integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
Lie integration assigns to a Lie algebra $\mathfrak{g}$ – or more generally an ∞-Lie algebra or ∞-Lie algebroid – a Lie group – or more generally ∞-Lie groupoid – that is infinitesimally modeled by $\mathfrak{g}$. The reverse operation to Lie differentiation.
If the ∞-Lie algebroids $\mathfrak{a}$ involved are incarnated dually in the form of their Chevalley-Eilenberg algebras $CE(\mathfrak{a})$ 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 $CE(\mathfrak{a})$.
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.
Let $\mathfrak{a}$ be an ∞-Lie algebroid (for instance a Lie algebra, or a Lie algebroid or an L-∞-algebra).
For $n \in \mathbb{N}$ write $\Delta^n$ for the $n$-simplex regarded as a smooth manifold (with boundary and corners).
For $d \in \mathbb{N}$, a $d$-path in the $\infty$-Lie algebroid is a morphism of $\infty$-Lie algebroids
from the tangent Lie algebroid $T \Delta^d_{Diff}$ of the standard smooth $d$-simplex to $\mathfrak{a}$.
Dually this a morphism of dg-algebra
from the Chevalley-Eilenberg algebra of $\mathfrak{a}$ to the de Rham complex.
Here we discuss the discrete ∞-groupoids underlying the smooth ∞-groupoids to which an ∞-Lie algebroid integrates.
For $\mathfrak{a}$ an $\infty$-Lie algebroid, the $d$-paths in $\mathfrak{a}$ naturally form a simplicial set
which is a Kan complex under mild technical fine-tuning of the definition of $d$-paths.
Since morphisms of ∞-Lie algebroids are dually equivalent to dg-algebra morphisms of their Chevalley-Eilenberg algebra, the above is equivalent to
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 $CE(\mathfrak{a})$.
(spurious homotopy groups)
For $\mathfrak{a}$ a Lie n-algebroid (an $n$-truncated $\infty$-Lie algebroid) this construction will not yield in general an $n$-truncated ∞-groupoid $\exp(\mathfrak{a})$.
To see this, consider the example (discussed in detail below) that $\mathfrak{a} = \mathfrak{g}$ is an ordinary Lie algebra. Then $\exp(\mathfrak{g})_n$ is canonically identified with the set of smooth based maps $\Delta^n \to G$ into the simply connected Lie group that integrates $\mathfrak{g}$ in ordinary Lie theory. This means that the simplicial homotopy groups of $\exp(\mathfrak{g})$ are the topological homotopy groups of $G$, which in general (say for $G$ the orthogonal group or unitary group) will be non-trivial in arbitrarily higher degree, even though $\mathfrak{g}$ 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 $\exp(\mathfrak{g})$ are generated from the generators in a minimal Sullivan model resolution of $\mathfrak{g}$.
For the purposes of $\infty$-Lie theory therefore instead one wants to truncate $\exp(\mathfrak{g})$ to its $(n+1)$-coskeleton
This divides out n-morphisms by $(n+1)$-morphisms and forgets all higher higher nontrivial morphisms, hence all higher homotopy groups.
We now discuss Lie integration of $\infty$-Lie algebroids to smooth ∞-groupoids, presented by the model structure on simplicial presheaves $[CartSp_{smooth}^{op}, sSet]_{proj,loc}$ over the site CartSp${}_{smooth}$.
For discussing smooth families of $d$-paths we need the following technical notion.
For $k \in \mathbb{N}$ regard the $k$-simplex $\Delta^k$ as a smooth manifold with corners in the standard way. We think of this embedded into the Cartesian space $\mathbb{R}^k$ in the standard way with maximal rotation symmetry about the center of the simplex, and equip $\Delta^k$ with the metric space structure induced this way.
A smooth differential form $\omega$ on $\Delta^k$ is said to have sitting instants along the boundary if, for every $(r \lt k)$-face $F$ of $\Delta^k$ there is an open neighbourhood $U_F$ of $F$ in $\Delta^k$ such that $\omega$ restricted to $U$ is constant in the directions perpendicular to the $r$-face on its value restricted to that face.
