The Geometry of Loop Spaces I: -Riemannian Metrics
A Riemannian metric on a manifold induces a family of Riemannian metrics on the loop space depending on a Sobolev space parameter . We compute the connection forms of these metrics and the higher symbols of their curvature forms, which take values in pseudodifferential operators (DOs). These calculations are used in the followup paper  to construct Chern-Simons classes on which detect nontrivial elements in the diffeomorphism group of certain Sasakian -manifolds associated to Kähler surfaces.
Dedicated to the memory of Prof. Shoshichi Kobayashi
The loop space ††MSC number: 58J40. Keywords: loop spaces, Levi-Civita connections, pseudodifferential operators. of a manifold appears frequently in mathematics and mathematical physics. In this paper, we develop the Riemannian geometry of loop spaces. In a companion paper , we describe a computable theory of characteristic classes for the tangent bundle .
Several new features appear for Riemannian geometry on the infinite dimensional manifold . First, for a fixed Riemannian metric on , there is a natural one-parameter family of metrics on associated to a Sobolev parameter . Our main goal is to compute the Levi-Civita connection for and the associated connection and curvature forms. For , this is the usual metric, whose connection one-form and curvature two-form are essentially the same as the corresponding forms for . Second, for these forms take values in zeroth order pseudodifferential operators (s) acting on sections of a trivial bundle over the circle. Thus the analysis of these s and their symbols is essential to understanding the geometry of . In contrast to the finite dimensional case, where a Riemannian metric on implements a reduction of the structure group of , is not compatible with the structure group of . This forces an extension of the structure group of from a gauge group to a group of invertible zeroth order s.
The paper is organized as follows. §2 discusses connections associated to After some preliminary material on s, we compute the Levi-Civita connection for (Lemma 2.1), (Theorem 2.2), (Theorem 2.10), and general (Theorem 2.12). The extension of the structure group is discussed in §2.6. In §3, we show that our results extend work of Freed and Larrain-Hubach on loop groups [3, 8]. In the Appendix, we compute the higher order symbols of the connection and curvature forms of these connections.
One main motivation for this paper is the construction of characteristic and secondary classes on , using the Wodzicki residue of s. In the companion paper , we use the main theorems in this paper to construct Chern-Simons classes on which detect that is infinite, where is the diffeomorphism group of infinite families of Sasakian -manifolds associated to integral Kähler surfaces. Thus this work relates to a major theme in Prof. Kobayashi’s many important papers, namely the relationship between Riemannian and complex geometry.
Our many discussions with Sylvie Paycha are gratefully acknowledged. We also thank Dan Freed for pointing out an error in an earlier version of the paper.
2. The Levi-Civita Connection for Sobolev Parameter
In this section, we compute the Levi-Civita connection on associated to a Riemannian metric on and a Sobolev parameter or The standard metric on is the case , and otherwise we avoid technical issues by assuming that is greater than the critical exponent for analysis on bundles over The main results are Lemma 2.1, Theorem 2.2, Theorem 2.10, and Theorem 2.12, which compute the Levi-Civita connection for , , , and general respectively.
2.1. Preliminaries on
Let be a closed, connected, oriented Riemannian -manifold with loop space of smooth loops. is a smooth infinite dimensional Fréchet manifold, but it is technically simpler to work with the smooth Hilbert manifold of loops in some Sobolev class as we now recall. For , the formal tangent space is , the space of smooth sections of the pullback bundle . The actual tangent space of at is the sections of of Sobolev class We will fix and use for , respectively.
For each we can complete with respect to the Sobolev inner product
Here , with the covariant derivative along . (We use this notation instead of the classical to keep track of .) We need the complexified pullback bundle , denoted from now on just as , in order to apply the pseudodifferential operator The construction of is reviewed in §2.2. By the basic elliptic estimate, the completion of with respect to (2.1) is . We can consider the metric on for any , but we will only consider or
A small real neighborhood of the zero section in is a coordinate chart near via the pointwise exponential map
The differentiability of the transition functions is proved in  and [4, Appendix A]. Here are close loops in the sense that a geodesically convex neighborhood of contains and vice versa for all Since is (noncanonically) isomorphic to the trivial bundle , the model space for is the set of sections of this trivial bundle. The metric is a weak Riemannian metric for in the sense that the topology induced on by the exponential map applied to is weaker than the topology.
The complexified tangent bundle has transition functions . Under the isomorphisms , the transition functions lie in the gauge group , so this is the structure group of
2.2. Review of Calculus
A linear operator is a of order if for every open chart and functions , is a of order on , where we do not distinguish between and its diffeomorphic image in . Let be a finite cover of with subordinate partition of unity Let have on supp and set Then is a on , and differs from by a smoothing operator, denoted . In particular, this sum is independent of the choices up to smoothing operators. All this carries over to s acting on sections of a bundle over .
