|analytic integration||cohomological integration|
|measurable space||Poincaré duality|
|measure||orientation in generalized cohomology|
|volume form||(virtual) fundamental class|
|Riemann/Lebesgue integration of differential forms||push-forward in generalized cohomology/in differential cohomology|
Lie integration is a process that assigns to a Lie algebra – or more generally to an ∞-Lie algebra or ∞-Lie algebroid – a Lie group – or more generally ∞-Lie groupoid – that is infinitesimally modeled by . It is essentially the reverse operation to Lie differentiation, except that there are in general several objects Lie integrating a given Lie algebraic datum, due to the fact that the infinitesimal data does not uniquely determine global topological properties.
Classically, Lie integration of Lie algebras is part of Lie's three theorems, which in particular finds an unique (up to isomorphism) simply connected Lie group integrating a given finite-dimensional Lie algebra.
One may observe that the simply connected Lie group integrating a (finite-dimensional) Lie algebra is equivalently realized as the collection of equivalence classes of Lie algebra valued 1-forms on the interval where two such are identified if they are interpolated by a flat Lie-algebra valued 1-form on the disk. (Duistermaat-Kolk 00, section 1.14, see also the example below).
In its evident generalization from Lie algebra valued differential forms to Lie algebroid valued differential forms this provides a means for Lie integration of Lie algebroids (e.g. Crainic-Fernandes 01).
In another direction, one may observe that L-∞ algebras are formally dually incarnated by their Chevalley-Eilenberg dg-algebras, and that under this identification the evident generalization of the path method to L-∞ algebra valued differential forms is essentially the Sullivan construction, known from rational homotopy theory, applied to these dg-algebras (Hinich 97, Getzler 04). Or rather, the bare such construction gives the geometrically discrete ∞-group underlying what should be the Lie integration to a smooth ∞-group. This is naturally obtained, as in the classical case, by suitably smoothly parameterizing the ∞-Lie algebroid valued differential forms (Henriques 08, Roytenberg 09, FSS 12).
Both these directions may be combined via the evident concept of ∞-Lie algebroid valued differential forms to yield a Lie integration of ∞-Lie algebroids to smooth ∞-groupoids. (Moreover, the same formula directly generalizes from -algebroids to A-infinity categories to yield the dg-nerve construction.)
While the construction exists and behaves as expected in examples, there is to date no good general theory providing higher analogs of, say, Lie's three theorems. But people are working on it.
for its Chevalley-Eilenberg algebra, a dg-algebra. Notice that, by the discussion at L-∞ algebra and at ∞-Lie algebroid, the Chevalley-Eilenberg dg-algebras is the formal dual of , in that the functor
is a fully faithful functor. Indeed, the following definition of Lie integration (being just a smooth refinement of the Sullivan construction) makes sense just as well for any dg-algebra, not necessarily in the essential image of this embedding. But only for dg-algebras in the essential image of this embedding do the examples come out as expected for higher Lie theory.
A -path in the -Lie algebroid is a morphism of -Lie algebroids of the form
See also at differential forms on simplices.
A -path in , def. 1, is equivalently
The Lie integration of is essentially the simplicial object whose -cells are the -paths in . However, in order for this to be well-behaved, it is possible and useful to restrict to -paths that are sufficiently well-behaved towards the boundary of the simplex:
A smooth differential form on is said to have sitting instants along the boundary if, for every -face of there is an open neighbourhood of in such that restricted to is constant in the directions perpendicular to the -face on its value restricted to that face.
More generally, for any CartSp a smooth differential form on is said to have sitting instants if there is such that for all points the pullback along is a form with sitting instants on -neighbourhoods of faces.
Smooth forms with sitting instants clearly form a sub-dg-algebra of all smooth forms. We write for this sub-dg-algebra.
The dimension of the normal direction to a face depends on the dimension of the face: there is one perpendicular direction to a codimension-1 face, and perpendicular directions to a vertex.
A smooth 0-form (a smooth function) has sitting instants on if in a neighbourhood of the endpoints it is constant.
A smooth function is in if there is such that for each the function is constant on .
A smooth 1-form has sitting instants on if in a neighbourhood of the endpoints it vanishes.
Let be a smooth manifold, be a smooth differential form. Let
Then the pullback form is a form with sitting instants.
The condition of sitting instants serves to make smooth differential forms not be affected by the boundaries and corners of . Notably for a collection of forms with sitting instants on the -cells of a horn that coincide on adjacent boundaries, and for
glue to a single smooth differential form (with sitting instants) on .
That having sitting instants does not imply that there is a neighbourhood of the boundary of on which is entirely constant. It is important for the following constructions that in the vicinity of the boundary is allowed to vary parallel to the boundary, just not perpendicular to it.
