The Radul cocycle, the Chern–Connes character, and manifolds with conical singularities

Let $H$ be a (complex) Hilbert space and let $\Delta$ be an unbounded, positive and self-adjoint operator acting on it, with domain $\mathrm{dom}{\Delta}$. To keep the exposition simple, we suppose that $\Delta$ has a compact resolvent.

We denote by $H^{\infty}$ the following intersection:

$$H^{\infty}=\bigcap_{k=0}^{\infty}\mathrm{dom}(\Delta^{k})$$

Having Gärding’s inequality in mind, the elliptic estimate exactly says that $\Delta^{1/r}$ should be thought as an "abstract elliptic operator" of order 1. It also says that any differential operator $X$ of order $q$ can be extended to a bounded operator form $H^{s+q}$ to $H^{s}$. This last property will be useful to define pseudodifferential calculus in this setting. The main example to keep in mind is of course the case in which $\Delta$ is a Laplace type operator on a closed Riemannian manifold $M$.

Let $(A,H,D)$ a spectral triple (cf. [1] or [3]). One may construct a algebra of abstract differential operators $\mathcal{D}=\mathcal{D}(A,D)$ recursively as follows :

	$$\displaystyle\mathcal{D}_{0}=\text{algebra generated by }A\text{ and }[D,A]$$
	$$\displaystyle\mathcal{D}_{1}=[\Delta,\mathcal{D}_{0}]+\mathcal{D}_{0}[\Delta,\mathcal{D}_{0}]$$
	$$\displaystyle\qquad\vdots$$
	$$\displaystyle\mathcal{D}_{k}=\sum_{j=1}^{k-1}\mathcal{D}_{j}\cdot\mathcal{D}_{k-j}+[\Delta,\mathcal{D}_{k-1}]+\mathcal{D}_{0}[\Delta,\mathcal{D}_{k-1}]$$

Let $\delta$ be the unbounded derivation $\mathrm{ad}|D|=[|D|,\,.\,]$ on $\mathcal{B}(H)$. The spectral triple is $(A,H,D)$ is said regular if $A,[D,A]$ are included in $\bigcap_{n=1}^{\infty}\mathrm{dom}\,\delta^{n}$. The following theorem of Higson relates algebras of abstract differential operators and spectral triples.

Let $\mathcal{D}(\Delta)$ be an algebra of abstract differential operators. For $z\in\mathbb{C}$, one defines the complex powers $\Delta^{-z}$ of $\Delta$ using functional calculus :

$$\Delta^{-z}=\frac{1}{2\pi i}\int\lambda^{-z}(\lambda-\Delta)^{-1}d\lambda$$

where the contour of integration is a vertical line pointing downwards separating $0$ and the (discrete) spectrum of $\Delta$. This converges in the operator norm when $\mathrm{Re}(z)>0$, and using the semi-group property, all the complex powers can be defined after multiplying by $\Delta^{k}$, for $k\in\mathbb{N}$ large enough. Moreover, since $\Delta$ has compact resolvent, the complex powers of $\Delta$ are well defined operators on $H^{\infty}$.

We will suppose that there exists a $d\geq 0$ such that for every $X\in\mathcal{D}_{q}(\Delta)$, the operator $X\Delta^{-z}$ extends to a trace-class operator on $H$ for $z$ on the half-plane $\mathrm{Re}(z)>\frac{q+d}{r}$. The zeta function of $X$ is

$$\zeta_{X}(z)=\mathrm{Tr}(X\Delta^{-z/r})$$

The smallest $d$ verifying the above property is called the analytic dimension of $\mathcal{D}(\Delta)$. In this case, the zeta function is holomorphic on the half-plane $\mathrm{Re}(z)>q+d$. We shall say that $\mathcal{D}(\Delta)$ has the analytic continuation property if for every $X\in\mathcal{D}(\Delta)$, the associated zeta function extends to a meromorphic function of the whole complex plane.

