synthetic differential geometry
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$(\Re \dashv \Im \dashv \&)$
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loop space object, free loop space object, derived loop space
The notion of the smooth loop space of a (smooth) manifold is a way to make the set of smooth maps from the circle to the target manifold into an object amenable to the tools of differential topology.
Let $M$ be a smooth finite dimensional manifold. Then the set of smooth maps $S^1 \to M$ is not usefully the underlying set of a finite dimensional manifold. Therefore to make it into an object that can be studied in differential topology, one has to allow for smooth spaces other than finite dimensional smooth manifolds. That is to say, one has to work in a larger category than just that of finite dimensional smooth manifolds.
If the category is cartesian closed then the smooth loop space of a smooth space, say $X$, is simply the result of applying the internal hom functor with that space as target and the circle as source:
(This is the smooth free loop space; for the smooth based loop space one applies the standard restriction.)
There are a variety of suitable categories listed at generalized smooth spaces.
The categories of smooth spaces discussed at generalized smooth spaces are all cartesian closed and thus all admit smooth loop spaces of all objects. However, by restricting to a subcategory that might not be cartesian closed (for example, finite dimensional smooth manifolds) one can ask for another subcategory that is large enough to contain all the resulting smooth loop spaces. For finite dimensional smooth manifolds it turns out that one can give their smooth loop spaces the structure of an infinite dimensional Fréchet manifold.
Depending on taste and application, certain smooth quotient objects of this free loop space may be and are often considered.
In this section, we will describe the structure of a smooth loop space of a smooth manifold as a Fréchet manifold in detail. The construction is quite standard, and a partial list of references and further reading can be found at the end of this page.
We start with a smooth manifold, $M$, of dimension $n$. Note that here manifolds definitely do not have a boundary. For simplicity, we assume that it is orientable. The point of this assumption is that it allows us to identify the model space of $L M$ with the Fréchet space $L \mathbb{R}^n = C^\infty (S^1, \mathbb{R}^n)$. In the unorientable case, some components would have this as model space whereas others would have a twisted version of this space.
The key piece of structure needed on $M$ is that of a local addition, $\eta \colon T M \to M$.
Let $\eta \colon T M \to M$ be a local addition on $M$. Let $V \subseteq M \times M$ be the image of the map $\pi \times \eta \colon T M \to M \times M$. Although (as yet) we know nothing about the topologies of $L T M$ or of $L V$, we can at least say that the looped map, $(\pi \times \eta)^L \colon L T M \to L V$, is a bijection.
Let $\alpha \in L M$. Define the set $U_\alpha \subseteq L M$ by:
Then the preimage of $\{\alpha\} \times U_\alpha$ under $(\pi \times \eta)^L$ is naturally identified with $\Gamma_{S^1}(\alpha^* T M)$. In particular, the zero section of $\alpha^* T M$ maps to $(\alpha, \alpha) \in \{\alpha\} \times U_\alpha$.
We claim that there is a diagram:
such that the bijection at the top takes the image of the left-hand vertical map to the image of the right-hand one. Both of the vertical maps are injective - the right-hand one obviously so, we shall investigate the left-hand one in a moment - and thus the bijection $(\pi \times \eta)^L$ induces a bijection from the lower-left to the lower-right.
The left-hand vertical map, $\Gamma_{S^1}(\alpha^* T M) \to L T M$, is defined as follows: the total space of $\alpha^* T M$ is:
It is an embedded submanifold of $S^1 \times T M$. Therefore, a map into $\alpha^* T M$ is smooth if and only if the compositions with the projections to $S^1$ and to $T M$ are smooth. Now a map $S^1 \to \alpha^* T M$ is a section if and only if it projects to the identity on $S^1$. Therefore, there is a bijection (of sets):
In particular, the map $\Gamma_{S^1}(\alpha^* T M) \to L T M$ is injective.
We apply $(\pi \times \eta)^L$ to the defining condition for $L_\alpha T M$ and see that $L_\alpha T M$ is the preimage under this map of everything of the form $(\alpha, \gamma)$ in $L V$. By construction, $\gamma \in L M$ is such that $(\alpha, \gamma) \in L V$ if and only if $\gamma \in U_\alpha$. Hence $(\pi \times \eta)^L$ identifies $L_\alpha T M$ with $\{\alpha\} \times U_\alpha$.
Finally, note that the zero section of $\alpha^* T M$ maps to the image of $\alpha$ under the zero section of $T M$. Since $\eta$ composed with the zero section of $T M$ is the identity on $M$, the image under the zero section of $\alpha^* T M$ in $V$ is $(\alpha, \alpha)$ as required.
