# Deformation Spaces, Rescaled Bundles and the Kirillov Character Formula

Maxim Braverman and Ahmad Reza Haj Saeedi Sadegh

(Date: November 2022)

###### Abstract.

In this paper, we construct a vector bundle over the deformation to the normal
cone $\textnormal{DNC}(V,M)$ through a rescaling of a vector bundle $E\to V$.
This method generalizes the construction of the spinor rescaled bundle over
the tangent groupoid by Nigel Higson and Zelin Yi. We use this generalized
construction to recover Kirillov’s character formula for the equivariant index
of Dirac-type operators. As another application, we show that the Witten and
the Novikov deformations of the de Rham-Dirac operator define a smooth
operator on certain rescaled bundles.

##  1\. Introduction

In the coadjoint orbit method of Kirillov, every character of a compact Lie
group is given as an integral over some orbits of the coadjoint action of $G$
on $\mathfrak{g}^{*}$. This is known as the Kirillov character formula.
Berline and Vergne [3] (see also [2, Ch. 8]) extended this formula to a
delocalized equivariant index theorem for a general equivariant Dirac-type
operator on a closed manifold $M$. In this version of the index theorem, also
known as the (generalized) Kirillov formula, the equivariant index is
expressed as an integral of an equivariant differential form over the whole
manifold. Applying the localization formula for the integral of an equivariant
form to the Kirillov formula, one recovers Atiyah-Segal fixed point formula
for the index, [1].

In [2, Ch. 8], a local version of the Kirillov form formula is proven by
adopting the Bismut’s generalization, [4], of the Getzler rescaling technique.
In this paper, we give a coordinate free version of the proof in [2] by using
the tangent groupoid $\mathbb{T}M$. This gives an equivariant generalization
of the method of Higson and Yi, [13].

We also study the _deformation to the normal cone_ $\textnormal{DNC}(V,M)$ for
an embedding $M\hookrightarrow V$ (see Subsection 4.1, or [12, 19, 20]), and
generalize the rescaled bundle construction of [13] to this deformation space.
In some sense this generalization makes the exposition clearer and more
conceptual, since it makes it evident which structures are needed for the
construction to work. It also allows us to study the Witten and the Novikov
deformations of the de Rham-Dirac operator using the rescaled bundle
construction, cf. Example 4.21.

We now give a slightly more detailed review of this construction.

To an embedding of smooth manifolds $M\hookrightarrow V$ on may associate the
deformation to the normal cone, denoted $\textnormal{DNC}(V,M)$, as a set is
given by the disjoint union

$\mathcal{N}\times\\{0\\}\bigsqcup_{t\neq 0}V\times\\{t\\}$

where $\mathcal{N}$ is the normal bundle of the embedding. There is smooth
structure on $\textnormal{DNC}(V,M)$ which can be given by an explicit atlas
([19]) or can be given using an algebraic point of view (see Subsection 4.1 or
[20]).

One important example of the deformation to the normal cone is the _tangent
groupoid_. Associated to the diagonal embedding $M\hookrightarrow M\times M$,
the deformation space $\textnormal{DNC}(M\times M,M)$, also denoted by
$\mathbb{T}M$, is called the tangent groupoid of $M$. As a set it is given by
the disjoint union

$TM\times\\{0\\}\bigsqcup_{t\neq 0}M\times M\times\\{t\\}.$

In [11], Connes used the tangent groupoid whose K-theory carries the
analytical index of elliptic operators. In [21], van Erp and Yuncken used the
tangent groupoid to completely characterize the algebra of pseudodifferential
operators.

In [13], Higson and Yi introduced a coordinate-free approach to Getzler’s
rescaling method. This was achieved by introducing the _rescaled spinor
bundle_ over the tangent groupoid of a spin manifold. So for a spin manifold
$M$, with the spinor bundle ${S}\to M$, Higson and Yi, [13], constructed a
vector bundle $\mathbb{S}\to\mathbb{T}M$ over the tangent groupoid. This
bundle is given by the following diagram

${\pi^{*}\Lambda^{*}T^{*}M}$${S\boxtimes
S^{*}}$${TM\times\\{0\\}}$${\bigsqcup_{t\neq 0}}$${M\times M\times\\{t\\}.}$

In this paper we generalized the construction of the rescaled bundle in [13]
to deformation to the normal cones and give some examples of the rescaled
bundles. As an application, we give a proof of the Kirillov formula by
introducing a family of rescaled bundles over the tangent groupoid of a
manifold $M$ that carries a $G$-equivariant Clifford module.

The paper is organized as follows: In Section 2 we give a quick overview of
Cartan’s model of the equivariant cohomology and equivariant characteristic
classes.

In Section 2 we state the (generalized) Kirillov formula and give an overview
of the heat kernel proof of this formula. In particular, we introduce the
Bismut Laplacian and explain its role in the proof.

In Section 4, we introduce the deformation to the normal cone
$\textnormal{DNC}(V,M)$ associated with an embedding of manifolds
$M\hookrightarrow V$. If there is a vector bundle $E\to V$ with connection
compatible with certain filtrations on $E|_{M}$ and on the bundle of
endomorphism $\textup{End}(E)$, then we introduce a vector bundle
$\mathbb{E}\to\textnormal{DNC}(V,M)$. In the construction of the deformation
to the normal cone and the bundle over it, we follow an algebraic method
introduced in [20] and [13].

In Section 5, we give an example of the rescaled bundle over the tangent
groupoid $\mathbb{T}M$, when $M$ carries a $G$-equivariant Clifford module
structure. We essential give a family of rescaled bundles
$\mathbb{E}_{J}\to\mathbb{T}M$ indexed by $J\in\mathbb{N}$. By studying the
Bismut Laplacian and its relation to the rescaled bundles, we then give a
proof of Kirillov’s character formula for the equivariant index of the Dirac
operator.

##  2\. The Equivariant Cohomology

In this section we recall the Cartan module for equivariant cohomology and the
construction of the equivariant characteristic classes. In our exposition we
roughly follow [2, §7.1].

