nLab
higher parallel transport

Context

\infty-Chern-Weil theory

Differential cohomology

Contents

Idea

A connection on a bundle induces a notion of parallel transport over paths . A connection on a 2-bundle induces a generalization of this to a notion of parallel transport over surfaces . Similarly a connection on a 3-bundle induces a notion of parallel transport over 3-dimensional volumes.

Generally, a connection on an ∞-bundle induces a notion of parallel transport in arbitrary dimension.

Definition

The higher notions of differential cohomology and Chern-Weil theory make sense in any cohesive (∞,1)-topos

(ΠDiscΓ):HΓDiscΠGrpdTop. (\Pi \dashv Disc \dashv \Gamma) : \mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd \simeq Top \,.

In every such there is a notion of connection on an ∞-bundle and of its higher parallel transport.

A typical context considered (more or less explicitly) in the literature is H=\mathbf{H} = ∞LieGrpd, the cohesive (,1)(\infty,1)-topos of smooth ∞-groupoids. But other choices are possible. (See also the Examples.)

Higher parallel transport

Let AA be an ∞-Lie groupoid such that morphisms XAX \to A in ∞LieGrpd classify the AA-principal ∞-bundles under consideration. Write A connA_{conn} for the differential refinement described at ∞-Lie algebra valued form, such that lifts

A conn X g A \array{ && A_{conn} \\ & {}^{\mathllap{\nabla}}\nearrow & \downarrow \\ X &\stackrel{g}{\to}& A }

describe connections on ∞-bundles.

Definition

For nn \in \mathbb{N} say that \nabla admits parallel nn-transport if for all smooth manifolds Σ\Sigma of dimension nn and all morphisms

ϕ:ΣX \phi : \Sigma \to X

we have that the pullback of \nabla to Σ\Sigma

ϕ *:ΣϕXA conn \phi^* \nabla : \Sigma \stackrel{\phi}{\to} X \stackrel{\nabla}{\to} A_{conn}

flat in that it factors through the canonical inclusion AA conn\mathbf{\flat}A \to A_{conn}.

In other words: if all the lower curvature kk-forms, 1kn1 \leq k \leq n of ϕ *\phi^* \nabla vanish (the higher ones vanish automatically for dimensional reasons).

Here A=[Π(),A]\mathbf{\flat}A = [\mathbf{\Pi}(-),A] is the coefficient for flat differential A-cohomology.

Remark

This condition is automatically satisfied for ordinary connections on bundles, hence for A=BGA = \mathbf{B}G with GG an ordinary Lie group: because in that case there is only a single curvature form, namely the ordinary curvature 2-form.

But for a principal 2-bundle with connection there is in general a 2-form curvature and a 3-form curvature. A 2-connection therefore admits parallel transport only if its 2-form curvature is constrained to vanish.

Notice however that if the coefficient object AA happens to be (n1)(n-1)-connected – for instance if it is an Eilenberg-MacLane object in degree nn, then there is no extra condition and every connection admits parallel transport. This is notably the case for circle n-bundles with connection.

Definition

For :XA conn\nabla : X \to A_{conn} an \infty-connection that admits parallel nn-transport, this is for each ϕ:ΣX\phi : \Sigma \to X the morphism

Π(Σ)A \mathbf{\Pi}(\Sigma) \to A

that corresponds to ϕ *\phi^* \nabla under the equivalence

H(Σ,A)H(Π(Σ),A). \mathbf{H}(\Sigma, \mathbf{\flat}A ) \simeq \mathbf{H}(\mathbf{\Pi}(\Sigma), A) \,.
Remark

The objects of the path ∞-groupoid Π(Σ)\mathbf{\Pi}(\Sigma) are points in Σ\Sigma, the morphisms are paths in there, the 2-morphisms surfaces between these paths, and so on. Hence a morphism Π(Σ)A\mathbf{\Pi}(\Sigma) \to A assigns fibers in AA to points in XX, and equivalences between these fibers to paths in Σ\Sigma, and so on.

Higher holonomy

We now define thee (DR - three? where are the other two?) higher analogs of holonomy for the case that Σ\Sigma is closed.

Definition

Let :XA conn\nabla : X \to A_{conn} be a connection with parallel nn-transport and ϕ:ΣX\phi : \Sigma \to X a morphism from a closed nn-manifold.

