Asymptotic pseudodifferential calculus and the rescaled bundle

Pseudodifferential operators have pseudo-local property, that is, their Schwartz kernels are smooth outside the diagonal. As a consequence, any pseudodifferential operator can be decomposed as a properly supported pseudodifferential operator plus a smoothing operator. Ordinary principal symbol calculus, which is captured by the construction of tangent groupoid, focus on the quotient space $\Psi^{m}(M)/\Psi^{-\infty}(M)$ where one cannot tell the difference between smoothing operator and the zero operator. However, as far as local index theory is concerned, it is precisely smoothing operators that carry index information. More precisely, according to the Mckean-Singer formula[MS67], the topological index can be calculated by supertrace of the smoothing operator given by the heat kernel of the Dirac operator. Therefore one need a full symbol calculus to explore the space of smoothing operators. Along this line, Widom have developed an asymptotic symbolic calculus where a smooth family of full symbols $a(x,\xi,t)$ of order $m$ is studied from the view-point of asymptotic expansion

(1.1)

$$a(x,\xi,t)\sim a_{0}(x,\xi)+ta_{1}(x,\xi)+t^{2}a_{2}(x,\xi)+\cdots.$$

where $a_{i}(x,\xi)$ is a symbol of order $m-i$ and the asymptotic expansion means for all $N$, the quantity

$$t^{-N}\left(a(x,\xi,t)-\sum_{k=0}^{N-1}t^{k}a_{k}(x,\xi)\right)$$

converges to zero in the space of symbols of order $m-N$ as $t\to 0$. It has the advantage that its calculus is easier than that of full symbols and, at the same time, useful aspect of smoothing operators, for example the asymptotic expansion of the operator trace at $t=0$, are preserved.

To adept this idea into the realm of local index theory, Block and Fox[BF90] developed the asymptotic pseudodifferential calculus on spinor bundles over compact spin manifold and use it, together with the JLO formula, to calculate Connes’ cyclic cocycle. In this paper, we recover the calculus from a geometric point-view by using a groupoid approach to pseudodifferential calculus by Van erp and Yuncken[EY17] in the context of rescaled bundle[HY19].

Let $M$ be an even dimensional spin manifold with spinor bundle $S\to M$. The rescaled bundle $\mathbb{S}$ is a vector bundle over the tangent groupoid $\mathbb{T}M$

(1.2)

whose restriction to $TM\times\{0\}$ is the pullback of the bundle of exterior algebras and restriction to $M\times M\times\mathbb{R}^{\ast}$ is the pullback of the tensor product $S^{\ast}\boxtimes S$. Moreover, there is an open neighborhood $\mathbb{U}$ of $TM\times\{0\}$ inside the tangent groupoid which is homeomorphic to an open neighborhood $\widetilde{\mathbb{U}}$ of $TM\times\{0\}$ inside $TM\times\mathbb{R}$. Let $\rho:TM\times\mathbb{R}\to M$ be the bundle projection, there is a natural isomorphism between the restricted rescaled bundle $\mathbb{S}|_{\mathbb{U}}$ and the bundle of exterior algebra $\rho^{\ast}\wedge T^{\ast}M|_{\widetilde{\mathbb{U}}}$. This fact together with the smooth structure of $S^{\ast}\boxtimes S$ over $M\times M\times\mathbb{R}^{\ast}$ completely determine the smooth structure of the rescaled bundle.

The rescaled bundle also carries a multiplicative structure which is given by the smoothly varying maps

(1.3)

$$\mathbb{S}_{\gamma}\otimes\mathbb{S}_{\eta}\to\mathbb{S}_{\gamma\circ\eta}$$

where $(\gamma,\eta)$ is a composable pair of elements in the tangent groupoid and $\gamma\circ\eta$ is their groupoid multiplication. When $\gamma=(x,X,0)$ and $\eta=(x,Y,0)$ come from $TM\times\{0\}$ part of the tangent groupoid, the multiplication map (1.3) is explicitly computable as

(1.4)

$$\wedge T^{\ast}_{x}M\otimes\wedge T^{\ast}_{x}M\ni\alpha\otimes\beta\mapsto\exp\left(-\frac{1}{2}\kappa(X,Y)\right)\wedge\alpha\wedge\beta$$

where $\kappa(X,Y)$ is a differential 2-form given by the symbol of curvature of the spinor bundle.