There properties are set for all the section, unless if it is explicitly mentioned.

These notions come from properties of the zeta function on a closed Riemannian manifold $M$ : it is well-known that the algebra of differential operators on $M$ has analytic dimension $\mathrm{dim}\,M$ and the analytic continuation property. Its extension to a meromorphic function has at most simple poles at the integers smaller that $\mathrm{dim}\,M$. In the case where $M$ is foliated, the dimension of the leaves appears in the analytic dimension when working in the suitable context. Hence, the zeta function provide informations not only on the topology of $M$, but also on its the geometric structure, illustrating the relevance of this abstraction.

Let $\mathcal{D}(\Delta)$ an algebra of abstract differential operators of analytic dimension $d$. To define the notion of pseudodifferential operators, we need a more general notion of order, not necessary integral, which covers the one induced by the filtration of $\mathcal{D}(\Delta)$.

That this notion of order covers the differential order is due to the elliptic estimate, as already remarked in Section 1.1. The space of such operators, denoted $\mathrm{Op}(\Delta)$, forms a $\mathbb{R}$-filtered algebra. There is also a notion of regularizing operators which are, as expected, the elements of the (two-sided) ideal of operators of all order.

The following notion is due to Uuye, cf. [11]. We just added an assumption on the zeta function which is necessary for what we do.

Of course, this is true for the algebra of (classical) pseudodifferential operators on a closed manifold. We shall recall later what happens in the example of Heisenberg pseudodifferential calculus on a foliation, as described by Connes and Moscovici in [1].

We end this part with a notion of asymptotic expansion for abstract pseudodifferential operators. This can be seen as "convergence under the residue".

In this context, asymptotic means that when taking values under the residue, such infinite sums, which have no reason to converge in the operator norm, are in fact finite sums. To this effect, the following lemma is crucial.

The proof relies of the following identity, for $z\in\mathbb{C}$ with $\mathrm{Re}(z)$ large enough (cf [3], Lemma 4.20) :

$$\Delta^{-z}Q-Q\Delta^{-z}=\sum_{k=1}^{N}\binom{-z}{k}Q^{(k)}\Delta^{-z-k}+\frac{1}{2\pi i}\int\lambda^{-z}(\lambda-\Delta)^{-1}Q^{(N+1)}(\lambda-\Delta)^{-N-1}\,d\lambda$$

(1.2)

We give in this paragraph a simple generalization of the Wodzicki residue trace in the case where the zeta function of the algebra $\mathcal{D}(\Delta)$ has poles of arbitrary order. This was already present in the work of Connes and Moscovici (see [1]).

Let $A$ be an associative algebra over $\mathbb{C}$, possibly without unit, and $I$ an ideal in $A$. The extension

$$0\to I\to A\to A/I\to 0$$

gives rise to the following diagram relating algebraic K-theory and periodic cyclic homology

(2.1)

The vertical arrows are respectively the odd and even Chern character.

We still denote $\partial:\mathrm{HP}^{0}(I)\rightarrow\mathrm{HP}^{1}(A/I)$ the boundary map in cohomology. As mentioned in [6], for $[\tau]\in\mathrm{HP}^{0}(I)$, $[u]\in K^{1}(A/I)$, one has the equality :

$$\langle[\tau],\mathrm{ch}_{0}\mathrm{Ind}[u]\rangle=\langle\partial[\tau],\mathrm{ch}_{1}[u]\rangle$$

(2.2)

A standard procedure to calculate the boundary map $\partial$ in cohomology associated to the extension as follows. If $[\tau]\in\mathrm{HP}^{0}(I)$ is represented by a hypertrace $\tau:I\to\mathbb{C}$, i.e a linear map satisfying the condition $\tau([A,I])=0$, then choose a lift $\widetilde{\tau}:A\to\mathbb{C}$ of $\tau$, such that $\widetilde{\tau}$ is linear (in general, this is not a trace), and a linear section $\sigma:A/I\to A$ such that $\sigma(1)=1$, after adjoining a unit where we have to. Then, $\partial[\tau]$ is represented by the following cyclic 1-cocycle :

$$c(a_{0},a_{1})=b\widetilde{\tau}(\sigma(a_{0}),\sigma(a_{1}))=\widetilde{\tau}([\sigma(a_{0}),\sigma(a_{1})])$$

where $b$ is the Hochschild coboundary.