Then the nn-holonomy of \nabla over Σ\Sigma is the image [ϕ *][\phi^* \nabla] of

ϕ *:Π(Σ)Γ(A) \phi^* \nabla : \Pi(\Sigma) \to \Gamma(A)

in the homotopy category

[ϕ *][Π(Σ),Γ(A)] [\phi^* \nabla] \in [\Pi(\Sigma), \Gamma(A)]

Examples

For trivial circle nn-bundles / for nn-forms

The simplest example is the parallel transport in a circle n-bundle with connection over a smooth manifold XX whose underlying B n1U(1)\mathbf{B}^{n-1}U(1)-bundle is trivial. This is equivalently given by a degree nn-differential form AΩ n(X)A \in \Omega^n(X). For ϕ:Σ nX\phi : \Sigma_n \to X any smooth function from an nn-dimensional manifold Σ\Sigma, the corresponding parallel transport is simply the integral of AA over Σ\Sigma:

tra A(Σ)=exp(i Σϕ *A)U(1). \tra_A(\Sigma) = \exp(i \int_\Sigma \phi^* A) \;\;\; \in \;\; U(1) \,.

One can understand higher parallel transport therefore as a generalization of integration of diifferential nn-forms to the case where

For circle nn-bundles with connection

We show how the nn-holonomy of circle n-bundles with connection is reproduced from the above.

Let ϕ *:Π(Σ)B nU(1)\phi^* \nabla : \mathbf{\Pi}(\Sigma) \to \mathbf{B}^n U(1) be the parallel transport for a circle n-bundle with connection over a ϕ:ΣX\phi : \Sigma \to X.

This is equivalent to a morphism

Π(Σ) nU(1),. \Pi(\Sigma) \to \mathcal{B}^n U(1) ,.

We may map this further to its (ndimΣ)(n-dim \Sigma)-truncation

:Grpd(Π(Σ), nU(1))τ ndimΣGrpd(Π(X), nU(1)). :\infty Grpd(\Pi(\Sigma), \mathcal{B}^n U(1)) \to \tau_{n-dim \Sigma} \infty Grpd(\Pi(X), \mathcal{B}^n U(1)) \,.
Theorem

We have

τ ndimΣGrpd(Π(Σ), nU(1))B ndimΣU(1). \tau_{n-dim\Sigma} \infty Grpd(\Pi(\Sigma), \mathcal{B}^n U(1)) \simeq \mathbf{B}^{n-dim \Sigma} U(1) \,.

(This is due to an observation by Domenico Fiorenza.)

Proof

By general abstract reasoning (recalled at cohomology and fiber sequence) we have for the homotopy groups that

(1)π iGrpd(Π(Σ), nU(1))H ni(Σ,U(1)). \pi_i \infty Grpd(\Pi(\Sigma),\mathcal{B}^n U(1)) \simeq H^{n-i}(\Sigma, U(1)) \,.

Now use the universal coefficient theorem, which asserts that we have an exact sequence

(2)0Ext 1(H ni1(Σ,),U(1))H ni(Σ,U(1))Hom(H ni(Σ,),U(1))0. 0 \to Ext^1(H_{n-i-1}(\Sigma,\mathbb{Z}),U(1)) \to H^{n-i}(\Sigma,U(1)) \to Hom(H_{n-i}(\Sigma,\mathbb{Z}),U(1)) \to 0 \,.

Since U(1)U(1) is an injective \mathbb{Z}-module we have

Ext 1(,U(1))=0. Ext^1(-,U(1))=0 \,.

This means that we have an isomorphism

(3)H ni(Σ,U(1))Hom Ab(H ni(Σ,),U(1)) H^{n-i}(\Sigma,U(1)) \simeq Hom_{Ab}(H_{n-i}(\Sigma,\mathbb{Z}),U(1))

that identifies the cohomology group in question with the internal hom in Ab from the integral homology group of Σ\Sigma to U(1)U(1).

For i<(ndimΣ)i\lt (n-dim \Sigma), the right hand is zero, so that

π iGrpd(Π(Σ),B nU(1))=0fori<(ndimΣ). \pi_i \infty Grpd(\Pi(\Sigma),\mathbf{B}^n U(1)) =0 \;\;\;\; for i\lt (n-dim \Sigma) \,.