###  2.1. The polynomial algebra

Consider the space of polynomials $\mathbb{C}[\mathfrak{g}]$ that comes with
the grading

$\mathbb{C}[\mathfrak{g}]=\bigoplus_{k}\mathbb{C}_{k}[\mathfrak{g}]$

with $\mathbb{C}^{k}[\mathfrak{g}]$ consists of homogeneous polynomials of
degree $k$. We also consider the corresponding filtration

$\mathbb{C}=\mathbb{C}^{0}[\mathfrak{g}]\subset\mathbb{C}^{1}[\mathfrak{g}]\subset\cdots\subset\mathbb{C}[\mathfrak{g}].$

Denote by $\mathbb{C}[\mathfrak{g}]_{(J)}$ the quotient of
$\mathbb{C}[\mathfrak{g}]$ by the ideal of polynomials of order $J+1$. Every
element $a\in\mathbb{C}[\mathfrak{g}]_{(J)}$ has a unique polynomial
representative of the form $\sum_{\alpha}a_{\alpha}X^{\alpha}$ and hence the
algebra $\mathbb{C}[\mathfrak{g}]_{(J)}$ inherits a grading (by degree of the
polynomials).

The polynomial algebra appears in the Kirillov formula context as

$\mathbb{C}[\mathfrak{g}]\otimes\Lambda V$

where $V$ is a finite dimensional vector space. This tensor space has a
grading given by

$\deg(P\otimes a)=2\deg(P)+\deg_{\Lambda}(a)$ (1)

where $P\in\mathbb{C}[\mathfrak{g}]$ and $a\in\Lambda V$. Similar grading will
descend to tensor spaces

$\mathbb{C}[\mathfrak{g}]_{(J)}\otimes\Lambda V.$

###  2.2. G-equivariant differential forms.

Asuume $G$ is a Lie group acting on the smooth manifold $M$. For
$X\in\mathfrak{g}$, we denote by $X^{M}$ or just $X$ the vector field on $M$
corresponding to the infinitesimal action on $C^{\infty}(M)$

$Xf(m):=\frac{d}{dt}|_{t=0}f(\exp(-tX)m).$

Let $\mathbb{C}[\mathfrak{g}]$ be the algebra of polynomial functions on
$\mathfrak{g}$ and consider the algebra

$\mathbb{C}[g]\otimes\mathcal{A}(M)$

which carries a left action of $G$: For
$\alpha\in\mathbb{C}[g]\otimes\mathcal{A}(M)$ and $g\in G$ we have

$(g.\alpha)(X)=g.\Big{(}\alpha(\textup{Ad}_{g^{-1}}X)\Big{)}.$

This algebra has a grading for $\alpha\in\mathbb{C}[g]\otimes\mathcal{A}(M)$
and $g\in G$ given by the equivariant degree

$o(\alpha)=o^{\textup{ext}}(\alpha)+2o^{\textup{poly}}(\alpha)$

where $o^{\textup{ext}}$ is the exterior degree of differential forms and
$o^{\textup{poly}}$ is the polynomial degree.

We define the equivariant differential $d_{\mathfrak{g}}$ on
$\mathbb{C}[g]\otimes\mathcal{A}(M)$ given by the formula

$(d_{\mathfrak{g}}\alpha)(X)=d(\alpha(X))-\iota(X^{M})\alpha(X).$

Note that the contraction $\iota(X^{M})$ reduces the differential degree by
$1$, increases the polynomial degree by $1$ and hence it increase the
equivariant degree by $1.$ Hence $d_{\mathfrak{g}}$ increases the equivariant
degree by $1.$ Note that $d^{2}_{\mathfrak{g}}\neq 0$, indeed

$(d^{2}_{\mathfrak{g}}\alpha)(X)=-\mathscr{L}(X)\alpha(X).$

However, square of the equivariant differential vanishes on the Cartan
subalgebra

$\mathcal{A}_{G}(M)=\Big{(}\mathbb{C}[\mathfrak{g}]\otimes\mathcal{A}(M)\Big{)}^{G}$

of invariant elements or equivariant differential forms. Indeed

  * •

For every $g\in G$

$(g.\alpha)(X)=\alpha(\textup{Ad}_{g}X)$

or equivalently

  * •

for every $X\in\mathfrak{g}$

$\mathscr{L}(X)\alpha(X)=0.$

Since $(\mathcal{A}_{G}(M),d_{\mathfrak{g}})$ is a complex, we can define its
cohomology, the equivariant cohomology of $M$, denoted by

$H_{G}(M).$

This is known as the Cartan model of equivariant cohomology [10]. When $M$ is
compact and oriented, we define the integration map

$\int_{M}:\mathbb{C}[\mathfrak{g}]\otimes\mathcal{A}(M)\to\mathbb{C}[\mathfrak{g]}$

by integrating only terms with top differential degree part. This map
restricts to

$\int_{M}:\mathcal{A}_{G}(M)\to\mathbb{C}[\mathfrak{g]}^{G}$

where the image is the algebra of invariant polynomials. This map indeed
descends to the cohomology level:

$\int_{M}:H_{G}(M)\to\mathbb{C}[\mathfrak{g]}^{G}$

###  2.3. The Equivariant Structure

Let $M$ be compact oriented Riemannian manifold of $M$. Assume a Lie group $G$
acts by positively oriented isometries on $M$. Assume $\mathscr{E}\to M$ is a
$G$-equivariant vector bundle with a connection $\nabla^{\mathscr{E}}$ that
commute with the $G$-action. We define the moment
$\mu^{\mathscr{E}}\in\mathcal{A}_{G}^{2}(M,\textup{End}(\mathscr{E}))$ of the
connection $\nabla^{\mathscr{E}}$ by the formula

$\mu^{\mathscr{E}}(X):=\mathscr{L}^{\mathscr{E}}(X)-\nabla_{X}^{\mathscr{E}}$

where $\mathscr{L}^{\mathscr{E}}(X)$ is the Lie derivative of
$X\in\mathfrak{g}$. In particular, when $\mathscr{E}=TM$ with the Levi-Civita
connection $\nabla$, we obtain the Riemannian moment

$\mu^{M}\in\Big{(}\Gamma(M,\mathfrak{so}(M))\otimes\mathfrak{g}^{*}\Big{)}^{G}$

so that for $X\in\mathfrak{g}$ and $\xi\in\Gamma(TM)$

$\displaystyle\mu^{M}(X)\xi$ $\displaystyle=[X,\xi]-\nabla_{X}\xi$
$\displaystyle=-\nabla_{\xi}X.$

Another example of moment map is the Kosmann’s formula for the spinor bundle
$\mathscr{E}=S$, when $M$ is spin:

######  Proposition 2.1.

[14] If $M$ is spin, and $\mathscr{E}=S$ is the spinor bundle then

$\mu^{S}(X)=-\frac{1}{4}c(d\theta_{X}).$

Assume $\mathscr{E}\to M$ is a G-equivariant Clifford module whose Clifford
action and Hermitian metric are compatible with the $G$-action. Locally, such
vector bundle is given as $S\otimes E$ where $S$ is the local spinor bundle
and $E$ is an equivariant auxiliary vector bundle with $G$-compatible
connection $\nabla^{E}$ such that $\nabla^{\mathscr{E}}=\nabla^{S}\otimes
1+1\otimes\nabla^{E}.$ Then we define the twisting moment map (locally) as the
moment of the auxiliary bundle $E$ or equivalently differnce

$\mu^{\mathscr{E}/S}:=\mu^{\mathscr{E}}-\mu^{S}.$

So we have $\mu^{\mathscr{E}/S}=\mu^{\mathscr{E}}+\frac{1}{4}c(d\theta_{X})$
and moreover we have
$\mu^{\mathscr{E}/S}\in\mathcal{A}_{G}^{2}(M,\textup{End}_{\textup{Cl}}(\mathscr{E}))$
(see [2]). Here $\textup{End}_{\textup{Cl}}$ denotes the space of
endomorphisms that commute with the Clifford action.