Following [LMV17] and [EY17], the space of properly supported $r$-fibered distribution on the rescaled bundle $\mathcal{E}^{\prime}_{r}(\mathbb{T}M,\mathbb{S})$ is defined to be the set of continuous $C^{\infty}(M\times\mathbb{R})$-module maps $C^{\infty}(\mathbb{T}M,\mathbb{S})\to C^{\infty}(M\times\mathbb{R})$ and we shall consider the subspace $\Psi^{m}(\mathbb{T}M,\mathbb{S})$ that consists of $\mathbb{P}\in\mathcal{E}^{\prime}_{r}(\mathbb{T}M,\mathbb{S})$ that satisfy the essentially homogeneous condition

(1.5)

$$\alpha_{\lambda,\ast}\mathbb{P}-\lambda^{m}\mathbb{P}\in C_{p}^{\infty}(\mathbb{T}M,\mathbb{S})$$

where $\alpha_{\lambda}$ is a smooth action on the tangent groupoid that sends $(x,y,t)$ to $(x,y,\lambda t)$ for $t\neq 0$ and $(x,Y,0)$ to $(x,\lambda^{-1}Y,0)$. In this paper, we shall focus on the Taylor’s expansion of $\mathbb{P}\in\Psi^{m}(\mathbb{T}M,\mathbb{S})$ at $t=0$. According to the smooth structure of the rescaled bundle, a distribution $\mathbb{P}\in\Psi^{m}(\mathbb{T}M,\mathbb{S})$ may be restricted to an open neighborhood of $TM\times\{0\}$ and full symbol of $\mathbb{P}$ is defined by its Fourier transformation.

Although the full symbol contains more information than the principal symbol, it is difficult to do calculation with (for example, its composition formula is complicated). In order to preserve useful aspects of smoothing operator, instead of the space of full symbols we shall study the space of Taylor’s expansion at $t=0$

(1.6)

$$a(x,\xi,t)\sim a(x,\xi,0)+t\partial_{t}a(x,\xi,0)+\frac{t^{2}}{2}\partial^{2}_{t}a(x,\xi,0)+\cdots$$

It has two advantages

• •

  the multiplicative structure (1.3) induces an algebra structure on $\Psi^{m}(\mathbb{T}M,\mathbb{S})$ which in turn induces an algebra structure on the space of Taylor’s expansion of full symbols. The crucial point is that the multiplication formula of Taylor’s expansion is explicitly computable and a lot easier than that of full symbols;

• •

  If $a(x,\xi,t)$ is a full symbol of order less than or equal to $-n$, than the supertrace of the corresponding pseudodifferential kernel $\mathbb{P}$ has asymptotic expansion

  $$\operatorname{Str}(\mathbb{P})\sim\sum_{i}t^{i}\cdot\left(\frac{2}{i}\right)^{n/2}(2\pi)^{-n}\int_{T^{\ast}M}a_{i}(x,\xi)d\xi dx$$

However, this space is restrictive in two ways:

• •

  it represents only properly supported distributions;

• •

  all $a_{i}(x,\xi)$’s in the expansion (1.6) are homogeneous modulo Schwartz functions.