We can finally come to the main theorem of this section. Let $\mathcal{D}(\Delta)$ be an algebra of abstract differential operators and $\Psi$ be an algebra of abstract pseudodifferential operators. We consider the extension

$$0\to\Psi^{-\infty}\to\Psi\to S\to 0$$

where $S$ is the quotient $\Psi/\Psi^{-\infty}$. The operator trace on $\Psi^{-\infty}$ is well defined, and $\mathrm{Tr}([\Psi^{-\infty},\Psi])=0$.

Here, the commutator $[\log\Delta^{1/r},a]$ is defined as the non-convergent asymptotic expansion

$$[\log\Delta^{1/r},a]\thicksim\dfrac{1}{r}\sum_{k\geq 1}\frac{(-1)^{k-1}}{k}a^{(k)}\Delta^{-k}$$

(2.3)

where $a^{(k)}$ has the same meaning as in Lemma 1.9. This expansion arises by first using functional calculus :

$$\log\Delta^{1/r}=\dfrac{1}{2\pi\textup{{i}}}\int\log\lambda^{1/r}(\lambda-\Delta)^{-1}\,d\lambda$$

and then, reproducing the same calculations made in the proof of Lemma 1.9 to obtain the formula (cf. [3] for details). In particular, note that $\log\Delta^{1/r}=\frac{1}{r}\log\Delta$.

Another equivalent expansion possible, that we will also use, is the following

$$[\log\Delta^{1/r},a]\thicksim\sum_{k\geq 1}\frac{(-1)^{k-1}}{k}a^{[k]}\Delta^{-k/r}$$

(2.4)

where $a^{[1]}=[\Delta^{1/r},a]$, and $a^{[k+1]}=[\Delta^{1/r},a^{[k]}]$. A heuristic explanation is the following. We first lift the trace on $\Psi^{-\infty}$ to a linear map $\widetilde{\tau}$ on $\Psi$ using a zeta function regularization by "Partie Finie" :

$$\widetilde{\tau}(P)=\mathrm{Pf}_{z=0}\mathrm{Tr}(P\Delta^{-z/r})$$

for any $P\in\Psi$. The "Partie Finie" $\mathrm{Pf}$ is defined as the constant term in the Laurent expansion of a meromorphic function. Let $Q\in\Psi$ be another pseudodifferential operator. Then, we have

$$\mathrm{Pf}_{z=0}\mathrm{Tr}([P,Q]\Delta^{-z/r})=\mathrm{Res}_{z=0}\mathrm{Tr}\left(P\cdot\frac{Q-\Delta^{-z/r}Q\Delta^{z/r}}{z}\Delta^{-z/r}\right)$$

by reasoning first for $z\in\mathbb{C}$ of sufficiently large real part to use the trace property, and then applying the analytic continuation property. Then, informally we can think of the complex powers of $\Delta$ as

$$\Delta^{z/r}=e^{\log\Delta\cdot z/r}=1+\dfrac{z}{r}\log\Delta+\ldots+\dfrac{1}{p!}\left(\dfrac{z}{r}\right)^{p}(\log\Delta)^{p}+O(z^{p+1})$$

which after some calculations, gives the expansion

$$(Q-\Delta^{-z/r}Q\Delta^{z/r})\Delta^{-z/r}=z\delta(Q)-\dfrac{z^{2}}{2}\delta^{2}(Q)+\ldots+(-1)^{p-1}\dfrac{z^{p}}{p!}\delta^{p}(Q)+O(z^{p+1})$$

For a complete proof, see [10]. But here is a standard example.

