k-morphisms are order higher gauge transformations.
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.
The canonical morphism
Here we are thinking of as a trivial bundle.
It is sufficient to show that for all we have
The first condition is evidently satisfied if already . The second condition follows with Cartan calculus and using that :
For a general -Lie algebra the curvature forms themselves are not closed, hence requiring them to have no component along the simplex does not imply that they descend. This is different for abelian -Lie algebras: for them the curvature forms themselves are already closed, and hence are themselves already curvature characteristics that do descent.
It is useful to organize the -valued form , together with its restriction to vertical differential forms and with its curvature characteristic forms in the commuting diagram (following Weil algebra – Characterization in the smooth infinity-topos)
The commutativity of this diagram is implied by .
Write for the -groupoid of -valued forms fitting into such diagrams.
If we just consider the top horizontal morphism in this diagram we obtain the object
discussed in detail at Lie integration. If we consider the top square of the diagram we obtain the object
We have an evident sequence of morphisms
where we label the objects by the structures they classify, as discussed at ∞-Chern-Weil theory.
Here the botton morphism is a weak equivalence and the others are monomorphisms.
Notice that in full ∞-Chern-Weil theory the fundamental object of interest is really – the object of pseudo-connections. The other objects only serve the purpose of picking particularly nice representatives:
the object may contain pseudo-connections, those for which at least the associated circle n-bundles with connection given by the -Chern Weil homomorphism are genuine connections.
This distinction is important: over objects ∞LieGrpd that are not smooth manifolds but for instance orbifolds, the genuine connections for higher Lie algebras do not exhaust all nonabelian differential cocycles. This just means that not every differential class in this case does have a nice representative in the usual sense.
Given a 1-morphism in , represented by -valued forms
consider the unique decomposition
with the horizonal differential form component and the canonical coordinate.
We describe now how this enccodes a gauge transformation
By the nature of the Weil algebra we have
where the sum is over all higher brackets of the ∞-Lie algebra .
Define the covariant derivative of the gauge parameter to be
In this notation we have
the general identity
This is known as the equation for infinitesimal gauge transformations of an -Lie algebra valued form.
(connections on ordinary bundles)
To see this, first note that the sheaves of objects on both sides are manifestly isomorphic, both are the sheaf of . For morphisms, observe that for a form which we may decompose into a horizontal and a verical pice as the condition is equivalent to the differential equation
For any initial value this has the unique solution
where is the parallel transport of :
(where for ease of notaton we write actions as if were a matrix Lie group).
In particular 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 .
For , we have that -valued differential forms are in natural bijection to ordinary closed differential forms in degree
Notice that under addition of differential forms, is over each an abelian simplicial group.
Under the Dold-Kan correspondence we may therefore identify with a presheaf of chain complexes.
The degreewise fiber integration of differential forms over simplices constitutes a morphism
that is a weak equivalence.
This is shown at circle n-bundle with connection – from Lie intgeration based on the discussion at ∞-Lie groupoid – Lie-integrated ∞-groups – differential coefficients.
What is called an “extended soft group manifold” in the literature on the D'Auria-Fre formulation of supergravity is really precisely a collection of -Lie algebroid valued forms with values in a super -Lie algebra such as the supergravity Lie 3-algebra (for 11-dimensional supergravity). The way curvature and Bianchi identity are read off from “extded soft group manifolds” in this literature is – apart from this difference in terminology – precisely what is described above.
-groupoid of ∞-Lie-algebra valued forms
The (obvious but conceptually important) observation that Lie algebra-valued 1-forms regarded as morphisms of graded vector spaces are equivalently morphisms of dg-algebras out of the Weil algebra and that one may think of as the identity as the universal -connection appears in early articles for instance highlighted on p. 15 of
but – somewhat implicitly – this construction appears earlier, notably in the D'Auria-Fre formulation of supergravity. A collection of such precursors to these notions is collected at
The structure of the formula (2) for infinitesimal gauge transformations of higher forms is widely known in the literature on supergravity and string theory, if maybe not formalized in terms of -Lie algebra theory as we do here. One exception is the remarkable book
The authors use the term extended soft group manifold for what here we identify as an -Lie algebra valued form .
With this terminological translation understood, and observing that all their constructions straightforwardly generalize to more general dg-algebras than these authors conisder explicitly, we find that
our equation (1) for the possibly shifted gauge transformation is their equation I.3.136;
In fact their full rheonomy constraint III.3.32 is essentialy the same horizontality constraint, but applied not just to the 1-simplex , but to the super simplex .