Formal Lie groupoids
For an ∞-Lie algebra (or more generally ∞-Lie algebroid), the -groupoid of -valued forms is the ∞-groupoid whose
This naturally refines to a non-concrete ∞-Lie groupoid whose -parameterized smooth families of objects are ∞-Lie algebroid-valued differential forms on .
A cocycle with coefficients in this is a connection on an ∞-bundle.
For an introduction see the section ∞-Lie algebra valued forms at ∞-Chern-Weil theory introduction.
For a smooth manifold and an ∞-Lie algebra or more generally an ∞-Lie algebroid, a -Lie algebroid valued differential form on is a morphism of dg-algebras
from the Weil algebra of to the de Rham complex of . Dually this is 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
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.
For a smooth manifold, the -groupoid of -valued forms is the Kan complex
whose k-morphisms are -valued forms on with sitting instants, and with the property that their curvature vanishes on vertical vectors.
The canonical morphism
to the untruncated Lie integration of is given by restriction of to vertical differential forms (see below).
For a -valued form on and for any invariant polynomial, the corresponding curvature characteristic form descends down to .
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
The commutativity of this diagram is implied by .
Write for the -groupoid of -valued forms fitting into such diagrams.
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
By the nature of the Weil algebra we have
where the sum is over all higher brackets of the ∞-Lie algebra .
In Cartan calculus for an ordinary one writes the corresponding second Ehremsnn condition equivalently
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 -Lie 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 transformatin .
Lie algebra valued 1-forms
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 .
Lie 2-algebra valued forms
Ordinary -forms and the de Rham complex
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.
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
- Franz W. Kamber; Philippe Tondeur, Semisimplicial Weil algebras and characteristic classes for foliated bundles in Čech cohomology , Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 1, pp. 283–294. Amer. Math. Soc., Providence, R.I., (1975).
following Eli Cartan’s influential work (see Weil algebra for more references).
The (evident) generalization to Weil algebras of ∞-Lie algebras and ∞-Lie algebroids is considered explicitly in
- Hisham Sati, Urs Schreiber, Jim Staasheff, -algebra valued connections (web)
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
In this old book no -Lie algebras are mentioned explicitly, but the dg-algebra computations that are considered are easily seen to be precisely their Chevalley-Eilenberg algebra-incarnations.
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;
our equation (2) for the genuine gauge transfomation is their equation for horizontal or rheonomic gauge transformations III.3.23 .
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 .