The equivariant curvature of $M$ is defined by

$R_{\mathfrak{g}}:=R+\mu^{M}\in\mathcal{A}_{G}^{2}(M,\mathfrak{so}(M)).$

where $R$ is the Riemannian curvature tensor. The equivariant
$\widehat{\textnormal{A}}$-genus is the closed equivariant differential form

$\widehat{\textnormal{A}}_{\mathfrak{g}}(X,M):=\textup{det}^{1/2}\Big{(}\frac{R_{\mathfrak{g}}/2}{\sinh(R_{\mathfrak{g}}/2)}\Big{)}\in\mathcal{A}_{G}(M).$

Using this we obtain the equivariant twisting curvature of $\mathscr{E}$ given
by

$F^{\mathscr{E}/S}_{\mathscr{g}}(X)=F^{\mathscr{E}}+\mu^{\mathscr{E}/S}\in\mathcal{A}_{G}^{2}(M,\textup{End}_{\textup{Cl}}(\mathscr{E})).$

The equivariant relative Chern character of $\mathscr{E}$ is then defined as

$\textnormal{Ch}_{\mathfrak{g}}(X,\mathscr{E}/S)=\textnormal{Str}_{\mathscr{E}/S}\Big{(}\exp(-F^{\mathscr{E}/S}_{\mathscr{g}}(X))\Big{)}\in\mathcal{A}_{G}^{2}(M).$

##  3\. The Equivarinat Index and the Kirillov Formula

In this section we formulate our main result – the Kirillov formula for the
index of the equivariant Dirac-type operator. Then we give an overview of the
heat kernel proof of this formula, [2, Ch. 8]. In particular, we define the
Bismut Laplacian [4] and explain its role in the proof.

###  3.1. The settings

Let $G$ be a compact Lie Group acting by positively oriented isometries on a
compact oriented even-dimensional Riemannian manifold $M^{n}$. Consider a
$\mathbb{Z}_{2}$-graded Clifford module $\mathscr{E}\to M$ which carries an
action of $G$ that commutes with the grading, the Hermitian product, and the
Clifford action. We fix a Clifford a $G$-equivariant connection
$\nabla^{\mathscr{E}}$ on $\mathscr{E}$ and consider the Dirac-type operator

$D:\Gamma(M,\mathscr{E})\xrightarrow[]{\nabla^{\mathscr{E}}}\Gamma(M,\mathscr{E}\otimes\Lambda^{1}T^{*}M)\xrightarrow[]{c}\Gamma(M,\mathscr{E}).$

Then $D$ commutes with the action of $G$. Any $g\in G$, takes
$\textnormal{ker}(D^{\pm})$ to themselves. The equivariant index
$\textup{ind}(g,D)$ of $D$ is defined as the supertrace

$\textup{ind}(g,D):=\textup{tr}(g,\textnormal{ker}(D^{+}))-\textup{tr}(g,\textnormal{ker}(D^{-})).$

######  Theorem 3.1 (Kirillov Formula).

For $g=e^{-X}$, with $X\in\mathfrak{g}$ sufficiently small,

$\textup{ind}(e^{-X},D)=(2\pi
i)^{-n/2}\int_{M}\widehat{\textnormal{A}}_{\mathfrak{g}}(X,M)\textnormal{Ch}_{\mathfrak{g}}(X,\mathscr{E}/S)$

In the rest of this section we give a brief overview of the heat kernel proof
of this theorem.

###  3.2. The modified McKean-Singer formula

The equivariant index of $D$ can be related to the heat kernel of $D$ via the
McKean-Singer-type theorem:

$\textup{ind}(g,D)=\textup{str}(ge^{-tD^{2}})$ (2)

for $t>0.$ However, it turns out that the asymptotic equation of the heat
kernel in the right hand side of this formula is complicated and the Getzler’s
rescaling method does not go through well for it. The calculation becomes
easier if we slightly modify the operator $D$. In fact, the following
generalization of (2) is true: Fix $X\in\mathfrak{g}$ and for $u\in\mathbb{C}$
define the deformed Dirac operator

$D_{u}=D+uc(X).$

Note that this is the Dirac operator associated with the connection

$\nabla^{\mathscr{E},-uX}:=\nabla^{\mathscr{E}}+u\theta_{X}.$ (3)

where $\theta_{X}\in\mathcal{A}^{1}(M)$ is the one-form dual to the vector
field $X^{M}$:

$\theta_{X}(\xi)=(X^{M},\xi),$

in other words

$D_{u}:\Gamma(M,\mathscr{E})\xrightarrow[]{\nabla^{\mathscr{E},-uX}}\Gamma(M,\mathscr{E}\otimes\Lambda^{1}T^{*}M)\xrightarrow[]{c}\Gamma(M,\mathscr{E}).$

######  Proposition 3.2.

For every $t>0$ and $u\in\mathbb{C}$ we have

$\textup{ind}(e^{-X},D)=\textup{str}(e^{-X}e^{-tD^{2}_{u}}).$

Note that, since $D$ commutes with the infinitesimal action of $\mathfrak{g}$,
we can write

$e^{-X}e^{-tD^{2}_{u}}=e^{-\mathscr{L}(X)}e^{-tD^{2}_{u}}=e^{-tD_{u}^{2}-\mathscr{L}(X)}.$

We denote the generalized Laplacian operator that appear in the exponent of
the right-hand side, for $t=1$, by

$H_{u}(X)=D_{u}^{2}+\mathscr{L}^{\mathscr{E}}(X).$ (4)

Therefore the equivariant index corresponding to $e^{-X}\in G$ is given by

$\textup{ind}(e^{-X},D)=\textup{str}(e^{-H_{u}(X)}).$

###  3.3. The Lichnerowicz formula

We now compute the operator $H_{u}$.

