Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Application to gauge theory
In every cohesive (∞,1)-topos there is an intrinsic notion of ∞-Chern-Weil theory that gives rise to a notion of connection on principal ∞-bundles. We describe here details of the realization of this general abstract structure in the cohesive -topos Smooth∞Grpd of smooth ∞-groupoids.
For an ∞-Lie group, a connection on a smooth -principal ∞-bundle is a structure that supports the Chern-Weil homomorphism in Smooth∞Grpd: it interpolates between the nonabelian cohomology class of the bundle and the refinements to ordinary differential cohomology of its characteristic classes: the curvature characteristic classes.
This generalizes the notion of connection on a bundle and the ordinary Chern-Weil homomorphism in differential geometry.
See the Motivation section at Chern-Weil theory in Smooth∞Grpd and the page ∞-Chern-Weil theory introduction for more background.
For braided -groups
Let be a cohesive (∞,1)-topos equippd with differential cohesion and let be a braided ∞-group. Write
for the Maurer-Cartan form on the delooping ∞-group .
be the morphism out of a 0-truncated object which is universal with the property that for any manifold, the induced internal hom map
is a 1-epimorphism.
Then write for the (∞,1)-pullback in
We say that is the moduli ∞-stack of -principal -connections.
For instance for the circle n-group the moduli -stack is presented by the Deligne complex for ordinary differential cohomology in degree , hence is the moduli -stack for circle n-bundles with connection.
For -groups obtained by Lie integration
We assume that the reader is familiar with the notation and constructions discussed at Smooth∞Grpd. The following definition may be understood as a direct generalization of the description of ordinary -connections as cocycles in the stack as discussed at connection on a bundle, in view of the characterization of Weil algebra in the smooth infinity-topos
Chevalley-Eilenberg algebra CE Weil algebra W invariant polynomials inv
differential forms on moduli stack of principal connections (Freed-Hopkins 13):
We discuss now connections on those -principal ∞-bundles for which Smooth∞Grpd is an smooth ∞-group that arises from Lie integration of an L-∞ algebra .
Let dgAlg be an L-∞ algebra over the real numbers and of finite type with Chevalley-Eilenberg algebra and Weil algebra .
For a smooth manifold, write for the de Rham complex of smooth differential forms. For let be the standard -simplex regarded as a smooth manifold with corners in the standard way. Write for the sub-dg-algebra of differential forms with sitting instants perpendicular to the boundary of the simplex, and for the further sub-dg-algebra of vertical differential forms with respect to the canonical projection .
in dgAlg we call an L-∞ algebra valued differential form with values in , dually a morphism of ∞-Lie algebroids
from the tangent Lie algebroid to the inner automorphism ∞-Lie algebra.
Its curvature is the composite of morphisms of graded vector spaces
that injects the shifted generators into the Weil algebra.
Precisely if the curvatures vanish does the morphism factor through the Chevalley-Eilenberg algebra
in which case we call flat.
The curvature characteristic forms of are the composite
where is the inclusion of the invariant polynomials.
We define now simplicial presheaves over the site CartSp SmoothMfd of Cartesian spaces and smooth functions between them.
Write for the simplicial presheaf given by
(the untruncated Lie integration of ).
Write for the simplicial presheaf given by
Write for the simplicial presheaf given by
Define the simplicial presheaf by
Here on the right we have in each case the sets of horizontal morphisms in dgAlg that make commuting diagrams in dgAlg as indicated, with the vertical morphisms being the canonical projections and inclusions. The functoriality in and is by the evident precomposition with the pullback of differential forms and .
There are canonical morphisms in between these objects
where the horizontal morphisms are monomorphisms of simplicial presheaves and the vertical morphism is over each an equivalence of Kan complexes (it is a weak equivalence between fibrant objects in the projective model structure on simplicial presheaves).
The inclusion is clear. The weak equivalence is discussed at Smooth∞Grpd (but is also directly verified).
To see the inclusion we need to check that the horizonality condition on the curvature of a -valued form for all vector fields tangent to the simplex implies that all the curvature characteristic forms are basic forms that “descend to ”, hence that are in the image of the inclusion .
For this it is sufficient to show that for all we have
where in the second line we have the Lie derivative along .
The first condition is evidently satisfied if already . The second condition follows with Cartan calculus and using that (which holds as a consequence of the definition of invariant polynomial):
For a general L-∞ algebra the curvature forms themselves are not necessarily closed (rather they satisfy the Bianchi identity), hence requiring them to have no component along the simplex does not imply that they descend. This is different for abelian L-∞ algebras: for them the curvature forms themselves are already closed, and hence are themselves already curvature characteristics that do descent.
For let be the simplicial coskeleton functor. Its prolongation to simplicial presheaves we denote here and write
etc. This is the delooping
of the universal Lie integration of to an smooth n-group .
For any and any cofibrant resolution in the local projective model structure on simplicial presheaves (see Smooth∞Grpd for details), we say that the sSet-hom object
is the ∞-groupoid of smooth -principal ∞-bundles on ;
is the ∞-groupoid of smooth -principal ∞-bundles on equipped with pseudo -connection;
is the ∞-groupoid of smooth -principal ∞-bundles on equipped with -connection.
1-Morphisms: integration of infinitesimal gauge transformations
The 1-morphisms in may be thought of as gauge transformations between -valued forms. We unwind what these look like concretely.
Given a 1-morphism in , represented by -valued forms
consider the unique decomposition
with the horizonal differential form component and the canonical coordinate.
We call the gauge parameter . This is a function on with values in 0-forms on for an ordinary Lie algebra, plus 1-forms on for a Lie 2-algebra, plus 2-forms for a Lie 3-algebra, and so forth.
We describe now how this enccodes a gauge transformation
where the sum is over all higher brackets of the L-∞ algebra .
This is the result of applying the contraction to the defining equation for the curvature of using the nature of the Weil algebra:
and inserting the above decomposition for .
Define the covariant derivative of the gauge parameter to be
In this notation we have
This is known as the equation for infinitesimal gauge transformations of an -algebra valued form.
By Lie integration we have that – and hence – defines an element in the ∞-Lie group that integrates .
The unique solution of the above differential equation at for the initial values we may think of as the result of acting on with the gauge transformation .
Ordinary connections on principal 1-bundles
(connections on ordinary bundles)
For an ordinary Lie algebra with simply connected Lie group and for the groupoid of Lie algebra-valued forms we have an equivalence
betweenn the 1-truncated coefficient object for -valued -connections and the coefficient objects for ordinary connections on a bundle (see there).
Notice that the sheaves of objects on both sides are manifestly isomorphic, both are the sheaf of .
On morphisms, we have by the above for a form decomposed into a horizontal and a verical pice as that the condition is equivalent to the differential equation
For any initial value this has the unique solution
(with the Maurer-Cartan form on ), where is the parallel transport of :
(where for ease of notaton we write actions as if were a matrix Lie group).
This implies that the endpoints of the path of -valued 1-forms are related by the usual cocycle condition in
In the same fashion one sees that given 2-cell in and any 1-form on at one vertex, there is a unique lift to a 2-cell in , obtained by parallel transporting the form around. The claim then follows from the previous statement of Lie integration that .
higher Atiyah groupoid
The local differential form data of -connections was introduced in
The global description was then introduced in
A more comprehensive account is in sections 3.9.6, 3.9.7 of