nLab geometry of physics -- flat connections

this entry is going to be one chapter of geometry of physics

previous chapters: smooth homotopy types

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Flat connections

Model Layer

Flat 1-connections

• flat connection

$X$ connected, $\pi_1(X) \in$Grp its fundamental group for any choice of basepoint, then the holonomy pairing

$$hol \colon [S^1,X]\times H^1_{conn}(X,G) \to G$$

descends to homotopy classes of (based) loops

$$hol \colon H^1_{conn,flat}(X,G) \stackrel{\simeq}{\to} Hom_{Grp}(\pi_1(X), G)/G$$

to a bijection from equivalence classes of flat? $G$-principal connections to the quotient set of group homomorphisms $\pi_1(X) \to G$ modulo the adjoint action of $G$ on itself.

Semantic Layer

Definition

For $G \in Grp(\mathbf{H})$ and $X \in \mathbf{H}$ a flat $G$-connection $\nabla$ on $X$ is a morphism

$$\nabla \colon X \to \flat \mathbf{B}G \,.$$

We write

$$\mathbf{H}_{flat}(X, \mathbf{B}G) \coloneqq \mathbf{H}(X, \flat \mathbf{B}G)$$

and accordingly

$$H^1_{flat}{X, G} \coloneqq \pi_0 \mathbf{H}_{flat}(X,G)$$

for the cohomology of $X \in \mathbf{H}$ with flat coefficients.

Remark

By adjunction,

$$\frac{X \stackrel{\nabla}{\to} \flat \mathbf{B}G}{\Pi(X) \stackrel{transport(\nabla)}{\to} \mathbf{B}G}$$

a flat $G$-connection is equivalently a morphism

$$transport(\nabla) \colon \Pi(X) \to \mathbf{B}G \,.$$

Since $\Pi(X)$ is the fundamental infinity-groupoid of $X$, this manifestly encodes the higher parallel transport of the flat connection.

Definition

Write

$$UnderlyingBundle_{\mathbf{B}G} \colon \flat \mathbf{B}G \to \mathbf{B}G$$

for the $(Disc \vdash \Gamma)$-counit-

Definition

For $\nabla \colon X \to \flat \mathbf{B}G$ the composite

$$UnderlyingBundle(\nabla) \colon X \stackrel{\nabla}{\to} \flat\mathbf{B}G \stackrel{UnderlyingBundle_{\mathbf{B}G}}{\to} \mathbf{B}G$$

modulates a $G$-principal ∞-bundle on $X$, by def. . This we call the underlying $G$-principal bundle of $\nabla$.

$$ConstantPaths_{X} \colon X \to \Pi(X)$$

Syntactic Layer

$$\mathbf{B}G \colon Type \;\vdash \; UnderlyingBundle \colon \flat \mathbf{B}G \to \mathbf{B}G$$