Odd characteristic classes in entire cyclic homology and equivariant loop space homology

Given a compact manifold $M$ and $g\in C^{\infty}(M,U(l;\mathbb{C}))$ we construct a Chern character $\mathrm{Ch}^-(g)$ which lives in the odd part of the equivariant (entire) cyclic Chen-normalized bar complex $\underline{\mathscr{C}}(\Omega_{\mathbb{T}}(M\times \mathbb{T}))$ of $M$, and which is mapped to the odd Bismut-Chern character under the equivariant Chen integral map. It is also shown that the assignment $g\mapsto \mathrm{Ch}^-(g)$ induces a well-defined group homomorphism from the $K^{-1}$ theory of $M$ to the odd homology group of $\underline{\mathscr{C}}(\Omega_{\mathbb{T}}(M\times \mathbb{T}))$

where ch − (g) ∈ Ω − (M) denotes the odd Chern character. The left hand side of (1) is precisely the spectral flow sf(D, D g ) [5]. Being motivated by the considerations of Atiyah and Bismut [1,2] for the even-dimensional case one finds that a very elegant, however purely formal, way to prove the latter formula is to assume the existence of a Duistermaat-Heckmann localization formula for the smooth loop space LM: indeed, with LM the smooth loop space, the spin structure on M induces an orientation on LM [1] and path integral formalism entails the elegant, however mathematically ill-defined, formula 1 1 2π where β = β 0 + β 2 ∈ Ω + (LM) denotes the even differential form defined on smooth vector fields X, Y on LM by β 0 (X) := 1 0 |X s | 2 ds, β 2 (X, Y ) := 1 0 (∇X s /∇s, Y s ) ds, and where Bch − (g) ∈ Ω − (M) denotes the odd Bismut-Chern character [3,14]. Now both differential forms exp(−β) and Bch − (g) are equivariantly closed (cf. Section 4 for the definition of the degree −1 differential P ), A direct implementation of the above arguments is not possible, as the right hand side of formula (2) is not well-defined for various reasons. For example, there exists no volume measure on LM, while smooth loops have Wiener measure zero, and, on the other hand, it is notoriously difficult to produce a variant of the complex (Ω(LM), d + P ) if one replaces LM with the smooth Banach manifold of continuous loops. Nevertheless and strikingly, the above formal manipulations lead to the highly powerful machinery of hypoelliptic Dirac operators, as is explained in [3] and the references therein. However, a possibe way out of these problems has been proposed by Getzler, Jones and Petrack (GJP) [8] [6]. In this approach, the idea is to take a model for Ω(LM) in terms of equivariant Chen integrals: this is a morphism of super complexes (cf. Section 4 below for the relevant definitions) where C (Ω T (M ×T)) denotes the Chen-normalized cyclic bar complex of the DGA Ω T (M × T). Now the GJP-program for infinite dimensional localization is as follows: here it is conjectured that the composition is mathematically well-defined (cf. [10] for first steps in this context), and that • LM e −β ∧ ρ(·) vanishes on exact elements of C (Ω T (M × T)) , • If w ∈ C (Ω T (M × T)) is closed, then one has the 'Duistermaat-Heckmann formula' If in addition one could canonically construct an element then from the above observations we would immediately obtain a proof of (1) within the GJP-program for infinite dimensional localization. Note that in the even dimensional case such a Chern character has been constructed in [8].
The aim of this paper is precisely to construct a canonically given element Ch − (g) ∈ C − (Ω T (M × T)) satisfying the above properties i), ii), iii). In fact, our main results Theorem 5.1 and Theorem 5.3 below construct Ch − (g) for M a compact Riemannian manifold (possibly with boundary), which satisfies i) and iii) and in addition ii) if M is closed (so that LM is a well-defined smooth Fréchet manifold). We also show in Theorem 5.1 that the assignment g → Ch − (g) induces a well-defined group homomorphism from the K −1 theory of M to the odd cyclic homology group of C (Ω T (M × T)). In fact, we show that Ch − (g) lives in a topological subcomplex of C (Ω T (M × T)) which is defined by requiring growth conditions in the spirit of Connes' entire growth conditions [9] [4]. This result suggests that LM e −β ∧ρ(·) should actually be a continuous functional, as integration should be.
Acknowledgements: The authors would like to thank J.-M. Bismut, Matthias Ludewig and Shu Shen for helpful discussions.
1. Cyclic bar complex of a differential graded algebra (DGA) In the sequel, we understand all our linear spaces to be over C. Assume we are given a unital DGA Ω, that is, • Ω is a unital algebra j=0 Ω j is graded into subspaces Ω j ⊂ Ω such that Ω i Ω j ⊂ Ω i+j for all i, j ∈ N, there is a degree +1 differential d : Ω → Ω which satisfies the graded Leibnitz rule.
The following notation will be useful in the sequel: to be the cochain which has a in its n-th slot and 0 anywhere else.