More generally, for any $U \in$ CartSp a smooth differential form $\omega$ on $U \times\Delta^k$ is said to have sitting instants if there is $0 \lt \epsilon \in \mathbb{R}$ such that for all points $u : * \to U$ the pullback along $(u, \mathrm{Id}) : \Delta^k \to U \times \Delta^k$ is a form with sitting instants on $\epsilon$-neighbourhoods of faces.
Smooth forms with sitting instants clearly form a sub-dg-algebra of all smooth forms. We write $\Omega^\bullet_{si}(U \times \Delta^k)$ for this sub-dg-algebra.
We write $\Omega_{si,vert}^\bullet(U \times \Delta^k)$ for the further sub-dg-algebra of vertical differential forms with respect to the projection $p : U \times \Delta^k \to U$, hence the coequalizer
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 $k$ perpendicular directions to a vertex.
A smooth 0-form (a smooth function) has sitting instants on $\Delta^1$ if in a neighbourhood of the endpoints it is constant.
A smooth function $f : U \times \Delta^1 \to \mathbb{R}$ is in $\Omega^0_{\mathrm{vert}}(U \times \Delta^1)$ if there is $0 \lt \epsilon \in \mathbb{R}$ such that for each $u \in U$ the function $f(u,-) : \Delta^1 \simeq [0,1] \to \mathbb{R}$ is constant on $[0,\epsilon) \coprod (1-\epsilon,1)$.
A smooth 1-form has sitting instants on $\Delta^1$ if in a neighbourhood of the endpoints it vanishes.
Let $X$ be a smooth manifold, $\omega \in \Omega^\bullet(X)$ be a smooth differential form. Let
be a smooth function that has sitting instants as a function: towards any $k$-face of $\Delta^n$ it eventually becomes perpendicularly constant.
Then the pullback form $\phi^* \omega \in \Omega^\bullet(\Delta^n)$ 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 $\Delta^n$. Notably for $\omega_j \in \Omega^\bullet(\Delta^{n-1})$ a collection of forms with sitting instants on the $(n-1)$-cells of a horn $\Lambda^n_i$ that coincide on adjacent boundaries, and for
a standard piecewise smooth retracts, the pullbacks
glue to a single smooth form (with sitting instants) on $\Delta^n$.
Notice that $\omega \in \Omega^\bullet(\Delta^n)$ having sitting instants does not imply that there is a neighbourhood of the boundary of $\Delta^n$ on which $\omega$ is entirely constant. It is important for the following constructions that in the vicinity of the boundary $\omega$ is allowed to vary parallel to the boundary, just not perpendicular to it.
For the following definition recall the presentation of smooth ∞-groupoids by the model structure on simplicial presheaves over the site CartSp${}_{smooth}$.
For $\mathfrak{a}$ an L-∞ algebra of finite type with Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ define the simplicial presheaf $\exp(\mathfrak{a}) : CartSp_{smooth}^{op} \to sSet$ by
for all $U \in$ CartSp and $[n] \in \Delta$.
Compared to the integration to discrete ∞-groupoids above this definition knows about $U$-parametrized smooth families of $n$-paths in $\mathfrak{g}$.
The underlying discrete ∞-groupoid is recovered as that of the $\mathbb{R}^0 = *$-parametrized family:
The objects $\exp(\mathfrak{g})$ are indeed Kan complexes over each $U \in$ CartSp.
Observe that the standard continuous horn retracts $f : \Delta^k \to \Lambda^k_i$ are smooth away from the preimages of the $(r \lt k)$-faces of $\Lambda[k]^i$.
For $\omega \in \Omega^\bullet_{si,vert}(U \times \Lambda[k]^i)$ a differential form with sitting instants on $\epsilon$-neighbourhoods, let therefore $K \subset \partial \Delta^k$ be the set of points of distance $\leq \epsilon$ from any subface. Then we have a smooth function
The pullback $f^* \omega \in \Omega^\bullet(\Delta^k \setminus K)$ may be extended constantly back to a form with sitting instants on all of $\Delta^k$.