There is an important class of pseudodifferential operators: the heat kernel that falls out of this category. We shall enlarge the space by looking at the space of symbols $S^{m}$ which is defined to be the subspace of $C^{\infty}(T^{\ast}M\times\mathbb{R},\rho^{\ast}\wedge T^{\ast}M)$ satisfy the classical symbol estimate

(1.7)

$$\left|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\partial_{t}^{\gamma}a(x,\xi,t)\right|\leq C\cdot(1+|\xi|+t)^{m-|\beta|},$$

for all $\alpha,\beta,\gamma$, here $\rho:T^{\ast}M\times\mathbb{R}\to M$ is the bundle projection. The space of Taylor’s expansion associate with this extended symbol space retains the two advantages. Moreover, the heat kernel, which can be expressed as the fundamental solution to the differential equation

$$\frac{\partial}{\partial\tau}+t^{2}D^{2}=0,$$

is captured by this class of symbols. By first passing to distributions level, second to the full symbol level and third to the Taylor’s expansion level of the differential equation, combine with the fact that multiplication formula of Taylor’s expansion is explicitly computable, the differential equations are simplified and the solutions are explicitly computable. Under this light, the asymptotic expansion of the heat kernel $e^{-\tau t^{2}D^{2}}$ can be calculated and the Mckean-Singer formula guarantee that the supertrace of the leading term is precisely the topological index. As a corollary, we obtain that the heat kernel $e^{-\tau t^{2}D^{2}}$ forms a smooth section of the rescaled bundle.

This paper is organized as follows: in section 2 we review some basic facts of the theory of distributions on manifolds with coefficient in vector bundles and their Fourier transformations; in section 3 we summarize the construction of the rescaled bundle over tangent groupoid for closed spin manifold; in section 4 we recall the theory of fibered distributions developed in [LMV17] and, as in [EY17], use it to define a class of pseudodifferential operators in section 5; in section 6, we study the space of Taylor’s expansion of full symbols of pseudodifferential operators and show that this space has an explicitly computable multiplication formula which can be applied to heat equation and gives the asymptotic expansion of the heat kernel in section 7.

Let $M$ be a closed Riemannian manifold, $E\to M$ a vector bundle over $M$ and $E^{\ast}$ the dual vector bundle. The space of distributions with coefficient in $E$ denoted by $\mathcal{D}^{\prime}(M,E)$ is defined to be the continuous dual of $C_{c}^{\infty}(M,E^{\ast})$. Notice that we have choose a Riemannian structure on $M$ so that the density is omitted in the discussion of distributions. Let $\{U_{i}\}$ be an open cover of $M$ that consists of finitely many open sets and $E$ is trivial over each member, denote by $\varphi_{i}:E|_{U_{i}}\to U_{i}\times\mathbb{R}^{n}$ the trivialization and $\rho_{i}$ the subordinate partition of unity. For any $u_{i}\in\mathcal{D}^{\prime}(M,E)$ that has supported within some $U_{i}$, we have $u_{i}\in\mathcal{D}^{\prime}(U_{i})\otimes\mathbb{R}^{n}$. In another word, $u_{i}$ can be written as $\sum_{I=1}^{n}u_{i,I}\otimes e_{i,I}$ where $u_{i,I}$ are distributions on $U_{i}$ and $e_{i,I}$ are basis of $\mathbb{R}^{n}$.

For general $u\in\mathcal{D}^{\prime}(M,E)$, we have the following decomposition:

(2.1)

$$u=\sum_{i}\rho_{i}u=\sum_{i}\sum_{I=1}^{n}\rho_{i}u_{i,I}\otimes e_{i,I}.$$

Let $u\in\mathcal{D}^{\prime}(M)$ be a distribution on manifold $M$ and $\varphi\in C^{\infty}(M,E)$ be a smooth section of $E$, the product $u\cdot\varphi\in\mathcal{D}^{\prime}(M,E)$ is defined so that

(2.2)

$$\langle u\cdot\varphi,s\rangle=\langle u,\langle\varphi,s\rangle\rangle$$

for all $s\in C^{\infty}_{c}(M,E^{\ast})$. Here $\langle\varphi,s\rangle$ denote the compactly supported function given by $\langle\varphi,s\rangle(m)=\langle\varphi(m),s(m)\rangle_{E,E^{\ast}}$. Moreover, it is straightforward to check that for any $f\in C^{\infty}(M)$, we have $uf\cdot\varphi=u\cdot f\varphi$. Under the light of (2.2), the equation (2.1) becomes $u=\sum_{i,I}\rho_{i}u_{i,I}\cdot e_{i,I}$. Therefore, we have the following proposition.