For an -Lie algebroid, the -paths in naturally form a simplicial set as varies:
(We are indicating only the face maps, not the degeneracy maps, just for notational simplicity).
If here instead of smooth differential forms one uses polynomial differential forms then this is precisely the Sullivan construction of rational homotopy theory applied to . We next realize smooth structure on this and hence realize this as an object in higher Lie theory.
(spurious homotopy groups)
To see this, consider the example (discussed in detail below) that is an ordinary Lie algebra. Then is canonically identified with the set of smooth based maps into the simply connected Lie group that integrates in ordinary Lie theory. This means that the simplicial homotopy groups of are the topological homotopy groups of , which in general (say for the orthogonal group or unitary group) will be non-trivial in arbitrarily higher degree, even though is just a Lie 1-algebra. This phenomenon is well familiar from rational homotopy theory, where a classical theorem asserts that the rational homotopy groups of are generated from the generators in a minimal Sullivan model resolution of .
This divides out n-morphisms by -morphisms and forgets all higher higher nontrivial morphisms, hence all higher homotopy groups.
for all CartSp and .
The underlying discrete ∞-groupoid is recovered as that of the -parameterized family:
For a differential form with sitting instants on -neighbourhoods, let therefore be the set of points of distance from any subface. Then we have a smooth function
The pullback may be extended constantly back to a form with sitting instants on all of .
The resulting assignment
See at smooth infinity-groupoid – structures – Lie groups for more details.
The operation of parallel transport yields a weak equivalence (in )
This follows from the Steenrod-Wockel approximation theorem and the following observation.
The bijection is given as follows. For a flat 1-form, the corresponding function sends to the parallel transport along any path from the base point to
Conversely, for every such function we recover as the pullback of the Maurer-Cartan form on
From this we obtain
The -groupoid is equivalent to the groupoid with a single object (no non-trivial 1-form on the point) whose morphisms are equivalence classes of smooth based paths (with sitting instants), where two of these are taken to be equivalent if there is a smooth homotopy (with sitting instant) between them.
Since is simply connected, these equivalence classes are labeled by the endpoints of these paths, hence are canonically identified with .
Write for the smooth line (n+1)-group.
The -Lie integration of is the circle n-group .
Moreover, with the standard presentation given under the Dold-Kan correspondence by the chain complex of sheaves concentrated in degree on the equivalence is induced by the fiber integration of differential -forms over the -simplex:
First we observe that the map
is a morphism of simplicial presheaves on CartSp. Since it goes between presheaves of abelian simplicial groups by the Dold-Kan correspondence it is sufficient to check that we have a morphism of chain complexes of presheaves on the corresponding normalized chain complexes.
The only nontrivial degree to check is degree . Let . The differential of the normalized chains complex sends this to the signed sum of its restrictions to the -faces of the -simplex. Followed by the integral over this is the piecewise integral of over the boundary of the -simplex. Since has sitting instants, there is such that there are no contributions to this integral in an -neighbourhood of the -faces. Accordingly the integral is equivalently that over the smooth surface inscribed into the -simplex, as indicated in the following diagram
Since is a closed form on the -simplex, this surface integral vanishes, by the Stokes theorem. Hence is indeed a chain map.
a smooth family of closed -forms with sitting instants on the boundary of may be extended to a smooth family of closed forms with sitting instants on precisely if their smooth family of integrals over the boundary vanishes;
Any smooth family of closed -forms with sitting instants on the boundary of may be extended to a smooth family of closed -forms with sitting instants on .
To demonstrate this, we want to work with forms on the -ball instead of the -simplex. To achieve this, choose again and construct the diffeomorphic image of inside the -simplex as indicated in the above diagram: outside an -neighbourhood of the corners the image is a rectangular -thickening of the faces of the simplex. Inside the -neighbourhoods of the corners it bends smoothly. By the Steenrod-Wockel approximation theorem the diffeomorphism from this -thickening of the smoothed boundary of the simplex to extends to a smooth function from the -simplex to the -ball.
By choosing smaller than each of the sitting instants of the given -form on , we have that this -form vanishes on the -neighbourhoods of the corners and is hence entirely determined by its restriction to the smoothed simplex, identified with the -ball.
It is now sufficient to show: a smooth family of smooth -forms extends to a smooth family of closed -forms that is radially constant in a neighbourhood of the boundary for all and for precisely if its smooth family of integrals vanishes, .
Notice that over the point this is a direct consequence of the de Rham theorem: an -form on is exact precisely if or if and its integral vanishes. In that case there is an -form with . Choosing any smoothing function (smooth, surjective, non,decreasing and constant in a neighbourhood of the boundary) we obtain an -form on , vertically constant in a neighbourhood of the ends of the interval, equal to at the top and vanishing at the bottom. Pushed forward along the canonical this defines a form on the -ball, that we denote by the same symbol . Then the form solves the problem.