The resulting assignment
Write $\mathbf{cosk}_{n+1} \exp(a)$ for the simplicial presheaf obtained by postcomposing $\exp(\mathfrak{a}) : CartSp^{op} \to sSet$ with the $(n+1)$-coskeleton functor $\mathbf{cosk}_{n+1} : sSet \stackrel{tr_n}{\to} sSet_{\leq n+1} \stackrel{cosk_{n+1}}{\to} sSet$.
See also at smooth ∞-groupoid the section Exponentiated ∞-Lie algebras.
Let $\mathfrak{g} \in L_\infty$ be an ordinary (finite dimensional) Lie algebra. Standard Lie theory (see Lie's three theorems) provides a simply connected Lie group $G$ integrating $\mathfrak{g}$.
With $G$ regarded as a smooth ∞-group write $\mathbf{B}G \in$ Smooth∞Grpd for its delooping. The standard presentation of this on $[CartSp_{smooth}^{op}, sSet]$ is by the simplicial presheaf
See Cohesive ∞-groups – Lie groups for details.
The operation of parallel transport $P \exp(\int -) : \Omega^1([0,1], \mathfrak{g}) \to G$ yields a weak equivalence (in $[CartSp^{op}, sSet]_{proj}$)
This follows from the Steenrod-Wockel approximation theorem and the following observation.
For $X$ a simply connected smooth manifold and $x_0 \in X$ a basepoint, there is a canonical bijection
between the set of Lie-algebra valued 1-forms on $X$ whose curvature 2-form vanishes, and the set of smooth functions $X\to G$ that take $x_0$ to the neutral element $e \in G$.
The bijection is given as follows. For $A \in \Omega^1_{flat}(X,\mathfrak{g})$ a flat 1-form, the corresponding function $f_A : X \to G$ sends $x \in X$ to the parallel transport along any path $x_0 \to x$ from the base point to $x$
Because of the assumption that the curvature 2-form of $A$ vanishes and the assumption that $X$ is simply connected, this assignment is independent of the choice of path.
Conversely, for every such function $f : X \to G$ we recover $A$ as the pullback of the Maurer-Cartan form on $G$
From this we obtain
The $\infty$-groupoid $\mathbf{cosk}_2 \exp(\mathfrak{g})$ 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 $\Delta^1 \to G$ (with sitting instants), where two of these are taken to be equivalent if there is a smooth homotopy $D^2 \to G$ (with sitting instant) between them.
Since $G$ is simply connected, these equivalence classes are labeled by the endpoints of these paths, hence are canonically identified with $G$.
We do not need to fall back to classical Lie theory to obtain $G$ in the above argument. A detailed discussion of how to find $G$ with its group structure and smooth structure from $d$-paths in $\mathfrak{g}$ is in (Crainic).
For $n \in \mathbb{N}, n \geq 1$ write $b^{n-1} \mathbb{R}$ for the L-∞-algebra whose Chevalley-Eilenberg algebra is given by a single generator in degree $n$ and vanishing differential. We may call this the line Lie $n$-algebra.
Write $\mathbf{B}^{n} \mathbb{R}$ for the smooth line (n+1)-group.
The discrete ∞-groupoid underlying $\exp(b^{n-1} \mathbb{R})$ is given by the Kan complex that in degree $k$ has the set of closed differential $n$-forms (with sitting instants) on the $k$-simplex
The $\infty$-Lie integration of $b^{n-1} \mathbb{R}$ is the line Lie n-group $\mathbf{B}^{n} \mathbb{R}$.