Entire cyclic homology of a metrizable unital DGA
The following definition is motivated by the fact a locally convex space is metrizable, if and only if its topology is induced by a countable sequence of seminorms: Definition 2.1. By a metrizable unital DGA we understand a unital DGA Ω, with a locally convex topology which is induced by a countable increasing family · k , k ∈ N, of seminorms such that • the differential is continuous, e.g., for every k there exists k ′ ≥ k and C > 0 with • the multiplication is jointly continuous, e.g., for every k there exists k ′ ≥ k and C > 0 with • the seminorms respect the grading, e.g., 2 Again,Ω inherits the above structure canonically, and we equip the algebraic tensor product Ω ⊗Ω ⊗n with the induced family · k;n , k ∈ N of the π-tensor seminorms, that is, each · k;n is defined as the smallest seminorm on Ω ⊗Ω ⊗n such that ω 0 ⊗ · · · ⊗ ω n k;n = ω 0 k · · · ω n k for all ω 0 ∈ Ω, ω 1 , . . . , ω k ∈Ω.
It is easily checked that C ǫ (Ω) becomes a locally convex space when equipped with the seminorms · k,l , k, l ∈ N. Our growth conditions are modelled on the entire growth conditions for ungraded Banach algebras by Getzler/Szenes from [9]. Note that C ǫ (Ω) is not complete (cf. Remark 4.1 below for a explanation of why in this paper we do not work with the completion of C ǫ (Ω)).
Proof. Fix k, l ∈ N. Clearly, one has Γw k,l ≤ w k,l for all w ∈ C ǫ (Ω). By the definition of a metrizable unital DGA, we may pick a constant C ′′ > 0 and a number k ′′ ≥ k, such that for all ω ∈ Ω one has dω k ≤ C ′′ ω k ′′ . Likewise, we may pick C ′ > 0 and a number k ′ ≥ k, such that for all ω 1 , ω 2 ∈ Ω one has ω 1 ω 2 k ≤ C ′ ω 1 k ′ ω 2 k ′ . Using this and n + 1 ≤ 2 n it is easily checked that for all w ∈ C ǫ (Ω).
Likewise, it follows immediately that Bw k,l ≤ 1 k w k,2l for all w ∈ C ǫ (Ω).
Defining the subspace D ǫ (Ω) ⊂ C ǫ (Ω) by D ǫ (Ω) := D(Ω) ∩ C ǫ (Ω), it follows automatically that the maps Γ, b, B map D ǫ (Ω) to itself continuously, too, producing with Finally we can give: Definition 2.4. The complex (9) is called the entire cyclic bar complex of Ω and its homology groups are denoted with HC ± ǫ (Ω). Likewise, the complex (10) is called the Chen-normalized entire cyclic bar complex of Ω and its homology groups are denoted with HC ± ǫ (Ω).