######  Lemma 3.3.

[2, Prop. 3.45] If $\theta\in\mathcal{A}^{1}(M)$ is dual to the vector field
$X\in\mathfrak{X}(M)$ then

$[D,c(d\theta)]=-2\nabla_{X}^{\mathscr{E}}+c(d\theta)+d^{*}\theta.$

######  Proposition 3.4.

We have a Lichnerowicz-type formula for $H_{u}(X)$:

$H_{u}(X)=\Big{(}\nabla^{\mathscr{E},uX}\Big{)}^{*}\nabla^{\mathscr{E},uX}+\frac{1}{4}{\kappa}_{M}+c(F^{\mathscr{E}/S})+\mu^{\mathscr{E}}(X)+uc(d\theta_{X})+(1-4u)\nabla_{X}^{\mathscr{E}}.$

With respect to a local orthonormal frame $\\{e_{i}\\}$

$\Big{(}\nabla^{\mathscr{E},uX}\Big{)}^{*}\nabla^{\mathscr{E},uX}=-\sum_{i}\Big{(}\nabla^{\mathscr{E},uX}_{e_{i}}\Big{)}^{2}-\nabla^{\mathscr{E},uX}_{\nabla_{e_{i}}e_{i}}.$

Note that when $u=\frac{1}{4}$, using the equality
$\mu^{\mathscr{E}/S}=\mu^{\mathscr{E}}+\frac{1}{4}c(d\theta_{X})$, we obtain

$H_{1/4}(X)=\Big{(}\nabla^{\mathscr{E},\frac{1}{4}X}\Big{)}^{*}\nabla^{\mathscr{E},\frac{1}{4}X}+\frac{1}{4}{\kappa}_{M}+c(F^{\mathscr{E}/S})+\mu^{\mathscr{E}/S}(X).$

###### Proof.

Note that since the $D_{u}$ is the Dirac operator associated to the connection
$\nabla^{\mathscr{E},-uX}$, and
$\nabla^{\mathscr{E},-uX}=\nabla^{\mathscr{E}}+u\theta_{X}$ is a Clifford
connection, then we have the Lichnerowicz formula

$\displaystyle D_{u}^{2}$
$\displaystyle=\Big{(}\nabla^{\mathscr{E},-uX}\Big{)}^{*}\nabla^{\mathscr{E},-uX}+\frac{1}{4}\kappa_{M}+c(F^{\mathscr{E}/S,-u})$
$\displaystyle=\Big{(}\nabla^{\mathscr{E},-uX}\Big{)}^{*}\nabla^{\mathscr{E},-uX}+\frac{1}{4}\kappa_{M}+c(F^{\mathscr{E}/S})+uc(d\theta_{X})$

We may write a similar formula for $D_{-u}^{2}$ and by subtracting we have

$\displaystyle D_{u}^{2}-D_{-u}^{2}$
$\displaystyle=\Big{(}\nabla^{\mathscr{E},-uX}\Big{)}^{*}\nabla^{\mathscr{E},-uX}-\Big{(}\nabla^{\mathscr{E},uX}\Big{)}^{*}\nabla^{\mathscr{E},uX}+2uc(d\theta_{X})$

we also have

$\displaystyle D_{u}^{2}-D_{-u}^{2}$ $\displaystyle=2u[D,c(\theta_{X})]$
$\displaystyle=-4u\nabla^{\mathscr{E}}_{X}+2uc(d\theta_{X})+2ud^{*}\theta_{X}$

where we used Lemma 3.3 in the second equality. Note that
$d^{*}\theta_{X}=\textnormal{tr}(\mu^{M}(X))=0$. Combining these formulas we
find

$\displaystyle H_{u}(X)$
$\displaystyle=D_{u}^{2}+\mathscr{L}^{\mathscr{E}}(X)$
$\displaystyle=\Big{(}\nabla^{\mathscr{E},uX}\Big{)}^{*}\nabla^{\mathscr{E},uX}+\frac{1}{4}\kappa_{M}++c(F^{\mathscr{E}/S})+uc(d\theta_{X})-4u\nabla_{X}^{\mathscr{E}}+\mathscr{L}^{\mathscr{E}}(X).$

Using the equality
$\mathscr{L}^{\mathscr{E}}(X)=\nabla^{\mathscr{E}}_{X}+\mu^{\mathscr{E}}(X)$
we then achieve the result.

∎

###  3.4. The Bismut Laplacian

As we see, for $u=1/4$, we obtain a simpler Lichnerowicz formula. The operator

$H_{1/4}=D^{2}_{1/4}+\mathscr{L}^{\mathscr{E}}(X)$ (5)

is called the Bismut Laplacian. From now on we drop the $1/4$ from the Bismut
Laplacian notation and denote it by $H(X):=H_{1/4}(X)$. Following [2] we
replace $D^{2}$ with $H_{1/4}$ in (2).

###  3.5. The heat kernel in normal coordinates

The proof of the Kirollov formula 3.1, is done through rescaling of the
smoothing kernel of the (a conjugate of the) heat operator $e^{-tH(X)}$ in
normal coordinates. Roughly one needs to trivialize the bundle $\mathscr{E}\to
M$ over a small neighborhood using the normal coordinates
$\bm{\mathrm{x}}\to\exp_{m}(\bm{\mathrm{x}})$, through parallel transport
along radial geodesics. Here $\bm{\mathrm{x}}$ belongs to a small neighborhood
of the origin in $T_{m}M$.

The Bismut Laplacian $H(X)$ is associated to the perturbed connection
$\nabla^{\mathscr{E},\frac{1}{4}X}=\nabla^{\mathscr{E}}-\frac{1}{4}\theta_{X}$.
So it is natural to use the parallel transport of this connection to define
the trivialization of $\mathscr{E}$. However, it is more convenient to do a
“two step” trivialization: first trivialize the bundle using the connection
$\nabla^{\mathscr{E}}$ and compute the heat kernel $e^{-tH(X)}$ in this
trivialization and then consider the conjugated kernel
$\rho_{\frac{1}{4}X}(\bm{\mathrm{x}})e^{-tH(X)}\rho_{\frac{1}{4}X}(\bm{\mathrm{x}})^{-1}$,
where $\rho_{\frac{1}{4}X}(\bm{\mathrm{x}})$ is the parallel transport map on
the trivial line bundle along the radial geodesics with respect to the
connection $d-\frac{1}{4}\theta_{X}$ (see (20)). In this approach, in
particular, the dependence of the result on $X$ is more explicit.

