---
tag: 888SJ2508Y
kind: object
name: Space of connections modulo gauge
msc: [58D27, 53C07, 81T13]
instance_of: [0BR3FY5Q8M]
uses: [707DEWCCFX, NJ21PN5ZGJ, 8E37E74XVP, 0BR3FY5Q8M, 0RGGF8ZS6D]
sources:
  - url: https://en.wikipedia.org/wiki/Gauge_theory_(mathematics)
---

## Description

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$, let $X$ be a compact manifold, and let $P \to X$ be a principal $G$-bundle.
The space $\mathcal{A}(P)$ of [[connections|NJ21PN5ZGJ]] on $P$ is an affine space over the vector space $\Omega^1(X; \mathrm{ad} P)$ of $1$-forms with values in the adjoint bundle.
The [[gauge group|707DEWCCFX]] $\mathcal{G}(P)$ acts on $\mathcal{A}(P)$, and the *space of connections modulo gauge* is the [[quotient space|0BR3FY5Q8M]]
$$\mathcal{B}(P) = \mathcal{A}(P) / \mathcal{G}(P).$$

## Remarks

The simplest case is $G = U(1)$, $X = S^1 = \R / 2\pi \Z$ and $P$ trivial.
A connection is $A = i a(t) \, dt$ with $a$ a smooth real function, and a gauge transformation $g = e^{i \phi}$, with $\phi \colon \R \to \R$ smooth and $\phi(t + 2\pi) - \phi(t) \in 2\pi \Z$, sends $A$ to $A + i \, d\phi$.
So the class of $A$ is determined by $\int_0^{2\pi} a(t) \, dt$ modulo $2\pi$, and $\mathcal{B}(P)$ is a circle; the map is the holonomy $A \mapsto \exp(-i \int_0^{2\pi} a \, dt) \in U(1)$.

In general $\mathcal{B}(P)$ is not a manifold, because the [[stabilisers|0RGGF8ZS6D]] of connections vary.
The stabiliser of $A$ always contains the centre of $G$, acting by constant gauge transformations.
The connection $A$ is called irreducible when its stabiliser is exactly the centre.

After completing in suitable Sobolev norms, the irreducible connections modulo gauge form a Hilbert manifold, with local charts given by the Coulomb slice $\set{A + a : d_A^* a = 0, \ a \text{ small}}$.

## Explanations

### L1 human

Two connections that differ only by a change of gauge describe the same physical field, so one wants to count them once.
The space of connections modulo gauge is the result: one point for each field, not for each description of it.
On a circle, a $U(1)$ connection modulo gauge is described by a single angle, its holonomy around the circle.
Moduli spaces of instantons and of other special connections sit inside this quotient space.