Let $\pi:TM\to M$ be the tangent bundle of $M$, the following result is parallel to proposition 2.1.

In this paper, we shall have occasion to consider the Fourier transformation of distributions with coefficient in vector bundles. Let $u\in\mathcal{S}^{\prime}(TM,\pi^{\ast}E)$ be a tempered distribution with coefficient in $E$, its Fourier transformation, which is denote by $\widehat{u}\in\mathcal{S}^{\prime}(T^{\ast}M,\pi^{\ast}E)$, is defined, according to [H0̈3, Chapter VII], to satisfy

$$\langle\widehat{u},\varphi\rangle=\langle u,\widehat{\varphi}\rangle$$

where $\varphi\in\mathcal{S}(T^{\ast}M,\pi^{\ast}E^{\ast})$ is any Schwartz section and $\widehat{\varphi}\in\mathcal{S}(TM,\pi^{\ast}E^{\ast})$ its Fourier transformation.

Let $U\subset TM$ be an open neighborhood of the zero section where the exponential map is well defined and injective. Let $\widetilde{\mathbb{U}}$ be the open neighborhood of $TM\times\{0\}\subset TM\times\mathbb{R}$ given by

$$\widetilde{\mathbb{U}}=\{(x,Y,t)\in TM\times\mathbb{R}\mid(x,-tY)\in U\}.$$

Then the following homeomorphism determines the smooth structure of the tangent groupoid near $t=0$.

(3.1)		$\displaystyle TM\times\mathbb{R}\supset\widetilde{\mathbb{U}}$	$\displaystyle\xrightarrow{\Phi}\mathbb{U}\subset\mathbb{T}M$
	$\displaystyle(x,Y,t)$	$\displaystyle\mapsto(x,\exp_{x}(-tY),t)$
	$\displaystyle(x,Y,0)$	$\displaystyle\mapsto(x,Y,0),$

where $\mathbb{U}\subset\mathbb{T}M$ is the image of the above map.

Let $M$ be a closed spin manifold with spinor bundle $S\to M$. The rescaled bundle $\mathbb{S}\to\mathbb{T}M$ is a vector bundle over the tangent groupoid whose restriction to $M\times M\times\mathbb{R}^{\ast}$ is the pullback of $S^{\ast}\boxtimes S$ and whose restriction to $TM$ is the pullback of the bundle of exterior algebras. This information is summarized in the diagram (1.2). Each part in the diagram has its own smooth structure, to specify the smooth structure of the rescaled bundle it is enough to specify how does these two parts fit together.

Laurent polynomials (3.2) can be evaluated at any $\gamma\in\mathbb{T}M$. The evaluation is given as follows: its value at $\gamma=(x,y,\lambda)\in M\times M\times\mathbb{R}^{\ast}$ is given by $\sum f_{p}(x,y)\lambda^{-p}$ and its value at $(m,X,0)\in TM\times\{0\}$ is given by$\sum\frac{1}{p!}X^{p}f_{p}$. Moreover, the character spectrum of the associative algebra $A$ is precisely the tangent groupoid $\mathbb{T}M$, namely for any point $\gamma\in\mathbb{T}M$ there is a maximal ideal $I_{\gamma}\subset A(\mathbb{T}M)$ given by the collection of elements whose evaluation is zero at $\gamma$, and this viewpoint can be used to determines the manifold structure of $\mathbb{T}M$ (see [HSSH18] or [HY19]).

One may take the algebra $A(\mathbb{T}M)$ as the space of ”algebraic functions” on the tangent groupoid. In the following, we shall define an $A(\mathbb{T}M)$-module denoted by $S(\mathbb{T}M)$ which can be taken as, with the previous analogy, the space of ”algebraic sections” of rescaled bundle.