An example is the for a positive order nonnegative elliptic and outside the spectrum of In each , we construct a parametrix for by formally inverting and then constructing a with the inverted symbol. By [1, App. A], is a parametrix for . Since , is itself a . For , by definition
where is a bump function with [5, p. 29]; the symbol depends on the choice of
The operator for Re, which exists as a bounded operator on by the functional calculus, is also a . To see this, we construct the putative symbol of in each by a contour integral around the spectrum of . We then construct a on with , and set By arguments in , , so is a .
2.3. The Levi-Civita Connection for
The smooth Riemannian manifold has tangent bundle with For the metric on (i.e., in (2.1)), the Levi-Civita connection exists and is determined by the six term formula
In general, the Sobolev parameter in (2.1) differs from the parameter defining the loop space. We discuss how this affects the existence of a Levi-Civita connection.
For general , the Levi-Civita connection for the metric is guaranteed to exist on the bundle , as above. However, it is inconvenient to have the bundle depend on the Sobolev parameter, for several reasons: (i) is strictly speaking not the tangent bundle of , (ii) for the () metric, the Levi-Civita connection should be given by the Levi-Civita connection on applied pointwise along the loop (see Lemma 2.1), and on this would have to be interpreted in the distributional sense; (iii) to compute Chern-Simons classes on in , we need to compute with a pair of connections corresponding to on the same bundle. These problems are not fatal: (i) and (ii) are essentially aesthetic issues, and for (iii), the connection one-forms will take values in zeroth order s, which are bounded operators on any , so can be fixed.
Thus it is more convenient to fix and consider the family of metrics on for . However, the existence of the Levi-Civita connection for the metric is trickier. For a sequence with in or in , the RHS of (2.3) goes to for fixed Since is dense in , the RHS of (2.3) extends to a continuous linear functional on . Thus the RHS of (2.3) is given by for some We set . Note that even if we naturally demand that , we only get without additional work. Part of the content of Theorem 2.12 is that the Levi-Civita connection exists in the strong sense: given a tangent vector and a smooth vector field for all , See Remark 2.6.
We need to discuss local coordinates on . For motivation, recall that
in local coordinates on a finite dimensional manifold. Note that in this notation.
Let be a vector field on , and let be a tangent vector at The local variation of in the direction of at is defined as usual: let be a family of loops in with Fix , and let be coordinates near . We call these coordinates manifold coordinates. Then
Note that by definition.
Having defined only near a fixed is inconvenient. We can find coordinates that work for all points of as follows. For fixed , there is an such that for all , is inside the cut locus of if has Fix such an Call short if for all Then
is a coordinate neighborhood of parametrized by
We know noncanonically, so is parametized by short sections of for a different In particular, we have a smooth diffeomorphism from to short sections of .
Put coordinates on , which we identify canonically with the fiber over in . For , we have As with finite dimensional coordinate systems, we will drop and just write These coordinates work for all near and for all The definition of above carries over to exponential coordinates.
We will call these coordinates exponential coordinates.
(2.4) continues to hold for vector fields on , in either manifold or exponential coordinates. To see this, one checks that the coordinate-free proof that for (e.g. [16, p. 70]) carries over to functions on . In brief, the usual proof involves a map of a neighborhood of the origin in into , where are parameters for the flows of resp. For , we have a map , where is the loop parameter. The usual proof uses only differentiations, so is unaffected. The point is that the are local functions on the parameter space, whereas the are not local functions on at points where loops cross or self-intersect.
We first compute the () Levi-Civita connection invariantly and in manifold coordinates.
Let be the Levi-Civita connection on . Let be Then is the Levi-Civita connection on . In manifold coordinates,
is a connection on . We have . If is a coordinate neighborhood on near some , then on ,
Since is a connection, for each fixed , and ,
has Leibniz rule with respect to functions on . Thus is a connection on
is torsion free, as from the local expression
To show that is compatible with the metric, first recall that for a function on , for (Here depends only on .) Thus (suppressing the partition of unity, which is independent of )
The local expression for also holds in exponential coordinates. More precisely, let be a global frame of given by the trivialization of Then is also naturally a frame of for all We use to pull back the metric on to a metric on :
Then the Christoffel symbols the term e.g., are computed with respect to this metric. For example, the term means , etc. The proof that has the local expression (2.5) then carries over to exponential coordinates.
The Levi-Civita connection on is given as follows.
The Levi-Civita connection on is given at the loop by
On the right hand side of this formula, the term denotes the vector field along whose value at is
We prove this in a series of steps. The assumption in the next Proposition will be dropped later.
The Levi-Civita connection for the metric is given by
where we assume that for , is well-defined by
By Lemma 2.1,
The six terms on the left hand side must sum up to in the sense of Remark 2.1. After some cancellations, we get
Now we compute the bracket terms in the Proposition. We have . For ,
Note that are locally defined functions on Let be a smooth map with , and Since are coordinate functions on , we have
At the loop , In particular, is a zeroth order operator.