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.
The higher notions of differential cohomology and Chern-Weil theory make sense in any cohesive (∞,1)-topos
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.)
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
describe connections on ∞-bundles.
For $n \in \mathbb{N}$ say that $\nabla$ admits parallel $n$-transport if for all smooth manifolds $\Sigma$ of dimension $n$ and all morphisms
we have that the pullback of $\nabla$ to $\Sigma$
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.
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.
For $\nabla : X \to A_{conn}$ an $\infty$-connection that admits parallel $n$-transport, this is for each $\phi : \Sigma \to X$ the morphism
that corresponds to $\phi^* \nabla$ under the equivalence
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.
We now define the higher analogs of holonomy for the case that $\Sigma$ is closed.
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
in the homotopy category
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 $\Sigma$:
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.
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
We may map this further to its $(n-dim \Sigma)$-truncation
We have
(This is due to an observation by Domenico Fiorenza.)
By general abstract reasoning (recalled at cohomology and fiber sequence) we have for the homotopy groups that
Now use the universal coefficient theorem, which asserts that we have an exact sequence
Since $U(1)$ is an injective $\mathbb{Z}$-module we have
This means that we have an isomorphism
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
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
The resulting morphism
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;.
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).
See parallel 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
i.e. morphisms in $2DiffeoGrpd$ are characterized by Lie 2-algebra valued differential forms on $X$.
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
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
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
Using this we obtain a smooth function
Then set
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.
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
Let
and consider the map $f : [0,1]^3 \to Z$ given by
and the map $g : Z \to X$ given away from $(x^2 + y^2) = 0$ by
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
as $[0,1]^3 \times [0,1]^2 \to Z \times [0,1]^2 \to Z \to X$
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
By the above construction we have the the push-forwards
and similarly for $\frac{\partial}{\partial y}$ are indendent of $z$. It follows by the chain rule that also
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.
see 3-groupoid of Lie 3-algebra valued forms
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
an equivalence. The abstract fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos $\Pi$ is here the ordinary fundamental ∞-groupoid
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
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
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
in Top, as described at principal ∞-bundle.
Using the fundamental ∞-groupoid functor we may send this equivalently to a fiber sequence in ∞Grpd
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
be the global section (∞,1)-geometric morphism of the cohesive (∞,1)-topos ∞LieGrpd.
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
now in the smooth context of $\infty LieGrpd$.
The ∞-groupoid of $F$-fibrations in Top is equivalent to the $\infty$-groupoid of $Disc Aut(F)$-principal ∞-bundles in ∞LieGrpd:
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
always exists.
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.
(…)
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
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).
In physics various action functionals for quantum field theories are nothing but higher parallel transport.
The gauge interaction part of the action functional for the particle charged under a background electromagnetic field, which is a circle bundle with connection $\nabla$, is the parallel 1-transport of $\nabla$.
The gauge interaction part of the action functional for the string charged under a background Kalb-Ramond field, which is a circle 2-bundle with connection $\nabla$, is the parallel 2-transport of $\nabla$.
The gauge interaction part of the action functional for the membrane charged under a background supergravity C-field, which is a circle 3-bundle with connection $\nabla$, is the parallel 3-transport of $\nabla$.
The action functional of Chern-Simons theory is the parallel 3-transport of a Chern-Simons circle 3-bundle.
connection on a bundle, connection on a 2-bundle, connection on an infinity-bundle,
parallel transport, higher parallel transport
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
John Baez, Urs Schreiber, Higher gauge theory, in A. Davydov et al. (eds.), Categories in Algebra, Geometry and Mathematical Physics, Contemp Math 431, AMS, Providence, Rhode Island (2007) pp 7-30 (arXiv:0511710, arXi:hep-th/0412325hep-th/0412325)
Urs Schreiber, Konrad Waldorf Connections on nonabelian gerbes and their holonomy, Theory Appl. Categ., Vol. 28, 2013, No. 17, pp 476-540 (arXiv:0808.1923, TAC)
Applications are discussed in
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)
David Lipsky, Cocycle constructions for topological field theories (2010) (pdf)
see also the discussion at fiber integration in ordinary differential cohomology.
Realization of this as an extended TQFT is discussed in
Parallel transport with coefficients in crossed complexes/strict infinity-groupoids is discussed in
The integration of flat differential forms with values in chain complexes to flat $\infty$-parallel transport on $\infty$-vector bundles is in
based on
in turn based on constructions in
Remarks on $\infty$-parallel transport in Top are in