We shall refer to [HY19, Definition 3.3.5] for a precise definition and detailed discussion for the notion of scaling order. However, we shall give an important example of sections with certain scaling order.

It is a fact that $S(\mathbb{T}M)$ is a module over $A(\mathbb{T}M)$ by ordinary multiplication of Laurent polynomials. The fiber of $S(\mathbb{T}M)$ over $\gamma$ is defined to be

$$S(\mathbb{T}M)|_{\gamma}=S(\mathbb{T}M)\big{/}I_{\gamma}\cdot S(\mathbb{T}M).$$

There is an isomorphism between $S(\mathbb{T}M)|_{(x,y,t)}$ and $S^{\ast}_{x}\otimes S_{y}$ which is induced by the map

$$\varepsilon_{(x,y,\lambda)}:\sum s_{p}t^{-p}\mapsto\sum s_{p}(x,y)\lambda^{-p}$$

for $t\neq 0$. For $t=0$, there is a similar isomorphism between $S(\mathbb{T}M)|_{(x,Y,0)}$ and $\wedge T^{\ast}_{x}M$ which is slightly more complicated and we refer the reader to [HY19, Proposition 3.4.9]. According to these two isomorphisms, there is a map

$$S(\mathbb{T}M)\to\prod_{\gamma\in\mathbb{T}M}S(\mathbb{T}M)|_{\gamma}$$

which sends $\sigma\in S(\mathbb{T}M)$ to its image $\widehat{\sigma}$ under the corresponding isomorphism.

It can be shown that this is a locally free sheaf and is the space of smooth sections of the rescaled bundle $\mathbb{S}\to\mathbb{T}M$. According to the construction, for any element $\sigma$ of $S(\mathbb{T}M)$, the assignment $\widehat{\sigma}$ which sends $\gamma\in\mathbb{T}M$ to $\widehat{\sigma}(\gamma)$ is a smooth section of the rescaled bundle. In fact, thanks to Example 3.3, the assignments $\widehat{e_{I}t^{\ell(I)}}$, as $I$ ranges over all multi-index $I\subset\{1,2,\cdots,n\}$, form a local frame of the rescaled bundle. The following Proposition clarify the smooth structure of the rescaled bundle over $\mathbb{U}$.

The rescaled bundle also has a multiplicative structure which we shall describe now. Let $G$ be a Lie groupoid, $G^{(2)}=\{(\gamma,\eta)\in G\times G\mid s(\gamma)=r(\eta)\}$ be the set of composable pairs. Denote by $p_{1},p_{2}:G^{(2)}\to G$ the two coordinate projections and $m:G^{(2)}\to G$ the multiplication map.

The multiplicative structure of the rescaled bundle $m:p_{1}^{\ast}\mathbb{S}\otimes p_{2}^{\ast}\mathbb{S}\to\mathbb{S}$ is given as follows:

• •

  the restriction of $m$ away from $t=0$ is the multiplication $\mathbb{S}_{(x,y,t)}\otimes\mathbb{S}_{(y,z,t)}\to\mathbb{S}_{(x,z,t)}$ given by contracting the middle two variables;

• •

  the restriction of $m$ to $t=0$ is the multiplication $\mathbb{S}_{(x,Y,0)}\otimes\mathbb{S}_{(x,Z,0)}\to\mathbb{S}_{(x,Y+Z,0)}$ given by sending $\alpha\otimes\beta\in\wedge T^{\ast}_{x}M\otimes\wedge T^{\ast}_{x}M$ to $\alpha\wedge\beta\wedge\exp(-\frac{1}{2}\kappa(Y,Z))$ where $\kappa(Y,Z)$ is the differential two form given by the symbol of curvature of the spinor bundle.