To complete the proof we have to show that this simple argument does extend to smooth families of forms, i.e., that we can choose the -form in a way depending smoothly on the the -form .
One way of achieving this is using Hodge theory. Fix a Riemannian metric on , and let be the corresponding Laplace operator, and the projection on the space of harmonic forms. Then the central result of Hodge theory for compact Riemannian manifolds states that the operator , seen as an operator from the de Rham complex to itself, is a cochain map homotopic to the identity, via an explicit homotopy expressed in terms of the adjoint of the de Rham differential and of the Green operator of . Since the -form is exact its projection on harmonic forms vanishes. Therefore
Hence is a solution of the differential equation depending smoothly on .
Let be the string Lie 2-algebra.
Then is equivalent to the 2-groupoid
with a single object;
whose morphisms are based paths in ;
whose 2-morphisms are equivalence class of pairs , where
is a smooth based map (where we use a homeomorphism which away from the corners is smooth, so that forms with sitting instants there do not see any non-smoothness, and the basepoint of is the 0-vertex of )
and , and where two such are equivalent if the maps coincides at their boundary and if for any 3-ball filling them the labels differ by the integral ,,
where is the Maurer-Cartan form, the 3-form obtained by plugging it into the cocycle.
Unlike finite dimensional Lie algebras, not every Lie algebroid may be integrated to a Lie groupoid. There exists topological obstruction coming from of the leaf of a Lie algebroid . The integrability criteria is completely classified through the behaviour of certain monodromy groups and that is the achievement of Crainic-Fernandes 01. These monodromy groups, providing obstructions of integration, may be seen as the image of a transgression map of the long exact sequence of Lie algebroid homotopy groups? induced by the natural Lie algebroid fibration? . (see Brahic-Zhu 10 ).
Let us now describe the construction of the universal groupoid for a Lie algebroid . This is a major step contained in Crainic-Fernandes 01 and was earlier discovered in the setting of Poisson manifolds as the phase space of Poisson sigma model in Cattaneo-Felder 00.
Given a Lie algebroid , an -path (a Lie algebroid path), is a Lie algebroid morphism from , that is, it is a path in such that where is the base path of (see example ).
An (end-fixing) -homotopy (a Lie algebroid homotopy) is a Lie algebroid morphism from satisfying certain boundary conditions. Let us be more precise: a vector bundle morphism can be denoted by (see example ). Then the boundary condition is that and . The fact that is a Lie algebroid morphism is equivalent to the following PDE:
where is a TM connection on and and are certain time dependent sections of extending and respectively. Notice that there is also a way writing down the right hand side independent of choice of sections. See Section 1 of Crainic-Fernandes 01.
Then the universal groupoid associated to is the space of -paths () dividing by -homotopies. It is naturally a topological groupoid. But only when the obstruction vanishes, it becomes a Lie groupoid and is the source-simply connected Lie groupoid integrating .
Fortunately -homotopies form finite codimensional foliation , even though the quotient might not be always representible. Thus, represents a differentiable stack which becomes a stacky Lie groupoid over . It turns out that there is a one-to-one correspondence between 'etale stacky Lie groupoid and Lie algebroid. This correspondence provides a positive answer to Lie's third theorem for Lie algebroids. Tseng-Zhu 04.
Examples of sequences of local structures
|geometry||point||first order infinitesimal||formal = arbitrary order infinitesimal||local = stalkwise||finite|
|smooth functions||derivative||Taylor series||germ||smooth function|
|curve (path)||tangent vector||jet||germ of curve||curve|
|smooth space||infinitesimal neighbourhood||formal neighbourhood||germ of a space||open neighbourhood|
|function algebra||square-0 ring extension||nilpotent ring extension/formal completion||ring extension|
|arithmetic geometry||finite field||p-adic integers||localization at (p)||integers|
|Lie theory||Lie algebra||formal group||local Lie group||Lie group|
|symplectic geometry||Poisson manifold||formal deformation quantization||local strict deformation quantization||strict deformation quantization|
The “path method” of integrating Lie algebras to simply connected Lie groups appears in
and for general ∞-Lie algebras in
(whose main point is the discussion of a gauge condition applicable for nilpotent -algebras that cuts down the result of the Sullivan construction to a much smaller but equivalent model) .
(whose origin possibly preceeds that of Getzler’s article).
For general ∞-Lie algebroids the general idea of the integration process by “-paths” had been indicated in
Discussion of Lie integration of Lie algebroids by the path method is due to
upgraded to the stacky version by
Obstruction interpreted as transgression of a Lie algebroid fibration by
Essentially the same integration prescription is considered in
Application to the problem of Lie integrating ordinary but infinite-dimensional Lie algebras is in