For i=(ndimΣ)i=(n-dim \Sigma), instead, H ni(Σ,)H_{n-i}(\Sigma,\mathbb{Z})\simeq \mathbb{Z}, since Σ\Sigma is a closed dimΣdim \Sigma-manifold and so

π (ndimΣ)Grpd(Π(Σ), nU(1))U(1). \pi_{(n-dim\Sigma)} \infty Grpd(\Pi(\Sigma),\mathcal{B}^n U(1))\simeq U(1) \,.
Definition

The resulting morphism

H(Σ,A conn)exp(iS())B ndimΣU(1) \mathbf{H}(\Sigma, A_{conn}) \stackrel{\exp(i S(-))}{\to} \mathbf{B}^{n-dim\Sigma} U(1)

in ∞Grpd we call the \infty-Chern-Simons action on Σ\Sigma.

Here in the language of quantum field theory

Nonabelian parallel transport in low dimension

At least in low categorical dimension one has the definition of the path n-groupoid P n(X)\mathbf{P}_n(X) of a smooth manifold, whose nn-morphisms are thin homotopy-classes of smooth functions [0,1] nX[0,1]^n \to X. Parallel nn-transport with only the (n+1)(n+1)-curvature form possibly nontrivial and all the lower curvature degree 1- to nn-forms nontrivial may be expressed in terms of smooth nn-functors out of P n\mathbf{P}_n (SWI, SWII, MartinsPickenI, MartinsPickenII).

1-Transport

See parallel transport.

2-Transport

We work now concretely in the category 2DiffeoGrpd2DiffeoGrpd of 2-groupoids internal to the category of diffeological spaces.

Let XX be a smooth manifold and write P 2(X)2DiffeoGrpd\mathbf{P}_2(X) \in 2DiffeoGrpd for its path 2-groupoid. Let GG be a Lie 2-group and BG2DiffeoGrpd\mathbf{B}G \in 2DiffeoGrpd its delooping 1-object 2-groupoid. Write 𝔤\mathfrak{g} for the corresponding Lie 2-algebra.

Assume now first that GG is a strict 2-group given by a crossed module (G 1G 0)(G_1 \to G_0). Corresponding to this is a differential crossed module (𝔤 1𝔤 0)(\mathfrak{g}_1 \to \mathfrak{g}_0).

We describe now how smooth 2-functors

tra:P 2(X)BG tra : \mathbf{P}_2(X) \to \mathbf{B}G

i.e. morphisms in 2DiffeoGrpd2DiffeoGrpd are characterized by Lie 2-algebra valued differential forms on XX.

Definition

Given a morphism F:P 2(X)BGF : \mathbf{P}_2(X) \to \mathbf{B}G we construct a 𝔤 1\mathfrak{g}_1-valued 2-form B FΩ 2(X,𝔤 1)B_F \in \Omega^2(X, \mathfrak{g}_1) as follows.

To find the value of B FB_F on two vectors v 1,v 2T pXv_1, v_2 \in T_p X at some point, choose any smooth function

Γ: 2X \Gamma : \mathbb{R}^2 \to X

with

  • Γ(0,0)=p\Gamma(0,0) = p

  • dds| s=0Γ(s,0)=v 1\frac{d}{d s}|_{s = 0} \Gamma(s,0) = v_1

  • ddt| t=0Γ(0,t)=v 2\frac{d}{d t}|_{t = 0} \Gamma(0,t) = v_2.

Notice that there is a canonical 2-parameter family

Σ : 22MorP 2( 2) \Sigma_{\mathbb{R}} : \mathbb{R}^2 \to 2Mor \mathbf{P}_2(\mathbb{R}^2)

of classes of bigons on the plane, given by sending (s,t) 2(s,t) \in \mathbb{R}^2 to the class represented by any bigon (with sitting instants) with straight edges filling the square

Σ (s,t)=((0,0) (0,t) (s,0) (s,t)). \Sigma_{\mathbb{R}}(s,t) = \left( \array{ (0,0) &\to& (0,t) \\ \downarrow && \downarrow \\ (s,0) &\to& (s,t) } \right) \,.