(See [HY19, Section 4])

Most of the statements in this section are valid for general vector bundle over Lie groupoid that has a multiplicative structure. At the end of this section, we shall specialize to the case when $G=\mathbb{T}M$ is the tangent groupoid and $E=\mathbb{S}$ is the rescaled bundle.

In the following discussion, we shall concern mainly with the case when $M$ is a Lie groupoid $G$, $B$ is the unit space $G^{(0)}$ and $s$ is the range map $r:G\to G^{(0)}$. We shall also have occasion to consider the case when $M=G^{(2)}$ with vector bundle $E\boxtimes E\to G^{(2)}$, $B=G^{(0)}$ and $s=r\circ m$ the composition of the multiplication map with the range map.

Now, let $E\to G$ be a vector bundle with multiplicative structure, we shall describe the algebra structure on the set of distributions $\mathcal{E}^{\prime}_{r}(G,E)$. The multiplicative structure (3.4) of $E$ induces a map $m^{\ast}:C^{\infty}(G,E^{\ast})\to C^{\infty}(G^{(2)},E^{\ast}\boxtimes E^{\ast})$ which is given by $m^{\ast}f(\gamma_{1},\gamma_{2})=m_{\gamma_{1},\gamma_{2}}^{\ast}f(\gamma_{1}\gamma_{2})$, here $f$ is any smooth section of $E^{\ast}\to G$ and $m_{\gamma_{1},\gamma_{2}}^{\ast}:E^{\ast}_{\gamma_{1}\gamma_{2}}\to E^{\ast}_{\gamma_{1}}\otimes E^{\ast}_{\gamma_{2}}$ is the dual of the multiplication map $m:E_{\gamma_{1}}\otimes E_{\gamma_{2}}\to E_{\gamma_{1}\gamma_{2}}$. At the level of fibered distributions, we can define $m_{\ast}:\mathcal{E}^{\prime}_{r\circ m}(G^{(2)},E\boxtimes E)\to\mathcal{E}^{\prime}_{r}(G,E)$ by $m_{\ast}u(f)=u(m^{\ast}f),$ for any $f\in C^{\infty}(G,E^{\ast})$ and any $u\in\mathcal{E}^{\prime}_{r\circ m}(G^{(2)},E\boxtimes E)$. Next, consider the commutative diagram

Notice that the fibers of $p_{1}:G^{(2)}\to G$ correspond to the range fibers of $G$. For any $u\in\mathcal{E}^{\prime}_{r}(G,E)$, its pullback $p_{1}^{\ast}u$ by the first projection $p_{1}$ is defined to be the continuous $C^{\infty}(G)$-module map $C^{\infty}(G^{(2)},E^{\ast}\boxtimes E^{\ast})\to C^{\infty}(G,E^{\ast})$ which is given by

$$\langle p_{1}^{\ast}u,g\rangle(\gamma)=\langle u^{s(\gamma)},g|_{(\gamma,G^{s(\gamma)})}\rangle$$

here the $C^{\infty}(G)$-module structure on $C^{\infty}(G^{(2)},E^{\ast}\boxtimes E^{\ast})$ is given by $p_{1}$, we take $(\gamma,G^{s(\gamma)})$ as a submanifold of $G^{(2)}$ and $u^{s(\gamma)}:C^{\infty}(G^{s(\gamma)},E^{\ast})\to\mathbb{C}$ is the distribution induced by $u$.

Let $u,v\in\mathcal{E}^{\prime}_{r}(G,E)$, then the convolution multiplication $u\ast v$ is defined to be $m_{\ast}(u\circ\operatorname{pr_{2}}^{\ast}v)\in\mathcal{E}^{\prime}_{r}(G,E)$. Explicitly the convolution is given by

(4.3)

$$\langle u\ast v,\varphi\rangle(x)=\left\langle u^{x}(\gamma),\left\langle v^{s(\gamma)}(\eta),m^{\ast}\varphi(\gamma,\eta)\right\rangle\right\rangle$$

for $\varphi\in C^{\infty}(G,E^{\ast})$. In fact, slightly more is true.