a 0-form, i.e. a function, such that $A\prime =A+d\lambda$.

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

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

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

A gauge fixing is a choice of a subgroupoid $X_{\mathrm{gf}}$ of $X$ such that the inclusion $X_{\mathrm{gf}}\hookrightarrow X$ is an equivalence. The basic idea of gauge fixing is that the gauge fixed partition functions $Z_{\mathrm{gf}}$, 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 $Z$ (since $X_{\mathrm{gf}}$ is equivalent to $X$).

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

Classical gauge fixings

• BRST complex

• BV theory