Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Application to gauge theory
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 ∞LieGrpd, the cohesive -topos of smooth ∞-groupoids. But other choices are possible. (See also the Examples.)
Higher parallel transport
Let be an ∞-Lie groupoid such that morphisms in ∞LieGrpd classify the -principal ∞-bundles under consideration. Write for the differential refinement described at ∞-Lie algebra valued form, such that lifts
describe connections on ∞-bundles.
For say that admits parallel -transport if for all smooth manifolds of dimension and all morphisms
we have that the pullback of to
flat in that it factors through the canonical inclusion .
In other words: if all the lower curvature -forms, of vanish (the higher ones vanish automatically for dimensional reasons).
Here is the coefficient for flat differential A-cohomology.
This condition is automatically satisfied for ordinary connections on bundles, hence for with 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 happens to be -connected – for instance if it is an Eilenberg-MacLane object in degree , then there is no extra condition and every connection admits parallel transport. This is notably the case for circle n-bundles with connection.
For an -connection that admits parallel -transport, this is for each the morphism
that corresponds to under the equivalence
The objects of the path ∞-groupoid are points in , the morphisms are paths in there, the 2-morphisms surfaces between these paths, and so on. Hence a morphism assigns fibers in to points in , and equivalences between these fibers to paths in , and so on.
We now define the higher analogs of holonomy for the case that is closed.
Let be a connection with parallel -transport and a morphism from a closed -manifold.
Then the -holonomy of over is the image of
in the homotopy category
For trivial circle -bundles / for -forms
The simplest example is the parallel transport in a circle n-bundle with connection over a smooth manifold whose underlying -bundle is trivial. This is equivalently given by a degree -differential form . For any smooth function from an -dimensional manifold , the corresponding parallel transport is simply the integral of over :
One can understand higher parallel transport therefore as a generalization of integration of diifferential -forms to the case where
For circle -bundles with connection
We show how the -holonomy of circle n-bundles with connection is reproduced from the above.
Let be the parallel transport for a circle n-bundle with connection over a .
This is equivalent to a morphism
We may map this further to its -truncation
(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 is an injective -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 to .
For , the right hand is zero, so that
For , instead, , since is a closed -manifold and so
The resulting morphism
in ∞Grpd we call the -Chern-Simons action on .
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 of a smooth manifold, whose -morphisms are thin homotopy-classes of smooth functions . Parallel -transport with only the -curvature form possibly nontrivial and all the lower curvature degree 1- to -forms nontrivial may be expressed in terms of smooth -functors out of (SWI, SWII, MartinsPickenI, MartinsPickenII).
See parallel transport.
We work now concretely in the category of 2-groupoids internal to the category of diffeological spaces.
Let be a smooth manifold and write for its path 2-groupoid. Let be a Lie 2-group and its delooping 1-object 2-groupoid. Write for the corresponding Lie 2-algebra.
Assume now first that is a strict 2-group given by a crossed module . Corresponding to this is a differential crossed module .
We describe now how smooth 2-functors
i.e. morphisms in are characterized by Lie 2-algebra valued differential forms on .
Given a morphism we construct a -valued 2-form as follows.
To find the value of on two vectors at some point, choose any smooth function
Notice that there is a canonical 2-parameter family
of classes of bigons on the plane, given by sending to the class represented by any bigon (with sitting instants) with straight edges filling the square
Using this we obtain a smooth function
This is well defined, in that 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 , proceed as follows.
For given vectors let be two choices of smooth maps as in the defnition. By restricting, if necessary, to a neighbourhood of the origin of , we may assume without restriction that these maps land in a single coordinate patch in . Using the vector space structure of defined by such a patch, define a smooth homotopy
and consider the map given by
and the map given away from by
Using Hadamard's lemma and the fact that by constructon 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 given by
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 that realizes a square-shaped bigon.
Under the hom-adjunction it corresponds to a factorization of into
By the above construction we have the the push-forwards
and similarly for are indendent of . It follows by the chain rule that also
is independent of . But at this equals , while at it equals . Therefore these two are equal.
see 3-groupoid of Lie 3-algebra valued forms
Flat -parallel transport in
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 Top lots of structure of cohesive -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 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 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 -parallel transport.
For instance: for and its automorphism ∞-group, -fibrations over a base space are classfied by morphisms
into the delooping of . The corresponding fibration 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 now as the -parallel transport coresponding to the original fibration:
to each point in it assigns the unique object of , which is the fiber itself;
to each path in it assigns an equivalence between the fibers etc.
If one presents by 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 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 into this using the constant ∞-stack-functor to get the discrete ∞-Lie group . Let then 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 .
The ∞-groupoid of -fibrations in Top is equivalent to the -groupoid of -principal ∞-bundles in ∞LieGrpd:
Moreover, all the principal ∞-bundles classified by the morphisms on the left have canonical extensions to Flat differential cohomology in , in that the flat parallel -transport in
Flat -parallel transport in
-Parallel transport from flat differential forms with values in chain complexes
A typical choice for an (∞,1)-category of “-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 for this stack, then the -parallel transport for a flat -vector bundle on some is a morphism
This is typically given by differential form data with values in .
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 -parallel transport is in (AbadSchaetz), building on a construciton in (Igusa).
In physics various action functionals for quantum field theories are nothing but 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 -transport in terms of -functors on the path n-groupoid for low 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)
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.
Parallel transport with coefficients in crossed complexes/strict infinity-groupoids is discussed in
The integration of flat differential forms with values in chain complexes toflat -parallel transport on -vector bundles is in
- Jonathan Block, Aaron Smith, A Riemann Hilbert correspondence for infinity local systems (arXiv)
in turn based on constructions in
Remarks on -parallel transport in Top are in