##  4\. Rescaled Bundles

In this paper, we achieve the rescaling calculus through a coordinate-free
method, i.e. through tangent groupoid of the manifold $M$. It is, however,
more general and arguably more natural to consider a more general deformation
space $\textnormal{DNC}(V,M)$. So in this and in the next sections, we
introduce a setup that gives vector bundles over the deformation space
$\textnormal{DNC}(V,M)$, generalizing the rescaled bundle of [13]. Specifying
to the case of the tangent groupoid we recover the Getzler symbol calculus.

###  4.1. Deformation to the normal cone

Assume $i:M\hookrightarrow V$ is an embedding of smooth manifolds. The normal
bundle of this embedding is the quotient bundle $\mathcal{N}=TV|_{M}/TM$. We
define the deformation to the normal cone $\textup{DNC}(V,M)$ associated with
the embedding in an algebraic way following [20].

Consider the subalgebra $\mathcal{A}(V,M)\subset C^{\infty}(V)[t,t^{-1}]$ that
consists of the Laurent polynomials

$\sum_{p}f_{p}t^{-p}$

where $f_{p}$ vanishes to $p$-th order along $M$.

######  Definition 4.1.

For a $\mathbb{C}$-algebra $\mathcal{A}$, the character spectrum,
$\textup{Spec}(\mathcal{A})$, is the space of all algebra homomorphisms

$\phi:\mathcal{A}\to\mathbb{C}$

with the weak topology of pointwise convergence.

It turns out the spectrum of the algebra $\mathcal{A}(V,M)$ consists of the
following characters:

  1. (a)

For every $(v,\lambda)\in V\times\mathbb{R}^{\neq 0}$

$\varepsilon_{(v,\lambda)}:\mathcal{A}(V,M)\to\mathbb{C}$
$\sum_{p}f_{p}t^{-p}\mapsto\sum_{p}f_{p}(v)\lambda^{-p}$ (6)

  2. (b)

For $X_{m}\in\mathcal{N}_{m}=T_{m}V/T_{m}M$

$\varepsilon_{X_{m}}:\mathcal{A}(V,M)\to\mathbb{C}$
$\sum_{p}f_{p}t^{-p}\mapsto\sum_{p}\frac{1}{p!}X^{p}f_{p}(m)$ (7)

where $X\in\mathfrak{X}(V)$ is a a vector field that represents
$X_{m}\in\mathcal{N}_{m}$ at $m\in M.$

Hence there is a one to one correspondence

$\textup{Spec}(\mathcal{A}(V,M))\longleftrightarrow\mathcal{N}\times\\{0\\}\bigsqcup_{t\neq
0}V\times\\{t\\}.$

This spectrum is indeed a manifold:

######  Theorem 4.2.

[20] The spectrum $\textup{Spec}(\mathcal{A}(V,M))$ is a smooth manifold of
dimension $\textnormal{dim}(V)+1$ that has a canonical submersion to
$\mathbb{R}$. This manifold is called the deformation to the normal cone and
is denoted by $\textnormal{DNC}(V,M)$. As a set it is given by disjoint union
of fibers over $\mathbb{R}$ as follows

$\mathcal{N}\times\\{0\\}\bigsqcup_{t\neq 0}V\times\\{t\\}.$

To gain insight into this theorem, it is useful to see the algebra
$\mathcal{A}(V,M)$ for $\textnormal{DNC}(V,M)$ as a “homogeneous coordinate
ring” for a variety. Note that the vanishing order along the submanifold $M$,
gives a filtration

$C^{\infty}(V)=I_{0}\supset I_{1}\supset\cdots$

where $I_{p}$ is the ideal of functions vanishing to order $p$ along $M$.
Denote by $\mathcal{A}_{0}(V,M)$ the quotient algebra
$\mathcal{A}(V,M)/t\mathcal{A}(V,M)$ which is naturally isomorphic to the
associated graded algebra:

$\mathcal{A}_{0}(V,M)\buildrel\simeq\over{\longrightarrow}\bigoplus_{p=0}I_{p}/I_{p+1}$
$\sum_{p}f_{p}t^{-p}\mapsto\sum_{q\geq 0}\langle f_{p}\rangle_{p}$

We have the inclusion $C^{\infty}(M)\hookrightarrow\mathcal{A}_{0}(V,M)$ as
the zero degree part. This corresponds to the following fact

######  Lemma 4.3.

The character spectrum of $\mathcal{A}_{0}(V,M)$ is the normal bundle
$\mathcal{N}\to M.$ The quotient $\mathcal{A}_{0,m}(V,M)$ of the algebra
$\mathcal{A}_{0}(V,M)$ by the ideal of functions vanishing at $m\in M$ is
isomorphic to the polynomial algebra over the normal space $\mathcal{N}_{m}$:

$\sum_{p}f_{p}t^{-p}\mapsto[X_{m}\mapsto\sum_{p}\frac{1}{p!}X^{p}.f_{p}|_{m}]$

for $X_{m}\in\mathcal{N}_{m}.$

One important example of the deformation to the normal cone is the _tangent
groupoid_ :

######  Example 4.4.

Let $M$ be a smooth manifold. The deformation to the normal cone associated to
the diagonal embedding $M\hookrightarrow M\times M$ is called the tangent
groupoid and is denote by

$\mathbb{T}M:=\textnormal{DNC}(M\times M,M).$

As a set, it is given by the disjoint union

$M\times M\times\mathbb{R}^{\neq 0}\bigsqcup TM\times\\{0\\}$

which is naturally fibered over $\mathbb{R}$.

The tangent groupoid is indeed a smooth groupoid,
$\mathbb{T}M\rightrightarrows{M\times\mathbb{R}}$. The source and target maps
are given so that for $t\neq 0$, they restrict to pair groupoid

$M\times M\times\\{t\\}\rightrightarrows{M}\times\\{t\\}$

and for $t=0$, the source and target map are the same as the projection map of
the vector bundle:

$TM\times\\{0\\}\to M\times\\{0\\}.$

###  4.2. Alternative formula for $\epsilon_{X_{m}}$

We now give an alternative presentation of the formula (7) which justifies
many formulas in the upcoming subsections and sections. For every vector field
$X\in\mathfrak{X}(V)$ we have the derivation map

$tX:\mathcal{A}(V,M)\to\mathcal{A}(V,M)$
$\sum_{p}f_{p}t^{-p}\mapsto\sum_{p}X.f_{p}t^{-p+1}$

which descends to the quotion algebra

$tX:\mathcal{A}_{0}(V,M)\to\mathcal{A}_{0}(V,M).$

The action of $tX$ on $\mathcal{A}_{0}(V,M)$ is locally nilpotent, in the
sense that for each $f\in\mathcal{A}_{0}(V,M)$ there exists an integer
$n=n(f)$ such that $(tX)^{n}f=0$. Hence, we may define the exponential
homomorphism

$\exp(tX):\mathcal{A}_{0}(V,M)\to\mathcal{A}_{0}(V,M)$

as the finite sum:

$\exp(tX)f:=\sum_{j}\frac{(tX)^{j}f}{j!},\qquad f\in\mathcal{A}_{0}(V,M).$

Now we may rewrite the formula (7) as the composition

$\varepsilon_{X_{m}}=\varepsilon_{m}\circ\exp(tX)$

where

$\varepsilon_{m}=\varepsilon_{0_{m}}:\mathcal{A}_{0}(V,M)\to\mathbb{C},\qquad\sum_{p}f_{p}t^{-p}\mapsto
f_{0}(m)$

is the evaluation map at $X_{m}=0$.

