# nLab higher parallel transport

### Context

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

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

## Theorems

#### Differential cohomology

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

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

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 $\mathbf{H} =$ ∞LieGrpd, the cohesive $(\infty,1)$-topos of smooth ∞-groupoids. But other choices are possible. (See also the Examples.)

### Higher parallel transport

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

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

describe connections on ∞-bundles.

###### Definition

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

$\phi : \Sigma \to X$

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

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

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

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

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

###### Remark

This condition is automatically satisfied for ordinary connections on bundles, hence for $A = \mathbf{B}G$ with $G$ 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 $A$ happens to be $(n-1)$-connected – for instance if it is an Eilenberg-MacLane object in degree $n$, 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 $\nabla : X \to A_{conn}$ an $\infty$-connection that admits parallel $n$-transport, this is for each $\phi : \Sigma \to X$ the morphism

$\mathbf{\Pi}(\Sigma) \to A$

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

$\mathbf{H}(\Sigma, \mathbf{\flat}A ) \simeq \mathbf{H}(\mathbf{\Pi}(\Sigma), A) \,.$
###### Remark

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

### Higher holonomy

We now define the higher analogs of holonomy for the case that $\Sigma$ is closed.

###### Definition

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

Then the $n$-holonomy of $\nabla$ over $\Sigma$ is the image $[\phi^* \nabla]$ of

$\phi^* \nabla : \Pi(\Sigma) \to \Gamma(A)$

in the homotopy category

$[\phi^* \nabla] \in [\Pi(\Sigma), \Gamma(A)]$

## Examples

### For trivial circle $n$-bundles / for $n$-forms

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

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

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

• the $n$-form is not globally defined;

• the $n$-form takes values not in $\mathbb{R}$ but more generally is an ∞-Lie algebroid valued differential form.

### For circle $n$-bundles with connection

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

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

This is equivalent to a morphism

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

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

$:\infty Grpd(\Pi(\Sigma), \mathcal{B}^n U(1)) \to \tau_{n-dim \Sigma} \infty Grpd(\Pi(X), \mathcal{B}^n U(1)) \,.$
###### Theorem

We have

$\tau_{n-dim\Sigma} \infty Grpd(\Pi(\Sigma), \mathcal{B}^n U(1)) \simeq \mathbf{B}^{n-dim \Sigma} U(1) \,.$

(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)$\pi_i \infty Grpd(\Pi(\Sigma),\mathcal{B}^n U(1)) \simeq H^{n-i}(\Sigma, U(1)) \,.$

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

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

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

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

This means that we have an isomorphism

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

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

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

$\pi_i \infty Grpd(\Pi(\Sigma),\mathbf{B}^n U(1)) =0 \;\;\;\; for i\lt (n-dim \Sigma) \,.$

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

$\pi_{(n-dim\Sigma)} \infty Grpd(\Pi(\Sigma),\mathcal{B}^n U(1))\simeq U(1) \,.$
###### Definition

The resulting morphism

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

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

Here in the language of quantum field theory

• the objects of $\mathbf{H}(\Sigma,A_{conn})$ are the gauge field on $\Sigma$;

• the morphisms in $\mathbf{H}(\Sigma, A_{conn})$ are the gauge transformations;.

### Nonabelian parallel transport in low dimension

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

#### 2-Transport

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

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

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

We describe now how smooth 2-functors

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

i.e. morphisms in $2DiffeoGrpd$ are characterized by Lie 2-algebra valued differential forms on $X$.

###### Definition

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

To find the value of $B_F$ on two vectors $v_1, v_2 \in T_p X$ at some point, choose any smooth function

$\Gamma : \mathbb{R}^2 \to X$

with

• $\Gamma(0,0) = p$

• $\frac{d}{d s}|_{s = 0} \Gamma(s,0) = v_1$

• $\frac{d}{d t}|_{t = 0} \Gamma(0,t) = v_2$.

Notice that there is a canonical 2-parameter family

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

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

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

Using this we obtain a smooth function

$F_\Gamma : \mathbb{R}^2 \stackrel{\Sigma_{\mathbb{R}}}{\to} 2Mor \mathbf{P}_2(\mathbb{R}^2) \stackrel{\Gamma_*}{\to} 2Mor \mathbf{P}_2(X) \stackrel{F}{\to} G_0 \times G_1 \stackrel{p_2}{\to} G_1 \,.$

Then set

$B_F(v_1, w_1) := \frac{\partial^2 F_\Gamma}{\partial x \partial y}|_{(0,0)} \,.$
###### Proposition

This is well defined, in that $B_F(v_1,v_2)$ 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 $\Gamma$, proceed as follows.

For given vectors $v_1,v_2 \in \T_X X$ let $\Gamma, \Gamma' : \mathbb{R}^2 \to X$ be two choices of smooth maps as in the defnition. By restricting, if necessary, to a neighbourhood of the origin of $\mathbb{R}^2$, we may assume without restriction that these maps land in a single coordinate patch in $X$. Using the vector space structure of $\mathbb{R}^n$ defined by such a patch, define a smooth homotopy

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

Let

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

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

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

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

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

Using Hadamard's lemma and the fact that by constructon $\tau$ 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 $[0,1]^3 \to 2Mor(\mathbf{P}_2(X))$ given by

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

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

$((x,y,z),(s,t)) \mapsto ((x, y, \frac{1}{2}(x^2 + y^2)), (s,t)) \mapsto (s x , t y, \frac{1}{2}((s x)^2 + (t y)^2)z) \mapsto \tau(s x, s y, z) \,.$

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 $[0,1]^2 \to [0,1]^2$ that realizes a square-shaped bigon.

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

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

By the above construction we have the the push-forwards

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

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

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

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

### Flat $\infty$-parallel transport in $Top$

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

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

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

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

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 $\Pi$ is presented by the singular simplicial complex functor in a Quillen equivalence

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

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 $F \in Top$ and $Aut(F) \in Top$ its automorphism ∞-group, $F$-fibrations over a base space $X \in Top$ are classfied by morphisms

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

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

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

in Top, as described at principal ∞-bundle.

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

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

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

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

• to each path $(x \to y)$ in $X$ it assigns an equivalence between the fibers $F_x to F_y$ etc.

If one presents $\Pi$ by $Sing : Top \to sSet_{Quillen}$ 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 $Aut(F)$ as a discrete ∞-Lie groupoid as discussed in the section Flat ∞-Parallel transport in ∞LieGrpd.

For that purpose let

$(\Pi_{smooth} \dashv Disc_{smooth} \dashv \Gamma_{smooth}) : \infty LieGrpd \stackrel{\overset{\Pi_{smooth}}{\to}}{\stackrel{\overset{Disc_{smooth}}{\leftarrow}}{\underset{\Gamma_{smooth}}{\to}}} \infty Grpd \simeq Top$

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

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

now in the smooth context of $\infty LieGrpd$.

###### Proposition

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

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

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

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

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 $Disc$ is a full and faithful (∞,1)-functor.

(…)

### $\infty$-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 $Mod$ for this stack, then the $\infty$-parallel transport for a flat $\infty$-vector bundle on some $X$ is a morphism

$\mathbf{\Pi}(X) \to Mod \,.$

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

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 $\mathbf{\Pi}(X) \to Mod$ 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 $n$-transport in terms of $n$-functors on the path n-groupoid for low $n$ 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)

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