Using this we obtain a smooth function

F Γ: 2Σ 2MorP 2( 2)Γ *2MorP 2(X)FG 0×G 1p 2G 1. F_\Gamma : \mathbb{R}^2 \stackrel{\Sigma_{\mathbb{R}}}{\to} 2Mor \mathbf{P}_2(\mathbb{R}^2) \stackrel{\Gamma_*}{\to} 2Mor \mathbf{P}_2(X) \stackrel{F}{\to} G_0 \times G_1 \stackrel{p_2}{\to} G_1 \,.

Then set

B F(v 1,w 1):= 2F Γxy| (0,0). B_F(v_1, w_1) := \frac{\partial^2 F_\Gamma}{\partial x \partial y}|_{(0,0)} \,.
Proposition

This is well defined, in that B F(v 1,v 2)B_F(v_1,v_2) does not depend on the choices made. Moreover, the 2-form defines this way is smooth.

Proof

To see that the definition does not depend on the choice of Γ\Gamma, proceed as follows.

For given vectors v 1,v 2T XXv_1,v_2 \in \T_X X let Γ,Γ: 2X\Gamma, \Gamma' : \mathbb{R}^2 \to X be two choices of smooth maps as in the defnition. By restricting, if necessary, to a neighbourhood of the origin of 2\mathbb{R}^2, we may assume without restriction that these maps land in a single coordinate patch in XX. Using the vector space structure of n\mathbb{R}^n defined by such a patch, define a smooth homotopy

τ:[0,1] 3X:(x,y,z)(1z)Γ(x,y)+zΓ(x,y) \tau : [0,1]^3 \to X : (x,y,z) \mapsto (1-z)\Gamma(x,y) + z \Gamma'(x,y)

Let

Z={(x,y,w)[0,1] 3|0w12(x 2+y 2)} Z = \{(x,y,w) \in [0,1]^3 | 0 \leq w \leq \frac{1}{2}(x^2 + y^2) \}

and consider the map f:[0,1] 3Zf : [0,1]^3 \to Z given by

f:(x,y,z)(x,y,12(x 2+y 2)z) f : (x,y,z) \mapsto (x,y, \frac{1}{2}(x^2 + y^2) z)

and the map g:ZXg : Z \to X given away from (x 2+y 2)=0(x^2 + y^2) = 0 by

g:(x,y,w)τ(x,y,2wx 2+y 2). g : (x,y,w) \mapsto \tau(x,y, 2 \frac{w}{x^2 + y^2}) \,.

Using Hadamard's lemma and the fact that by constructon τ\tau has vanishing 0th and 1st order differentials at the origin it follows that this is indeed a smooth function.

We want to similarly factor the smooth family of bigons [0,1] 32Mor(P 2(X))[0,1]^3 \to 2Mor(\mathbf{P}_2(X)) given by

[0,1] 3×[0,1] 2X [0,1]^3 \times [0,1]^2 \to X
((x,y,z),(s,t))τ(sx,ty,z) ((x,y,z),(s,t)) \mapsto \tau(s x, t y, z)

as [0,1] 3×[0,1] 2Z×[0,1] 2ZX[0,1]^3 \times [0,1]^2 \to Z \times [0,1]^2 \to Z \to X

((x,y,z),(s,t))((x,y,12(x 2+y 2)),(s,t))(sx,ty,12((sx) 2+(ty) 2)z)τ(sx,sy,z). ((x,y,z),(s,t)) \mapsto ((x, y, \frac{1}{2}(x^2 + y^2)), (s,t)) \mapsto (s x , t y, \frac{1}{2}((s x)^2 + (t y)^2)z) \mapsto \tau(s x, s y, z) \,.

As before using Hadamard’s lemma this is a sequence of smooth functions. To make this qualify as a family of bigons (which are maps from the square that are constant in a neighbourhood of the left and right boundary of the square) furthermore precompose this with a suitable smooth function [0,1] 2[0,1] 2[0,1]^2 \to [0,1]^2 that realizes a square-shaped bigon.

Under the hom-adjunction it corresponds to a factorization of G Γ:[0,1] 32Mor(P 2(X))G_\Gamma : [0,1]^3 \to 2 Mor(\mathbf{P}_2(X)) into

G Γ:[0,1] 3fZ2Mor(P 2(X)). G_\Gamma : [0,1]^3 \stackrel{f}{\to} Z \to 2 Mor(\mathbf{P}_2(X)) \,.