The resulting map, let us write it $\Psi_\alpha \colon \Gamma_{S^1}(\alpha^* T M) \to U_\alpha$, has the following concrete description. Let $\beta \colon S^1 \to \alpha^* T M$ be a section and let $\tilde{\beta} \colon S^1 \to T M$ be the corresponding loop in $T M$ (so that $\beta(t) = (t, \tilde{\beta}(t))$ when viewing $\alpha^* T M$ as a submanifold of $S^1 \times T M$). Then $(\pi \times \eta)^L (\tilde{\beta}) = (\alpha, \eta^L (\tilde{\beta}))$ so $\Psi_\alpha(\beta) = \eta^L (\tilde{\beta})$.
As we have assumed $M$ to be orientable, $\alpha^* T M$ can be trivialised. A smooth such trivialisation defines a linear homeomorphism $\Gamma_{S^1}(\alpha^* T M) \cong L \mathbb{R}^n$. We use this to impose a smooth structure on $\Gamma_{S^1}(\alpha^* T M)$, noting that any two such trivialisations induce the same structure.
To investigate the transition functions, we need two loops. In fact, let’s have two of everything.
Let $\eta_1, \eta_2 \colon T M \to M$ be local additions with corresponding neighbourhoods $V_1$, $V_2$ of the diagonal in $M \times M$. Let $\alpha_1$, $\alpha_2$ be smooth loops in $M$. Let $\Psi_1 \colon \Gamma_{S^1}(\alpha_1^* T M) \to U_1$ and $\Psi_2 \colon \Gamma_{S^1}(\alpha_2^* T M) \to U_2$ be the corresponding maps defined as above.
The transition function:
is a diffeomorphism.
We start by characterising the space $\Psi_1^{-1}(U_1 \cap U_2)$ within $\Gamma_{S^1}(\alpha_1^* T M)$. Let $W_{1 2} \subseteq \alpha_1^* T M$ be the set:
Note that this is open in $\alpha_1^* T M$ as it is the preimage of the open set $V_2$ by the continuous map $\alpha_2 \times \eta_1$.
Let $\gamma \in \Gamma_{S^1}(\alpha_1^* T M)$ and let $\tilde{\gamma} \in L T M$ be the image of $\gamma$ (so that $\gamma(t) = (t, \tilde{\gamma}(t))$. Then $\gamma(t) \in W_{1 2}$ for all $t$ if and only if $(\alpha_2(t), \eta_1(\tilde{\gamma}(t))) \in V_2$ for all $t$. That is to say, if and only if $(\alpha_2, \eta_1^L(\tilde{\gamma})) \in L V_2$. This is precisely the condition that $\eta_1^L(\tilde{\gamma}) \in U_2$. Since $\eta_1^L(\tilde{\gamma}) = \Psi_1(\gamma)$, we see that $\gamma$ takes values in $W_{1 2}$ if and only if $\Psi_1(\gamma) \in U_2$. Since $\operatorname{Im} \Psi_1 = U_1$, we conclude that $\Gamma_{S^1}(W_{1 2}) = \Psi_1^{-1}(U_1 \cap U_2)$.
Let us define $W_{2 1} \subseteq \alpha_2^* T M$ similarly. The idea of the proof that $\Phi_{1 2}$ is a diffeomorphism is to show that it is induced by a diffeomorphism $W_{1 2} \cong W_{2 1}$.
Let $\theta_1 \colon W_{1 2} \to T M$ be the map:
The definition of $W_{1 2}$ ensures that $(\alpha_2(t) ,\eta_1(v)) \in V_2$ for $(t,v) \in W_{1 2}$ and this is the image of $\pi \times \eta_2$. Hence $\theta_1$ is well-defined. Define $\theta_2 \colon W_{2 1} \to T M$ similarly. These are both smooth maps.
Notice that $\pi(\pi \times \eta_i)^{-1} \colon V_i \subseteq M \times M \to M$ is the projection on to the first factor and $\eta_i(\pi \times \eta_i)^{-1} \colon V_i \to M$ is the projection on to the second. Thus $\pi \theta_1(t,v) = \alpha_2(t)$. Hence $\theta_1 \colon W_{1 2} \to T M$ is such that $(t, \theta_1(t,v)) \in \alpha_2^* T M$ for all $(t,v) \in W_{1 2}$. Then:
so $(t, \theta_1(t,v)) \in W_{2 1}$. Hence we have a map $\phi_{1 2} \colon W_{1 2} \to W_{2 1}$ given by:
Similarly we have a map $\phi_{2 1} \colon W_{2 1} \to W_{1 2}$. These are both smooth since the composition with the inclusion into $S^1 \times T M$ is smooth.
