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geometry of physics -- flat connections

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Flat connections

Model Layer

Flat 1-connections

XX connected, π 1(X)\pi_1(X) \in Grp its fundamental group for any choice of basepoint, then the holonomy pairing

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

descends to homotopy classes of (based) loops

hol:H conn,flat 1(X,G)Hom Grp(π 1(X),G)/G 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? GG-principal connections to the quotient set of group homomorphisms π 1(X)G\pi_1(X) \to G modulo the adjoint action of GG on itself.

Semantic Layer

Definition

For GGrp(H)G \in Grp(\mathbf{H}) and XHX \in \mathbf{H} a flat GG-connection \nabla on XX is a morphism

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

We write

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

and accordingly

H flat 1X,Gπ 0H flat(X,G) H^1_{flat}{X, G} \coloneqq \pi_0 \mathbf{H}_{flat}(X,G)

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

Remark

By adjunction,

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

a flat GG-connection is equivalently a morphism

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

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

Definition

Write

UnderlyingBundle BG:BGBG UnderlyingBundle_{\mathbf{B}G} \colon \flat \mathbf{B}G \to \mathbf{B}G

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

Definition

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

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

modulates a GG-principal ∞-bundle on XX, by def. . This we call the underlying GG-principal bundle of \nabla.

ConstantPaths X:XΠ(X) ConstantPaths_{X} \colon X \to \Pi(X)

Syntactic Layer

BG:TypeUnderlyingBundle:BGBG \mathbf{B}G \colon Type \;\vdash \; UnderlyingBundle \colon \flat \mathbf{B}G \to \mathbf{B}G

Created on May 13, 2015 at 13:28:58. See the history of this page for a list of all contributions to it.