By the above construction we have the the push-forwards

f *:x(x=0,y=0,z)x(x=0,y=0,w=0) f_* : \frac{\partial}{\partial x}(x=0,y=0,z) \mapsto \frac{\partial}{\partial x}(x= 0, y = 0, w = 0)

and similarly for y\frac{\partial}{\partial y} are indendent of zz. It follows by the chain rule that also

2G Γxy| (x=0,y=0) \frac{\partial^2 G_\Gamma}{\partial x \partial y}|_{(x=0,y=0)}

is independent of zz. But at z=0z = 0 this equals 2F Γxy| (x=0,y=0)\frac{\partial^2 F_\Gamma}{\partial x \partial y}|_{(x=0,y=0)}, while at z=1z = 1 it equals 2F Γxy| (x=0,y=0)\frac{\partial^2 F_{\Gamma'}}{\partial x \partial y}|_{(x=0,y=0)}. Therefore these two are equal.

3-Transport

see 3-groupoid of Lie 3-algebra valued forms

Flat \infty-parallel transport in TopTop

Even though it is a degenerate case, it can be useful to regard the (∞,1)-topos Top explicitly a cohesive (∞,1)-topos. For a discussion of this see discrete ∞-groupoid.

For H=\mathbf{H} = Top lots of structure of cohesive (,1)(\infty,1)-topos theory degenerates, since by the homotopy hypothesis-theorem here the global section (∞,1)-geometric morphism

(ΠΔΓ):TopΓΔΠGrpd (\Pi \dashv \Delta \dashv \Gamma) : Top \stackrel{\overset{\Pi}{\leftarrow}}{\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}} \in \infty Grpd

an equivalence. The abstract fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos Π\Pi is here the ordinary fundamental ∞-groupoid

Π:TopGrpd. \Pi : Top \stackrel{\simeq}{\to} \infty Grpd \,.

If both (∞,1)-toposes here are presented by their standard model category models, the standard model structure on simplicial sets and the standard model structure on topological spaces, then Π\Pi is presented by the singular simplicial complex functor in a Quillen equivalence

(||Sing):Top QuillenTop. (|-| \dashv Sing) : Top \stackrel{\leftarrow}{\overset{\simeq_{Quillen}}{\to}} Top \,.

This means that in this case many constructions in topology and classical homotopy theory have equivalent reformulations in terms of \infty-parallel transport.

For instance: for FTopF \in Top and Aut(F)TopAut(F) \in Top its automorphism ∞-group, FF-fibrations over a base space XTopX \in Top are classfied by morphisms

g:XBAut(F) g : X \to B Aut(F)

into the delooping of Aut(F)Aut(F). The corresponding fibration PXP \to X itself is the homotopy fiber of this cocycles, given by the homotopy pullback

P * X g BAut(F) \array{ P &\to& * \\ \downarrow && \downarrow \\ X &\stackrel{g}{\to}& B Aut(F) }

in Top, as described at principal ∞-bundle.

Using the fundamental ∞-groupoid functor we may send this equivalently to a fiber sequence in ∞Grpd

Π(P)Π(X)BAut(Π(F)). \Pi(P) \to \Pi(X) \to B Aut(\Pi(F)) \,.

One may think of the morphism Π(X)BAut(Π(F))\Pi(X) \to B Aut(\Pi(F)) now as the \infty-parallel transport coresponding to the original fibration:

  • to each point in XX it assigns the unique object of BAut(Π(F))B Aut(\Pi(F)), which is the fiber FF itself;

  • to each path (xy)(x \to y) in XX it assigns an equivalence between the fibers F xtoF yF_x to F_y etc.

If one presents Π\Pi by Sing:TopsSet QuillenSing : Top \to sSet_{Quillen} as above, then one may look for explicit simplicial formulas that express these morphisms. Such are discussed in Stasheff.

We may embed this example into the smooth context by regarding Aut(F)Aut(F) as a discrete ∞-Lie groupoid as discussed in the section Flat ∞-Parallel transport in ∞LieGrpd.