In our context, manifolds with conical singularities are just manifolds with boundary with an additional structure given by a suitable algebra of differential operators.

More precisely, let $M$ be a compact manifold with (connected) boundary, and $r:M\to\mathbb{R}_{+}$ be a boundary defining function, i.e a smooth function vanishing on $\partial M$ and such that its differential is non-zero on every point of $\partial M$. We work in a collar neighbourhood $[0,1)_{r}\times\partial M_{x}$ of the boundary, the subscripts are the notations for local coordinates.

$\mathrm{Diff}_{b}^{m}(M)$ denotes the algebra of $b$-differential operators in Melrose’s calculus for manifolds with boundary. We now recall the associated small $b$-pseudodifferential calculus $\Psi_{b}(M)$.

Let $M_{b}^{2}$ be the $b$-stretched product of $M$, i.e the manifold with corners whose local charts are given by the usual charts on $M^{2}\smallsetminus\partial M^{2}$, and parametrized by polar coordinates over $\partial M$ in $M^{2}$. More precisely, writing $M\times M$ near $r=r^{\prime}=0$ as

$$M^{2}\simeq[0,1]_{r}\times[0,1]_{r^{\prime}}\times\partial M^{2}$$

this means that we parametrize the part $[0,1]_{r}\times[0,1]_{r^{\prime}}$ in polar coordinates

$$r=\rho\cos\theta,\quad r^{\prime}=\rho\sin\theta$$

for $\rho\in\mathbb{R}_{+}$, $\theta\in[0,\pi/2]$. The right and left boundary faces are respectively the points where $\theta=0$ and $\theta=\pi/2$.

Remark that $\log(r/r^{\prime})$ should be singular at $r=r^{\prime}=0$ if we would have considered kernels defined on $M^{2}$. Introducing the $b$-stretch product $M^{2}_{b}$ has the effect of blowing-up this singularity.

The algebra of conic pseudodifferential operators is then the algebra $r^{-\mathbb{Z}}\Psi_{b}^{\mathbb{Z}}(M)$. The opposed signs in the filtrations are only to emphasize that $r^{\infty}\Psi_{b}^{-\infty}(M)$ is the associated ideal of regularizing operators.

To such an operator $A=r^{-p}P\in r^{-p}\Psi_{b}^{m}$ , we define on the chart $U$ the local density

$$\omega(P)(r,x)=\left(\int_{|\nu|=1}p_{-n}(r,x,\tau,\xi)\iota_{L}d\tau d\xi\right)\cdot\,\dfrac{dr}{r}\,dx$$

where $\nu=(\tau,\xi)$ and $L$ is the generator of the dilations.

It turns out (but this is not obvious) that this a priori local quantity does not depend on the choice of coordinates on $M$, and hence, define a globally defined density $\omega(P)$, smooth on $M$, that we call the Wodzicki residue density. Unfortunately, the integral on $M$ of this density does not converge in general, as the boundary introduces a term in $1/r$ in the density. However, we can regularize this integral, thanks to the following lemma. Here, $\Omega_{b}$ denote the bundle of $b$-densities on $M$, that is, the trivial line bundle with local basis on the form $(dr/r)dx$. The following lemma from Gil and Loya is proved in [2].

Applying this regularization to the Wodzicki residue density is useful to many "residues traces" that we immediately study.

