Vector fields on path spaces

I've been reading Chen's original works on iterated integrals and in order to consider differential forms on the path space $PM$ of a smooth manifold $M$ he gives $PM$ the following "differentiable space" structure:

Let $N$ be a smooth manifold. A continuous map $\alpha: N \to PM$ (where $PM$ has the compact open topology) is said to be smooth if the adjoint map $\tilde{\alpha}: N \times I \to M$ defined by $(n,t)\mapsto \alpha(n)(t)$ is smooth in the usual sense. The smooth map $\alpha$ is a plot if $N$ is an open convex subset of $\mathbb{R}^n$ for some $n$. Thus, we are modeling $PM$ locally with plots of varying dimension.

Chen defines a differential n-form $\omega$ on $PM$ as a rule which assigns to every plot $\alpha: U \to PM$ a differential form $\omega_{\alpha} \in \Omega^n(U)$. We define $(d\omega)_{\alpha}=d\omega_{\alpha}$. It turns out that for Chen's purposes one does not need to develop more calculus tools on $PM$. He shows a De Rham type result: the cohomology of the complex $\Omega^*(LM)$ (where $LM$ is the free loop space) is isomorphic to the real singular cohomology of $LM$.

However, for other purposes it is useful to consider forms as alternating tensors and to do this in this context we need a notion of vector fields on $PM$. I've always thought of a tangent vector at $\gamma \in PM$ as a vector field $T_\gamma$ along $\gamma$ on $M$, so a vector field on $PM$ assings each point $\gamma \in PM$ a vector field along $\gamma$. However, following Chen, the natural way to define vector fields to make it compatible with his notion of differential forms is as follows: a vector field $T$ on $PM$ is a rule which assings to each plot $\alpha: U \to PM$ a vector field $T_{\alpha}$ on $U$.

How do we reconcile these two notions of vector fields on $PM$? Are they equivalent?

• Differential forms pull back, but vectors fields don't (and they also don't push forward). This is the reason why Chen's definition makes sense only for differential forms. Hence there is no reason to expect that your second definition is related to vector fields on PM in any way. – Dmitri Pavlov Aug 11 '12 at 16:40
• Right, but shouldn't we be able to describe vector fields locally via plots? For example how would you define operations like interior multiplication, i.e. if $T$ is a vector field on $PM$, $\omega$ an n-form, and $\alpha: U \to PM$ a plot how would you define $(\iota_T \omega)_{\alpha}$? – Manuel Rivera Aug 11 '12 at 17:40
• Vector fields can be described locally, in fact your first definition can be easily adapted to the language of sheaves (I prefer the term ‘sheaf’ to the somewhat archaic term ‘plot’). If U→PM is a smooth map (a plot in your terminology), then a (smooth) section of the pullback of the tangent bundle of PM to U can be defined as a (smooth) section of the pullback of the tangent bundle of M to the smooth manifold U×I via the canonical (smooth) map U×I→M induced by the original smooth map U→PM. However, one cannot expect that the interior product can be described locally in this language. – Dmitri Pavlov Aug 11 '12 at 18:18
• Indeed, the pullback of a differential form on PM via the map U→PM simply does not have enough information to allow us to evaluate it on arbitrary section of the pullback of the tangent bundle of PM to U, because the new form only contains information about vectors that are tangent to the image of the map U→PM, which does not allow us to evaluate it on any other vectors. – Dmitri Pavlov Aug 11 '12 at 18:22
• This makes sense. For these purposes we should then define an n-form on $PM$ to be a function which assigns to each $\gamma \in PM$ an alternating n-tensor on $T_{\gamma}PM$, as usual. Then we can define interior multiplication in the usual sense. If we want to obtain forms in Chen's sense we can do by pulling back along plots. – Manuel Rivera Aug 11 '12 at 20:13

Good question!

There is a fairly well developed theory of diffeological spaces, with many recent references. The only difference between diffeological spaces and Chen's smooth spaces is that the plots of a diffeological space have open subsets $U \subset \mathbb{R}^n$ as their domains, while the ones of a smooth space have convex subsets. For questions concerning differential forms and tangent vectors, this seems to be irrelevant, so I'll continue with diffeological spaces.

As Dimitri pointed out in his comments, a differential form on a diffeological space $X$ is not just a form $\psi_c$ on the domain $U_c$ of each plot $c: U_c \to X$. It must be required for two plots $c_1: U_1 \to X$ and $c_2:U_2 \to X$ such that one has a smooth map $f: U_1 \to U_2$ with $c_2 \circ f = c_1$, the forms satisfy $$\psi_{c_1} = f^{*}\psi_{c_2}.$$ As also remarked by Dimitri, vector fields can neither be pulled back nor be pushed forward. Hence there is no analogous definition of vector fields.

Yet the concept of a tangent space to a general diffeological space $X$ can be defined. A good reference is the paper "Diffeological spaces" by Martin Laubinger (Proyecciones Journal of Mathematics, Vol. 25, No2, pp. 151-178). Roughly, the tangent space at a point $x\in X$ is the sum of all tangent spaces $T_0U_c$ over all plots $c:U_c \to X$ centered at $x$, i.e. $0 \in U_c$ and $c(0)=x$. Then there is an equivalence relation that enforces the chain rule for maps.

Smooth manifolds form a full subcategory of diffeological spaces, by regarding a smooth manifold $M$ as a diffeological space with plots all smooth maps $c:U \to M$, for all open subset $U \subset \mathbb{R}^n$ and all $n\in \mathbb{N}$. Thus, for a smooth manifold $M$ we have now the classical tangent space, and the diffeological tangent space. Laubinger proves that both tangent spaces are isomorphic.

Apparently less well-known is a theorem of M. Losik showing that Fréchet manifolds also form a full subcategory of diffeological spaces (with the plots defined in the very same way as for smooth manifolds). Thus, for a Fréchet manifold we also have two concepts of a tangent space, the classical one and the diffeological one. I don't know a reference for the statement that both concepts coincide, as in the smooth case, but I have looked at Laubinger's proof and I don't see why the same proof should not work for Fréchet manifolds.

Now let's look at the path space $PM$ or the loop space $LM$ or at any other mapping space of a compact smooth manifold into a smooth manifold $M$. I continue with $LM$ for simplicity. There are, a priori, two diffeologies on the loop space $LM$. The first one is the one from the theorem of Losik, i.e. the one where the plots are the Fréchet-smooth maps from open sets $U$ to $LM$. The second one is the one described in the question, it is called the functional diffeology, with plots those maps $c: U \to LM$ whose adjoint $U \times S^1 \to M$ is smooth (demanding continuity for $c$ is superfluous). Luckily, the two diffeologies coincide (see Lemma A.1.7 in my paper Transgression to Loop Spaces and its Inverse, Part I).

And here is the result: classical Fréchet theory says that the tangent space of $LM$ at a loop $\tau$ consists of vector fields along $\tau$. The earlier claimed generalization of Laubinger's theorem shows that it coincides with the diffeological tangent space, and the coincidence of the diffeologies on $LM$ shows that it can be defined via plots $c$ whose adjoints $U \times S^1 \to M$ are smooth maps. This is, I think, the best one could hope for.