For that purpose let

(Π smoothDisc smoothΓ smooth):LieGrpdΓ smoothDisc smoothΠ smoothGrpdTop (\Pi_{smooth} \dashv Disc_{smooth} \dashv \Gamma_{smooth}) : \infty LieGrpd \stackrel{\overset{\Pi_{smooth}}{\to}}{\stackrel{\overset{Disc_{smooth}}{\leftarrow}}{\underset{\Gamma_{smooth}}{\to}}} \infty Grpd \simeq Top

be the global section (∞,1)-geometric morphism of the cohesive (∞,1)-topos ∞LieGrpd.

We may reflect the ∞-group Aut(F)Aut(F) into this using the constant ∞-stack-functor DiscDisc to get the discrete ∞-Lie group DiscAut(F)Disc Aut(F). Let then XX be a paracompact smooth manifold, regarded naturally as an object of ∞LieGrpd. Then we can consider cocycles/classifying morphisms

XBDiscAut(F), X \to \mathbf{B} Disc Aut(F) \,,

now in the smooth context of LieGrpd\infty LieGrpd.

Proposition

The ∞-groupoid of FF-fibrations in Top is equivalent to the \infty-groupoid of DiscAut(F)Disc Aut(F)-principal ∞-bundles in ∞LieGrpd:

LieGrpd(X,BDiscAut(F))Top(X,BAut(F)). \infty LieGrpd(X, \mathbf{B} Disc Aut(F)) \simeq Top(X, B Aut(F)) \,.

Moreover, all the principal ∞-bundles classified by the morphisms on the left have canonical extensions to Flat differential cohomology in LieGrpd\infty LieGrpd, in that the flat parallel \infty-transport flat\nabla_{flat} in

X g BDiscAut(F) flat Π(X) \array{ X &\stackrel{g}{\to}& \mathbf{B} Disc Aut(F) \\ \downarrow & \nearrow_{\nabla_{flat}} \\ \mathbf{\Pi}(X) }

always exists.

Proof

The first statement is a special case of that spelled out at ∞LieGrpd and nonabelian cohomology. The second follows using that in a connected locally ∞-connected (∞,1)-topos the functor DiscDisc is a full and faithful (∞,1)-functor.

Flat \infty-parallel transport in LieGrpd\infty LieGrpd

(…)

\infty-Parallel transport from flat differential forms with values in chain complexes

A typical choice for an (∞,1)-category of “\infty-vector spaces” is that presented by the a model structure on chain complexes of modules. In a geometric context this may be replaced by some stack of complexes of vector bundles over some site.

If we write ModMod for this stack, then the \infty-parallel transport for a flat \infty-vector bundle on some XX is a morphism

Π(X)Mod. \mathbf{\Pi}(X) \to Mod \,.

This is typically given by differential form data with values in ModMod.

A discussion of how to integrate flat differential forms with values in chain complexes – a representation of the tangent Lie algebroid as discussed at representations of ∞-Lie algebroids – to flat \infty-parallel transport Π(X)Mod\mathbf{\Pi}(X) \to Mod is in (AbadSchaetz), building on a construciton in (Igusa).

Applications

In physics

In physics various action functionals for quantum field theories are nothing but higher parallel transport.

References

For references on ordinary 1-dimensional parallel transport see parallel transport.

For references on parallel 2-transport in bundle gerbes see connection on a bundle gerbe.

The description of parallel nn-transport in terms of nn-functors on the path n-groupoid for low nn is in

The description of connections on a 2-bundle in terms of such parallel 2-transport

Parallel transport for circle n-bundles with connection is discussed generally in

  • Kiyonori Gomi and Yuji Terashima, Higher dimensional parallel transport Mathematical Research Letters 8, 25–33 (2001) (pdf)

and

  • David Lipsky, Cocycle constructions for topological field theories (2010) (pdf)

see also the discussion at fiber integration in ordinary differential cohomology.

The integration of flat differential forms with values in chain complexes toflat \infty-parallel transport on \infty-vector bundles is in

based on

  • Jonathan Block, Aaron Smith, A Riemann Hilbert correspondence for infinity local systems (arXiv)

in turn based on constructions in

Remarks on \infty-parallel transport in Top are in

Revised on May 27, 2013 17:36:27 by Urs Schreiber (82.113.98.67)