We first begin by defining different algebras of pseudodifferential operators, introduced by Melrose and Nistor in [5]. The main algebra that we shall consider is

$$A=r^{-\mathbb{Z}}\Psi^{\mathbb{Z}}(M)=\bigcup_{p\in\mathbb{Z}}\bigcup_{m\in\mathbb{Z}}r^{-p}\Psi^{m}(M)$$

which clearly contains the algebra of Fuchs type operators. The ideal of regularizing operators is

$$I=r^{\infty}\Psi^{-\infty}(M)=\bigcup_{p\in\mathbb{Z}}\bigcup_{m\in\mathbb{Z}}r^{-p}\Psi^{m}(M)$$

and this explains why we note the two filtrations by opposite signs in $A$. Consider the following quotients

$$I_{\sigma}=r^{\infty}\Psi^{\mathbb{Z}}(M)/I,\quad I_{\partial}=r^{\mathbb{Z}}\Psi^{-\infty}(M)/I$$

Here, $I_{\sigma}$ should be thought as an extension of the algebra of pseudodifferential operators in the interior of $M$, whereas $I_{\partial}$ are regularizing operators up to the boundary. We finally define

$$A_{\partial}=\mathcal{A}/I_{\sigma},\quad A_{\sigma}=A/I_{\partial},\quad A_{\partial,\sigma}=A/(I_{\partial}+I_{\sigma})$$

The "Partie Finie" regularization of a trace does not give in general a trace, and this is indeed the same for the functional $\mathrm{Tr}_{\sigma}(P)$ acting on these algebras, the obstruction to that is precisely the presence of the boundary. However, by definition, $\mathrm{Tr}_{\sigma}(P)$ clearly defines an extension of the Wodzicki residue for pseudodifferential operators, one can expect that it is a trace on $I_{\sigma}=r^{\infty}\Psi^{\mathbb{Z}}(M)/I$.

By Lemma 4.3 and the definition above, the defect of $\mathrm{Tr}_{\sigma}$ to be a trace is precisely measured by $\mathrm{Tr}_{\partial,\sigma}(P)$, which can therefore be viewed as a restriction of the Wodzicki residue to the boundary $\partial M$. Then, the following proposition seems natural.

These two traces may be seen as "local" terms, since they only depend on the symbol of the pseudodifferential operator considered. The first can be seen as a trace on interior of $M$, the second is related to the boundary $\partial M$. There is one last trace to introduce, less easy to deal with because this one is not local.

Fix a holomorphic family $Q(z)\in r^{\alpha z}\Psi_{b}^{\beta z}(M)$, with $\alpha,\beta\in\mathbb{R}$, such that $Q$ is the identity at $z=0$. Take $P\in r^{-p}\Psi_{b}^{m}$, with $p,m\in\mathbb{Z}$ and let $(PQ(z))_{\Delta}$ be the restriction to the diagonal $\Delta$ of $M^{2}$ of the Schwartz kernel of $PQ(z)$. Melrose and Nistor noticed in [5] that $(PQ(z))_{\Delta}$ is meromorphic in $\mathbb{C}$, with values in $r^{\alpha z-p}C^{\infty}(M)$ with possible simple poles in the set

$$\left\{\dfrac{-n-m}{\beta},\dfrac{-n-m+1}{\beta},\ldots\right\}$$

There is an interpretation of $\mathrm{Tr}_{\partial}$ analogous to those of $\mathrm{Tr}_{\partial,\sigma}$ : If the order of $P$ is less than the dimension of $M$, then $\mathrm{Tr}_{\partial}(P)$ is a kind of $L^{2}$ of $P$ restricted to the boundary. This is precisely the content of the following result.

Now, let $\Delta\in r^{-2}\mathrm{Diff}_{b}^{2}(M)$ be fully elliptic, or parameter elliptic with respect to a parameter $\alpha$. We refer to [2] for the definition, what we need to know is just that this condition ensures the existence of the heat kernel $e^{-t\Delta}$ of $A$, and that operators of the type $P\Delta^{-z}$, with $P\in r^{-p}\Psi_{b}^{m}$, are of trace-class on $r^{\alpha-m}L^{2}_{b}(M)$ for $z$ in the half-plane $\mathrm{Re}z>max\{\frac{m+n}{2},\frac{p}{2}\}$, $n=\mathrm{dim}\,M$.

Then, the traces introduced in the previous paragraph gives the coefficients of the expansion of $\mathrm{Tr}(Pe^{-t\Delta})$.