Equivariant cyclic bar complex of a manifold
Assume N is a compact manifold (possibly with boundary) and denote with T the 1-sphere. We denote by Ω T (N × T) the smooth T-invariant differential forms on N × T, where T acts trivially on N and by rotation on itself. Every element of Ω T (N × T) can be uniquely written in the form α+β ∧α T for some α, β ∈ Ω(N), where α T denotes the canonical 1-form on T. We turn Ω T (N × T) into a unital algebra by means of Ω T (N × T) ⊂ Ω(N × T), and give Ω T (N × T) the grading . With ∂ T the canonical vector field on T, we have the differential finally turning Ω T (N × T) into a unital DGA. Pick now a Riemannian structure on N and consider the Levi-Civita connection ∇ acting in Ω(N). With | · | the fiber metric on ∧T * N, we define a family of seminorms on Ω T (N × T) by setting We have: Lemma 3.1. If N is a compact Riemannian manifold (possibly with boundary), then Ω T (N × T) becomes a metrizable unital DGA with respect to (11).
As a consequence, we get the short complexes

Equivariant Chen integrals
Let us consider a compact manifold N without boundary, and the space LN of smooth loops γ : T → N, where in the sequel we read T as T = [0, 1]/ ∼. This becomes an infinite dimensional Fréchet manifold which is locally modelled on the Fréchet space LR dim N of smooth loops T → R dim N . Then LN carries a natural smooth T-action, given by rotating each loop, and the fixed point set of this action is precisely N ⊂ LN, embedded as constant loops. Given γ ∈ LN the tangent space T γ LN is given by linear space of smooth vector fields on N along γ, that is, and the generator of the T-action on LN is the vector field γ →γ on LN. Let ι denote the contraction with respect to the latter vector field. In the sequel, we understand Ω(LN) to be the space of sequences (α 0 , α 1 , . . . ) such that α k ∈ Ω k (LN) for all k ∈ N. For fixed s ∈ T one has the diffeomorphism called the equivariant de Rham complex of LN. The induced homology groups are denoted with H ± T (LN). Given t ∈ T and α ∈ Ω k (N) one denotes with α(t) ∈ Ω k (LN) the form obtained by pulling α back with respect to the evaluation map γ → γ(t). With this notation at hand, one has the equivariant Chen integral map The map ρ is a morphism of short complexes from (12) to (16), which in turn descends to a map ρ : C (Ω T (N × T)) −→ Ω(LN) of short complexes from (13) to (16) (cf. [8] for these results).
Remark 4.1. It is essential to work with our definition of C ǫ (Ω T (N × T)) in order to be able to restrict ρ to C ǫ (Ω T (N × T)) and to get the induced map which is defined on C ǫ (Ω T (N × T)). There seems to be no useful way to extend ρ to the completion of C ǫ (Ω T (N × T)), as this will lead to certain infinite series of tensor products whose image under ρ will lead to infinite series of elements of Ω(LN) having a fixed degree (noting that there seems to be no canonic way to turn Ω(LN) into a nice Fréchet space).

Construction of cycles in C − ǫ (Ω T (M × T)) and the induced cycles in
Let now M be a compact Riemannian manifold (possibly with boundary). Given g ∈ C ∞ (M, U(l; C)) our aim is to construct a canonically given element Then for all s ∈ I we can form the covariant derivative d + sω on the trivial vector bundle U(l; C) × C l → U(l; C). Let A s ∈ Ω 1 U(l; C), Mat(l; C) , R s ∈ Ω 2 U(l; C), Mat(l; C) denote the connection 1-form of d + sω and the curvature of d + sω, respectively, and