Moreover, with $\mathbf{B}^n \mathbb{R}_{chn} \in [CartSp_{smooth}^{op}, sSet]$ the standard presentation given under the Dold-Kan correspondence by the chain complex of sheaves concentrated in degree $n$ on $C^\infty(-, \mathbb{R})$ the equivalence is induced by the fiber integration of differential $n$-forms over the $n$-simplex:
First we observe that the map
is a morphism of simplicial presheaves $\exp(b^{n-1} \mathbb{R}) \to \mathbf{B}^{n}\mathbb{R}_{chn}$ on CartSp${}_{smooth}$. 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 $n$. Let $\lambda \in \Omega_{si,vert,cl}^n(\Delta^{n+1})$. The differential of the normalized chains complex sends this to the signed sum of its restrictions to the $n$-faces of the $(n+1)$-simplex. Followed by the integral over $\Delta^n$ this is the piecewise integral of $\lambda$ over the boundary of the $n$-simplex. Since $\lambda$ has sitting instants, there is $0 \lt \epsilon \in \mathbb{R}$ such that there are no contributions to this integral in an $\epsilon$-neighbourhood of the $(n-1)$-faces. Accordingly the integral is equivalently that over the smooth surface inscribed into the $(n+1)$-simplex, as indicated in the following diagram
Since $\lambda$ is a closed form on the $n$-simplex, this surface integral vanishes, by the Stokes theorem. Hence $\int_{\Delta^\bullet}$ is indeed a chain map.
It remains to show that $\int_{\Delta^\bullet} : \exp(b^{n-1} \mathbb{R}) \to \mathbf{B}^{n}\mathbb{R}_{chn}$ is an isomorphism on all the simplicial homotopys group over each $U \in CartSp$. This amounts to the statement that
a smooth family of closed $n$-forms with sitting instants on the boundary of $\Delta^{n+1}$ may be extended to a smooth family of closed forms with sitting instants on $\Delta^{n+1}$ precisely if their smooth family of integrals over the boundary vanishes;
Any smooth family of closed $n \lt k$-forms with sitting instants on the boundary of $\Delta^{k+1}$ may be extended to a smooth family of closed $n$-forms with sitting instants on $\Delta^{k+1}$.
To demonstrate this, we want to work with forms on the $(k+1)$-ball instead of the $(k+1)$-simplex. To achieve this, choose again $0 \lt \epsilon \in \mathbb{R}$ and construct the diffeomorphic image of $S^k \times [1,1-\epsilon]$ inside the $(k+1)$-simplex as indicated in the above diagram: outside an $\epsilon$-neighbourhood of the corners the image is a rectangular $\epsilon$-thickening of the faces of the simplex. Inside the $\epsilon$-neighbourhoods of the corners it bends smoothly. By the Steenrod-Wockel approximation theorem the diffeomorphism from this $\epsilon$-thickening of the smoothed boundary of the simplex to $S^k \times [1-\epsilon,1]$ extends to a smooth function from the $(k+1)$-simplex to the $(k+1)$-ball.
By choosing $\epsilon$ smaller than each of the sitting instants of the given $n$-form on $\partial \Delta^{k+1}$, we have that this $n$-form vanishes on the $\epsilon$-neighbourhoods of the corners and is hence entirely determined by its restriction to the smoothed simplex, identified with the $(k+1)$-ball.
It is now sufficient to show: a smooth family of smooth $n$-forms $\omega \in \Omega^n_{vert,cl}(U \times S^k)$ extends to a smooth family of closed $n$-forms $\hat \omega \in \Omega^n_{vert,cl}(U \times B^{k+1})$ that is radially constant in a neighbourhood of the boundary for all $n \lt k$ and for $k = n$ precisely if its smooth family of integrals vanishes, $\int_{S^k} \omega = 0 \in C^\infty(U, \mathbb{R})$.
Notice that over the point this is a direct consequence of the de Rham theorem: an $n$-form $\omega$ on $S^k$ is exact precisely if $n \lt k$ or if $n = k$ and its integral vanishes. In that case there is an $(n-1)$-form $A$ with $\omega = d A$. Choosing any smoothing function $f : [0,1] \to [0,1]$ (smooth, surjective, non,decreasing and constant in a neighbourhood of the boundary) we obtain an $n$-form $f \wedge A$ on $(0,1] \times S^k$, vertically constant in a neighbourhood of the ends of the interval, equal to $A$ at the top and vanishing at the bottom. Pushed forward along the canonical $(0,1] \times S^k \to D^{k+1}$ this defines a form on the $(k+1)$-ball, that we denote by the same symbol $f \wedge A$. Then the form $\hat \omega := d (f \wedge A)$ 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 $(n-1)$-form $A$ in a way depending smoothly on the the $n$-form $\omega$.
