# nLab gauge fixing

### Context

#### $\infty$-Chern-Weil theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

# 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 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 $\lambda: A \to A'$ is a 0-form, i.e. a function, such that $A' = A + d \lambda$.

• 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.

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

where $X$ is some $(\infty,n)$-groupoid called the space of fields. The space of fields comes equipped with a projection $\pi:X\to M$ to an $(\infty,n)$-groupoid $M$ called the moduli space of the quantum theory. Then the (path integral) quantization is, if it exists, the Kan extension $Z:M\to nVect$ 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 $X_{gf}$ of $X$ such that the inclusion $X_{gf}\hookrightarrow X$ is an equivalence. The basic idea of gauge fixing is that the gauge fixed partition functions $Z_{gf}$, 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 $X_{gf}$ 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

## Examples

Revised on December 28, 2013 14:30:06 by Urs Schreiber (89.204.130.243)