We set
A s (g) := g * A s , R s g := g * R s , ω g := g * ω, so that A s (g) = sω g and by the Maurer-Cartan equation R s g = (s/2)ω 2 g . Then we can define A s (g) := A s g − R s g ∧ α T ∈ Ω T (M × T, Mat(l; C)). By varying s, the forms A s (g) induce a form A(g) ∈ Ω T (M × I × T, Mat(l; C)) and we set B(g) := ι ∂ I A(g) ∈ Ω T (M × I × T, Mat(l; C)).
Then we can define B s (g) ∈ Ω T (M × T, Mat(l; C)), to be the pullback of B(g) with respect to the embedding In fact, by a simple calculation one finds so that B s (g) actually does not depend on s. With these preparations, we can define an element Ch − (g) = (Ch − 0 (g), Ch − 1 (g), . . . ) ∈ C (Ω T (M × T)) by setting 3 We refer the reader to the paper [12] by Simons and Sullivan, where a construction of the usual odd Chern character ch − (g) ∈ Ω − (M) (cf. below) has been given that influenced our definition of Ch − (g). 3 Given linear spaces V 0 , . . . , V n , and v (j) ∈ Mat(l; V i ), j = 0, . . . , n, the generalized trace is defined by in,i0 .
Theorem 5.1. Let M be a compact Riemannian manifold, possibly with boundary. a) One has Ch − (g) ∈ C − ǫ (Ω T (M × T)), and (b + B)Ch − (g) = 0, in particular, Ch − (g) induces a homology class Proof. a) It is easily seen that ΓCh − (g) = −Ch − (g). To show that It is then easily checked that It remains to prove (b + B)Ch − (g) ∈ D(Ω T (M × T)).
In fact, as every Ch − n (g) contains the 0-form 1 and so is of the form (4) with f = 1. It remains to show that bCh − (g) ∈ D(Ω T (M × T)).
In order to see the latter, let us first notice that Using (17) and the explicit definition of b, we get whose sum is Thus, we finally have (bCh − (g)) n = Tr n 1 ⊗ ω ⊗n g , n = 1, 2, . . . .
To this end we have simply to employ the properties of the generalized trace. Indeed, for n ≥ 2 we can write where the last two terms cancel each other because of the trace property, which is precisely of the form (5) for f = g −1 . Similarly, for n = 1 it is sufficient to notice that which is of the form (4) with f = g −1 , completing the proof of bCh − (g) ∈ D(Ω T (M × T)). b) We have to prove the following two facts: i) If g, h ∈ C ∞ (M, U(l; C)), then one has Ch − (g ⊕ h) = Ch − (g) + Ch − (h). ii) If g 0 , g 1 ∈ C ∞ (M, U(l; C)) are connected by a smooth homotopy g · ∈ C ∞ (M × I, U(l; C)), then one has Ch − (g 1 ) − Ch − (g 0 ) = (b + B)w for some w ∈ C + (Ω T (M × T)).
Here, property i) is an immediate consequence of the properties of the generalized trace Tr n using the block diagonal form of g ⊕ h. To see ii), for any t ∈ I, we define the embedding Then again it is clear that Bw ∈ D(Ω T (M × T)), so that Bw = 0. On the other hand, by using the identity , and similarly for B s , and the same computations as in part a) we get This completes the proof.
We have the projection map // s ti (g)R s g (t i ) // s tj (g)Ȧ s g (t j ) n l=j+1 // s t l (g)R s g (t l )// s 1 dt 1 · · · dt n ds   , whereȦ s g = dA s g /ds = ω g ∈ Ω 1 (M, Mat(l; C)), and where // s · (g) denotes the parallel transport with respect to the connection d + sω g on M × C l → M. Theorem 5.3. Assume M is a compact Riemannian manifold, possibly with boundary, and let g ∈ C ∞ (M, U(l; C)). Then one has π(Ch − (g)) = ch − (g), and if M has no boundary then Bch − (g) = ρ(Ch − (g)).
Note that in view of Corollary 5.2, Theorem 5.3 provide a new proof of (d + P )Bch − (g) = 0 (see [14] for a variant of this result).
Then obviously one has Bch − (g) = Tr