###  4.3. The scaling order

Let $E\to V$ be a vector bundle such that the restricted vector bundle
$F=E|_{M}\to M$ has a filtration $F^{1}\subset F^{2}\subset\cdots\subset
F^{q}=F$. We assume the bundle of endomorphisms $\textup{End}(E)$ has an
algebra filtration

$\textup{End}(E)=\textup{End}(E)^{0}\subset\textup{End}(E)^{1}\subset\cdots\subset\textup{End}(E)^{q}=\textup{End}(E),$
(8)

whose restriction to $M$ is compatible with the filtration on $F$ in the sense
that

$A:F^{\bullet}\to F^{\bullet+j},\qquad\text{for}\quad
A\in\textup{End}(E)^{j}.$ (9)

We consider a connection $\nabla$ on $E\to V$ with the following properties:

  * •

The induced connection $i^{*}\nabla$ is compatible with the filtration in the
sense that

$i^{*}\nabla:\Gamma(F^{j})\to\Gamma(F^{j})\otimes\Omega^{1}(M),\qquad
j=1\ldots q.$ (10)

  * •

The curvature $K$ of $\nabla$ has filtration order at most 2:

$K\in\Gamma(\textup{End}(E)^{2})\otimes\Omega^{2}(M).$ (11)

  * •

The induced connection $\nabla^{\textup{End}(E)}$ has filtration order 0:

$\nabla^{\textup{End}(E)}:\Gamma(\textup{End}(E)^{j})\to\Gamma(\textup{End}(E)^{j})\otimes\Omega^{1}(M).$
(12)

The space of sections $\Gamma(\textup{End}(E))$ is filtered by the algebra
filtration of $\textup{End}(E)$. We use the notation $o^{g}(\phi)$ for the
order of $\phi\in\Gamma(\textup{End}(E))$ in this filtration. By definition we
have

$o^{g}(\phi_{1}\phi_{2})\leq o^{g}(\phi_{1})+o^{g}(\phi_{2}).$ (13)

The filtration $F^{\bullet}$ induces a filtration on $\textup{End}(E|_{M})$,
which might differ from the filtration induced by (8). For a section
$\phi\in\Gamma(\textup{End}(E))$ we denote by $o^{f}(\phi)$ the order of this
filtration of the restriction $\phi|_{M}$. Then it follows from (9) that

$o^{f}(\phi)\leq o^{g}(\phi).$

Note, however, that the order function $o^{f}$ fails to satisfy the property
(13), in general.

######  Definition 4.5.

A differential operator $D$ acting on $\Gamma(E)$ is of _Getzler order_ at
most $p$ if locally it can be written as the sums of terms of the form

$\phi\nabla_{X_{1}}\cdots\nabla_{X_{l}}$

where $\phi\in\Gamma(\textup{End}(E))$ and
$X_{1},\cdots,X_{l}\in\mathfrak{X}(M)$ where $o^{g}(\phi)+l\leq p.$ We denote
the Getzler order of $D$ by $o^{g}(D).$

We had to use the filtration order $o^{g}$, rather than $o^{f}$, on
$\Gamma(\textup{End}(E))$ in the definition of Getzler order to obtain the
following:

######  Lemma 4.6.

For two differential operator $D_{1},D_{2}$ acting on $\Gamma(E)$ we have

$o^{g}(D_{1}D_{2})\leq o^{g}(D_{1})+o^{g}(D_{2}).$

∎

The space of sections $\Gamma(E)$ has natural filtration obtained by
restricting to the submanifold $M$. We denote the corresponding filtration
order by $o^{f}(\sigma)$ for $\sigma\in\Gamma(E)$ (if $\sigma=0$ we set
$o^{f}(\sigma)=-\infty$).

We need yet another filtration on $\Gamma(E)$, defined by the _scaling order_

$o^{sc}(\sigma)=\min_{D}\\{o^{g}(D)-o^{f}(D\sigma)\\}.$

######  Remark 4.7.

The definition of the scaling order might seem unnatural at first. But it is
modeled on the following formula for the vanishing order of a function:

For $f\in C^{\infty}(V)$, define $o^{{val}}(f)$ to be $0$ if $f|_{M}$ is not
the zero function, otherwise define $o^{val}(f)=-\infty.$ One easily sees,
that the vanishing order of $f$ along the submanifold $M$, $o^{van}(f)$, is
given by the equality

$o^{van}(f)=\min_{D}\\{o(D)-o^{val}(Df)\\}$

where $o(D)$ is the ordinary differential order and the minimum is taken over
all differential operators on $V$.

It follows from the definition of the scaling order that

######  Lemma 4.8.

For a differential operator $D$ acting on $\Gamma(E)$ with $o^{g}(D)\leq q$
and $\sigma\in\Gamma(E)$ we have

$o^{sc}(D\sigma)\geq o^{sc}(\sigma)-q.$

∎

###  4.4. The Taylor order

In this subsection, we obtain a formula for the scaling order in local
coordinates.

######  Definition 4.9.

Let $M$ be an embedded submanifold of a manifold $V$. A vector field
$\mathcal{R}\in\mathfrak{X}(V)$ is called Euler-like for the embedding $M\to
V$, if for every function $f\in C^{\infty}(V)$ vanishing to $p$-th order along
$M$

$\mathcal{R}f=pf+g$

where $g\in C^{\infty}(V)$ vanishes to order $p+1$ along $M$.

The Euler-like vector fields can be used to obtain a tubular neighborhood
embeddings:

######  Theorem 4.10.

[9, 20] There is a bijection between the germs of Euler-like vector fields and
germs of tubular neighborhood embeddings.