One way of achieving this is using Hodge theory. Fix a Riemannian metric on $S^n$, and let $\Delta$ be the corresponding Laplace operator, and $\pi$ the projection on the space of harmonic forms. Then the central result of Hodge theory for compact Riemannian manifolds states that the operator $\pi$, seen as an operator from the de Rham complex to itself, is a cochain map homotopic to the identity, via an explicit homotopy $P := d^* G$ expressed in terms of the adjoint $d^*$ of the de Rham differential and of the Green operator $G$ of $\Delta$. Since the $k$-form $\omega$ is exact its projection on harmonic forms vanishes. Therefore
Hence $A := P\omega$ is a solution of the differential equation $d A=\omega$ depending smoothly on $\omega$.
Let $\mathfrak{string} = \mathfrak{g}_\mu$ be the string Lie 2-algebra.
Then $\mathbf{cosk}_3 \exp(\mathfrak{g}_\mu)$ is equivalent to the 2-groupoid $\mathbf{B}String$
with a single object;
whose morphisms are based paths in $G$;
whose 2-morphisms are equivalence class of pairs $(\Sigma,c)$, where
$\Sigma : D^2_* \to G$ is a smooth based map (where we use a homeomorphism $D^2 \simeq \Delta^2$ which away from the corners is smooth, so that forms with sitting instants there do not see any non-smoothness, and the basepoint of $D^2_*$ is the 0-vertex of $\Delta^2$)
and $c \in U(1)$, and where two such are equivalent if the maps coincides at their boundary and if for any 3-ball $\phi : D^3 \to G$ filling them the labels $c_1, c_2 \in U(1)$ differ by the integral $\int_{D^3} \phi^* \mu(\theta) \;\; mod \;\; \mathbb{Z}$,,
where $\theta$ is the Maurer-Cartan form, $\mu(\theta) = \langle \theta\wedge [\theta \wedge \theta]\rangle$ the 3-form obtained by plugging it into the cocycle.
This is the string Lie 2-group. It’s construction in terms of integration by paths is due to (Henriques)
Examples of sequences of local structures
geometry | point | first order infinitesimal | $\subset$ | formal = arbitrary order infinitesimal | $\subset$ | local = stalkwise | $\subset$ | finite |
---|---|---|---|---|---|---|---|---|
$\leftarrow$ differentiation | integration $\to$ | |||||||
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 | $\mathbb{F}_p$ finite field | $\mathbb{Z}_p$ p-adic integers | $\mathbb{Z}_{(p)}$ localization at (p) | $\mathbb{Z}$ 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 basic idea of identifying the Sullivan construction applied to Chevalley-Eilenberg algebras as Lie integration to discrete ∞-groupoids appears in
and for general ∞-Lie algebras in
(whose main point is the discussion of a gauge condition applicable for nilpotent $L_\infty$-algebras that cuts down the result of the Sullivan construction to a much smaller but equivalent model) .
This was refined from integration to bare $\infty$-groupoids to an integration to internal ∞-groupoids in Banach manifolds in
(whose origin possibly preceeds that of Getzler’s article).
For general ∞-Lie algebroids the general idea of the integration process by “$d$-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 $d$-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
A characterization of the ∞-stacks obtained by Lie integration as above is in theorem 5.3 of
The Lie integration- of Lie algebroid representations $\mathfrak{a} \to end(V)$ to morphisms of ∞-categories $A \to Ch_\bullet^\circ$ / higher parallel transport is discussed in
Application to the problem of Lie integrating ordinary but infinite-dimensional Lie algebras is in
A generalization of Lie integration to conjectural Leibniz groups has been conjectured by J-L. Loday. A local version via local Lie racks has been proposed in