Consider the composition $\phi_{2 1}\phi_{1 2}(t,v)$. Expanding this out yields:
The penultimate line is because $(t,v) \in \alpha_1^* T M$ so $\pi(v) = \alpha_1(t)$. Hence $\phi_{2 1}$ is the inverse of $\phi_{1 2}$ and so $\phi_{1 2}$ is a diffeomorphism. Thus the map $\phi_{1 2}^L$ is a diffeomorphism from $\Psi_1^{-1}(U_1 \cap U_2)$ to $\Psi_2^{-1}(U_1 \cap U_2)$. We just need to show that this is the transition function. To do this, we show that $\Psi_2 \psi_{1 2}^L = \Psi_2 \Phi_{1 2}$. The right-hand side is, by definition, $\Psi_1$ which satisfies:
On the other side,
Hence
Thus $\phi_{1 2}^L = \Phi_{1 2}$ and so the transition functions are diffeomorphisms.
This construction easily generalises quite widely. Very little of the structure of $S^1$ was used at all: that mainly came in in the smooth structure of $L \mathbb{R}^n$. The key structure of $M$ was the local addition and thus one could regard this as a construction of locally additive spaces. For more on the possible extensions, see the references.
For $X$ a smooth manifold, the Fréchet manifold structure on $L X$, discussed above agrees with the standard mapping space diffeology on $L X$ in the following sense.
There is a functor
from Fréchet manifolds to diffeological spaces defined in the same way as the well-known functor from smooth manifolds to diffeological spaces: a plot is precisely a smooth map $c : U \to L X$, where $U$ is an object in the domain category, e.g. an open subset of some $\mathbb{R}^n$.
(Notice that a theorem of M. Losik says that the functor F is full and faithful, just like that including manifolds into diffeological spaces!)
Now there are two diffeologies on $L X$: this is the structure of a sheaf on the category CartSp obtained as regarding $L X$ as the internal hom with respect to the closed monoidal structure on sheaves
(where $Y$ denotes the Yoneda embedding).
This means that a map $c : U \to L X$ is a plot if and only if the associated map $U \times S^1 \to X$ is smooth. The second diffeology is the one obtained from the functor $F$.
These two diffeologies coincide – in the sense that every plot of one is a plot of the other. In particular, they have the same sets of smooth functions.
A detailed proof of this is in (Waldorf, lemma A.1.7)
The usual notions of G-structures for manifolds, such as orientation, spin structure, string structure, etc. do not carry over directly to their smooth loop spaces, but they are closely related by transgression: a spin structure on $X$ is supposed to induce a kind of orientation structure on $\Omega X$, a string structure on $X$ is supposed to induce a kind of spin structure on $\Omega X$.
Formalizations of such “smooth loop space $G$-structures” have been proposed in (Stolz-Teichner 2005), (Waldorf 2010) and (Waldorf 2012). In particular an equivalence between spin structures on smooth loop space and string structure on the underlying space is discussed in (Waldorf 14).
Given the notion of the smooth path groupoid $\mathcal{P}_1(X)$ of the smooth space $X$, we may think of the smooth loop space as the corresponding loop space object.
We say this in detail now.
First briefly recall the situation for topological spaces and their loop space objects, which are the topological loop spaces:
The ordinary procedure is that we regard a topological space as an object in the (∞,1)-category Top and produce its loop space at a chosen base point $x : {*} \to X$ as the (∞,1)-pullback of the base point along itself:
In words this is the simple statement that $\Omega_x X$ is the space of all homotopies in the (∞,1)-category Top from the map $x : {*} \to X$ to itself. Any one such homotopy is itself a continuous map $\gamma : I \to X$ from the standard interval $I = [0,1]$ to $X$, such that restricted to its endpoints it produces the map $x$. Clearly, these are precisely the loops in $X$, based at $x$.
You might think: well, now let $X \in Sh_{(\infty,1)}(Diff)$ be a smooth manifold regarded as a representable object in the (∞,1)-category of (∞,1)-sheaves of ∞-stacks on Diff or maybe better in $Sh_{(\infty,1)}(CartSp)$ on CartSp – i.e. of Lie ∞-groupoids – , we play the same trick and compute the homotopy pullback $Q$
to obtain the smooth loop space. But it doesn’t: the $Q$ here is $Q = {*}$! That’s because $X$ regarded as a representable object in $Sh_{(\infty,1)}(Diff)$ is really a categorically discrete Lie groupoid: it has a smooth space $X$ of objects, but no nontrivial morphisms. And the “homotopies” in the homotopy pullback are homotopies as seen by the morphisms in $X$. There is just the identity morphism in $X$ going from ${*} \to X$ to itself, so the homotopy pullback is the point.
