|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 assigns to a Lie algebra – or more generally an ∞-Lie algebra or ∞-Lie algebroid – a Lie group – or more generally ∞-Lie groupoid – that is infinitesimally modeled by . The reverse operation to Lie differentiation.
If the ∞-Lie algebroids involved are incarnated dually in the form of their Chevalley-Eilenberg algebras then the bare ∞-groupoid (that is: without the smooth structure) integrating them is effectively given by the Sullivan construction from rational homotopy theory which turns a dg-algebra into a simplicial set (and then into a topological space by geometric realization) applied here to the dg-algebra .
This construction applied to an ordinary Lie algebra reproduces the integration method by paths in standard Lie theory (maybe less widely known than other integration methods). See our first example below.
For , a -path in the -Lie algebroid is a morphism of -Lie algebroids
from the tangent Lie algebroid of the standard smooth -simplex to .
Dually this a morphism of dg-algebra
For an -Lie algebroid, the -paths in naturally form a simplicial set
which is a Kan complex under mild technical fine-tuning of the definition of -paths.
This is (up to fine-tuning of the nature of the differential forms on the simplices) the Sullivan construction of rational homotopy theory that tuns a dg-algvebra into a simplicial set, applied to the dg-algebra .
(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 discussing smooth families of -paths we need the following technical notion.
For regard the -simplex as a smooth manifold with corners in the standard way. We think of this embedded into the Cartesian space in the standard way with maximal rotation symmetry about the center of the simplex, and equip with the metric space structure induced this way.
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.
Note that 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
a standard piecewise smooth retracts, the pullbacks
glue to a single smooth form (with sitting instants) on .
Notice 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 all CartSp and .
The underlying discrete ∞-groupoid is recovered as that of the -parametrized 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 Cohesive ∞-groups – Lie groups for 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 line Lie 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.
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||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|
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
A detailed review of how the traditional Lie integration of Lie algebras and Lie algebroids to Lie groups and Lie groupoids (including the smooth structure) is reproduced in terms of -pathis is given in
The description of Lie integration with values in [[smooth ∞-groupoid]s] regarded as simplicial presheaves on CartSp is in
Essentially the same integration prescription is considered in
Application to the problem of Lie integrating ordinary but infinite-dimensional Lie algebras is in