The correspondence given by this theorem can be described as follows: Given a
tubular neighborhood embedding we can choose local coordinates

$(x^{1},\cdots,x^{l},y^{1},\cdots,y^{k}):\mathcal{N}\to\mathbb{R}^{l}\times\mathbb{R}^{k}$
(14)

near a point of $M$, where $x_{i}$’s form a local coordinates on $M$ and
$y_{j}$’s are the linear coordinates for the fibers of the normal bundle.
Under the correspondence in Theorem 4.10, the associated Euler-like vector
field in the trivializing neighborhood is given by

$\mathcal{R}=\sum_{j}y^{j}\partial_{y^{j}}.$

Using the trivialization (14), we call a vector field _horizontal_ if it is
linear combination of coordinate vector fields $\partial_{x^{i}}$’s and
_vertical_ if it is linear combination of coordinate vector fields
$\partial_{y^{j}}$’s. In particular, the Euler-like vector field is a vertical
vector field.

######  Definition 4.11.

A section in $\sigma\in\Gamma(E)$ is called $\mathcal{R}$-synchronous if

$\nabla_{\mathcal{R}}\sigma=0.$

Using this concept, we define the Taylor expansion of a section
$\sigma\in\Gamma(E)$ in the trivializing neighborhood: The formal sum
$\sum_{I}\sigma_{I}y^{I}$ is a Taylor expansion for $\sigma$ if

  * •

$\sigma_{I}\in\Gamma(E)$ is an $\mathcal{R}$-synchronous section.

  * •

For every $N\geq 0$, the difference

$\sigma-\sum_{|I|<N}\sigma_{I}y^{I}$

vanishes to $N$-th order.

######  Definition 4.12.

For a section $\sigma\in\Gamma(E)$ with support in the trivializing
neighborhood, with Taylor expansion $\sum_{I}\sigma_{I}y^{I}$ we define the
_Taylor order_ by the formula

$o^{t}(\sigma):=\min_{|I|}\\{|I|-o^{f}(\sigma_{I})\\}.$

We will show that the Taylor order equals the scaling order. To prove this we
need the following lemma:

######  Lemma 4.13.

Let $\sigma\in\Gamma(E)$ be a $\mathcal{R}$-synchronous section.

  * •

If $Y$ is a vertical vector field, then we have the Taylor expansion

$\nabla_{Y}\sigma\simeq\sum_{|I|>0}\omega_{I}(\sigma)y^{I}$

where $\omega_{I}$’s are endomorphisms of the vector bundle $E$ of order at
most $2$.

  * •

If $X$ is a horizontal vector field, then we have the following Taylor
expansion

$\nabla_{X}\sigma\simeq\sigma_{0}+\sum_{|I|>0}\eta_{I}(\sigma)y^{I}$

where $\eta_{I}$’s are endomorphisms of the vector bundle $E$ of order at most
$2$ and $o^{f}(\sigma_{0})\leq o^{f}(\sigma)$.

###### Proof.

For the first bullet point, without loss of generality, we may assume
$Y=\partial_{i}$, where we have $[\mathcal{R},Y]=-Y$. Consider the equation

$\nabla_{\mathcal{R}}\nabla_{Y}\sigma-\nabla_{Y}\nabla_{\mathcal{R}}\sigma-\nabla_{[\mathcal{R},Y]}\sigma=K(\mathcal{R},Y)\sigma.$

Since $\nabla_{\mathcal{R}}\sigma=0$, we obtain

$\nabla_{\mathcal{R}}\nabla_{Y}\sigma+\nabla_{Y}\sigma=K(\mathcal{R},Y)\sigma.$
(15)

Let $\sum_{I}\sigma_{I}y^{I}$ be the Taylor expansion for $\nabla_{Y}\sigma$
and let

$\sum_{|I|>0}\omega_{I}y^{I}$

be Taylor expansion of the endomorphism $K(\mathcal{R},Y)$, where
$\omega_{I}$’s are endomorphisms of $E$ where
$\nabla_{\mathcal{R}}\omega_{I}=0$. Now by writing Taylor expansion of both
sides of (15), we obtain

$\sum_{I}(1+|I|)\sigma_{I}y^{I}=\sum_{|I|>0}\omega_{I}(\sigma)y^{I}$

from which the statement follows.

The argument for the second bullet point is quite similar, except for the
inequality $o^{f}(\sigma_{0})\leq o^{f}(\sigma)$, where it follows from the
compatibility of the connection with filtration of the restricted bundle
$E|_{M}$. ∎

######  Corollary 4.14.

If $\sigma\in\Gamma(E)$ is $\mathcal{R}$-synchronous section and $D$ a
differential operator of Getzler order $k$. Then for the Taylor expansion

$D\sigma\sim\sum_{I}y^{I}\sigma_{I}$

we have

$o^{f}(\sigma_{I})\leq o^{f}(\sigma)+|I|+k.$

So in particular,

$o^{f}(D\sigma)\leq o^{f}(\sigma)+k.$

###### Proof.

If the $D$ is an endomorphism of the bundle $E\to V$, the statement is
obvious. Consider a vector field $X$ on $V$. We will prove the statement for
$D=\nabla_{X}$, and by induction on the differential order, the full
generality follows.

We can write $X=X^{\textup{h}}+X^{\textup{v}}$ where $X^{\textup{h}}$ and
$X^{\textup{v}}$ are the horizontal and vertical components of $X$. Now from
Lemma 4.13 the corollary follows. ∎

Recall that the Taylor order was defined in Definition 4.12.

######  Theorem 4.15.

For a section $\sigma\in\Gamma(E)$ with support in the trivializing
neighborhood (14), we have

$o^{sc}(\sigma)=o^{t}(\sigma).$

###### Proof.

Choose $I=(\alpha_{1},\cdots,\alpha_{k})$ with smallest $|I|$ such that

$o^{t}(\sigma)=|\alpha|-o^{f}(\sigma_{I}),$

and define the differential operator
$D=\nabla^{\alpha_{1}}_{\partial_{y^{1}}}\cdots\nabla^{\alpha_{n}}_{\partial_{y^{n}}}$.
We claim that

$o^{f}(D\sigma)=o^{f}(\sigma_{\alpha}).$

For any $J\neq I$ with $|J|\geq|I|$ we have

$D(y^{J}\sigma_{J})|_{M}=0.$

If $|J|<|I|$, by the way $I$ is chosen, we have

$|J|-o^{f}(\sigma_{J})>|I|-o^{f}(\sigma_{I}).$

Also by corollary 4.14, we have

$o^{f}(Dy^{J}\sigma_{J})\leq o^{f}(\sigma_{J})+|I|-|J|$

and combining this with the previous inequality gives

$o^{f}(Dy^{J}\sigma_{J})<o^{f}(\sigma_{I})$

from which our claim follows. Therefore

$o^{sc}(\sigma)\leq o^{t}(\sigma).$

Now we prove the opposite inequality. From Corollary 4.14, we have

$o^{t}(D\sigma)\geq o^{t}(\sigma)-o^{f}(D).$

By definition we have

$o^{t}(D\sigma)\leq-o^{f}(D\sigma),$

and consequently

$o^{sc}(\sigma)\geq o^{t}(\sigma).$

∎

###  4.5. Rescaled module

We are ready to define a module over the algebra $\mathcal{A}(V,M)$ which will
be later shown to be a module of sections of a certain vector bundle – the
rescaled bundle – over the deformation space.