Why then did it work in Top? Because, if you look closely, there really we did something different! Regarded as an (∞,1)-category, Top is really the collection of ∞-stacks on the point :
Under this identification, a topological space is not identified with a representable object! The only representable object in $Sh_{(\infty,1)}({*})$ is of course the point itself, the terminal object. Instead, as the notation above already suggests, under this identification a topological space is really identified with its singular simplicial complex $Sing(X)$. But that’s really to be thought of as the topological fundamental ∞-groupoid of $X$. We should write
instead of $X$ when we regard the topological space $X$ as an object of the (∞,1)-category of topological spaces! For more on this, see the discussion at homotopy hypothesis.
If instead we had interpreted the topological space $X$ as a representable object, hence as a categorically discrete object in the $(\infty,1)$-category of topological ∞-groupoids, we would have seen the same phenomenon as for the smooth $X$ above: its loop space object would have been the point. From this perspective now it is clear how the abstract notion of loop space object corresponds to the geometrically expected one: for a geometric space $X$, its loop space is the loop space object of its fundamental ∞-groupoid.
This statement can be given sense in all contexts where the underlying topos of our ambient (∞,1)-topos of spaces is a lined topos: we need to know which object $R$ is the standard line or interval object. This determines the geometric paths in a space. Taking these geometric paths to be the morphisms of a fundamental ∞-groupoid then makes the geometric paths into “categorical paths”, i.e. into morphisms. These then are what the abstract definition of loop space object can see.
And indeed, whenever the underlying topos of spaces that we are looking at is a lined topos the corresponding (∞,1)-topos comes equipped with a generalization of the topological fundamental ∞-groupoid construction: we can associate to every space $X$ its path ∞-groupoid $\Pi(X)$: the morphisms of $\Pi(X)$ are given by paths in $X$ as seen by the given interval object $I$. All entirely analogous to the familiar situation for Top, only that now we are testing our generalized spaces over test objects in an arbitrary site and are using a correspondingly different notion of interval object.
So for $X$ a smooth manifold, regarded as a representable object in the (∞,1)-topos $Sh_{(\infty,1)}(CartSp)$ of Lie ∞-groupoids we have now that the homotopy pullback of any point ${*} \to X \hookrightarrow \Pi(X)$ along itself in the path ∞-groupoid $\Pi(X)$ does indeed produce the expected Lie ∞-groupoid $\Omega_x X$ in
whose
smooth space of objects is the smooth space of smooth loops in $X$ based at $x$;
smooth space of morphisms is the smooth space of smooth $I$-homotopies between smooth loops in $X$
etc;
Here we want not the full loop ∞-groupoid, but just some sort of truncation to a 0-groupoid just of loops. There are several choices for how exactly to do this, depending on which higher morphisms we just discard, and which we use to identify 1-morphisms. Whatever we do, we end up with some notion of smooth path groupoid $\mathcal{P}_1(X)$ of $X$, whose 1-morphisms are certain smooth quotient space of the smooth space of 1-morphisms in $\Pi(X)$.
Then, accordingly, forming the loop space object of this path groupoid $\mathcal{P}_1(X)$ yields the smooth space $LoopSpace(X)$
which is the smooth subspace of the smooth space of morphisms in $\mathcal{P}_1(X)$ of those morphisms that start and end at $x$.
When unwrapping what all this means, one sees that the object $LoopSpace_x(X) \in Sh_{(\infty,1)}(CartSp)$ that we obtain this way is nothing but the image under the embedding $Sh(CartSp) \hookrightarrow Sh_{(\infty,1)}(CartSp)$ of ordinary sheaves into $\infty$-stacks of some quotient of the internal hom $[I,X]$ in the closed monoidal structure on sheaves. Being an internal hom of representables, this is a concrete sheaf and as such it is precisely the smooth loop space regarded as a diffeological space.
A general standard reference on generalized smooth spaces is
The structure of loop spaces as Fréchet manifolds is covered in chapter 42 of KM and in various other articles, many of which cover extensions of the basic construction to other mapping spaces. In particular,
The relation of this Fréchet manifold structure to the canonical diffeological space structure is discussed in
Discussion of G-structures on smooth loop spaces is in the following articles.
For orientation structure on loop space and its transgression from spin structure on target space:
For spin structure on loop spaces and its transgression from string structure on target space:
Konrad Waldorf, Spin structures on loop spaces that characterize string manifolds (arXiv:1209.1731)
Konrad Waldorf, String geometry vs. spin geometry on loop spaces (arXiv:1403.5656)
There are also sketchy notes in
For the motivation of much of this via the Dirac-Ramond operator and the Witten genus see the references there.
This entry was created in parallel with this MO thread from which parts of it is taken.