The space of Laurent polynomials $\Gamma(V,E)[t,t^{-1}]$ is a module over the
algebra $\mathcal{A}(V,M)$ and, hence, can be viewed as a sheaf over the
deformation space $\textnormal{DNC}(V,M)$. We are interested in the following
submodule of $\Gamma(V,E)[t,t^{-1}]$:

######  Definition 4.16.

The subspace $\mathcal{S}(E,\nabla)\subset\Gamma(V,E)[t,t^{-1}]$ of the space
of Laurent polynomials, consisting of the sections of the form

$s=\sum_{p}s_{p}t^{-p},\qquad\text{with}\quad o^{sc}(s_{p})\geq p,$

is called the rescaled module.

The rescaled module is also a sheaf over $\textnormal{DNC}(V,M)$. In order to
show that $\mathcal{S}(E,\nabla)$ is isomorphic to a subspace of a space of
section of a smooth bundle over $\textnormal{DNC}(V,M)$, we need to study the
restriction

$\mathcal{S}_{0}(E,\nabla):=\mathcal{S}(E,\nabla)/t\mathcal{S}(E,\nabla)$

of $\mathcal{S}(E,\nabla)$ to the zero fiber of $\textnormal{DNC}(V,M)$. We
denote by $\mathcal{S}_{0,m}(E,\nabla)$ the quotient of
$\mathcal{S}_{0}(E,\nabla)$ by the ideal of functions vanishing at $m\in M$.

Let
$\mathcal{P(N)}\subset\Gamma(\mathcal{N},\bigoplus_{p=1}^{q}F^{p}/F^{p-1})$
denote the space of section whose restriction to each fiber $\mathcal{N}_{m}$
of $\mathcal{N}$ is a vector-valued polynomial functions
$\mathcal{N}_{m}\to\bigoplus_{p=1}^{q}F_{m}^{p}/F_{m}^{p-1}$.

######  Lemma 4.17.

Let $I_{p}(E)\subset\Gamma(V,E)$ denote the submodule of sections with scaling
order at least $p$. There is a canonical isomorphism

$\mathcal{S}_{0}(E,\nabla)\buildrel\simeq\over{\longrightarrow}\bigoplus_{p}I_{p}(E)/I_{p+1}(E),\qquad\sum_{p}s_{p}t^{-p}\mapsto\sum_{p\geq
0}\langle s_{p}\rangle_{p}.$

Further, the map

$\sum_{p}s_{p}t^{-p}\mapsto\big{[}\,X_{m}\mapsto\sum_{p}\frac{1}{p!}\langle\nabla_{X}^{p}s_{p}|_{m}\rangle_{p}\,\big{]},$

where $\langle.\rangle_{p}$ denotes the corresponding class in the quotient
space $F_{m}^{p}/F_{m}^{p-1}$, defines an isomorphism between
$\mathcal{S}_{0,m}(E,\nabla)$ and the space of vector-valued polynomial
functions $\mathcal{P(N)}$.

∎

###  4.6. The construction of the rescaled bundle

We now give a rundown of how to obtain a smooth vector bundle for which a
“big” subspace of the space of sections is isomorphic to
$\mathcal{S}(E,\nabla)$.

As a set this bundle is given by the union

$\mathbb{E}=\Big{(}\bigoplus_{p=1}^{q}F^{p}/F^{p-1}\times\\{0\\}\Big{)}\bigsqcup_{t\neq
0}\big{(}E\times\\{t\\}\big{)}.$

To identify the elements of $\mathcal{S}(E,\nabla)$ with sections of
$\mathbb{E}$, we define for each element $\mu\in\textnormal{DNC}(V,M)$ an
evaluation map $\varepsilon_{\mu}:\mathcal{S}(E,\nabla)\to\mathbb{E}$. Then
the fiber $\mathbb{E}_{\mu}$ of $\mathbb{E}$ over $\mu$ is given by

$\mathbb{E}_{\mu}=\left\\{\begin{array}[]{ll}E_{v}&\mbox{if}\quad\mu=(v,t)\in
V\times\mathbb{R}^{\neq 0},\\\
\bigoplus_{p=1}^{q}F_{m}^{p}/F_{m}^{p-1}&\mbox{if}\quad\mu=(X_{m},0)\in\mathcal{N}\times\\{0\\}.\end{array}\right.$

For $(v,\lambda)$, $v\in V$, $\lambda\neq 0$, the evaluation map is simply

$\varepsilon_{(v,\lambda)}:\mathcal{S}(E,\nabla)\to E_{v}$
$\sum_{p}s_{p}t^{-p}\mapsto\sum_{p}s_{p}(v)\lambda^{-p}.$

This map will be the evaluation map on the nonzero fiber element
$(v,t)\in\textup{DNC}(V,M)$.

To define the evaluation over the zero fiber, we need a bit more work. First
note that for a vector field $X\in\mathfrak{X}(V)$ we have a well-define map

$t\nabla_{X}:\mathcal{S}(E,\nabla)\to\mathcal{S}(E,\nabla).$

This map descends to the quotient space $\mathcal{S}_{0}(E,\nabla)$ as a
locally nilpotent map:

$t\nabla_{X}:\mathcal{S}_{0}(E,\nabla)\to\mathcal{S}_{0}(E,\nabla),$

cf. Subsection 4.2. Hence we have a well-defined module homomorphism

$\exp(t\nabla_{X}):\mathcal{S}_{0}(E,\nabla)\to\mathcal{S}_{0}(E,\nabla).$

For $m\in M$ define the map

$\varepsilon_{m}:\mathcal{S}(E,\nabla)\to\bigoplus_{p=1}^{q}F_{m}^{p}/F_{m}^{p-1}$
$\sum_{p}s_{p}t^{-p}\mapsto\sum_{p\leq 0}\langle s_{p}(m)\rangle_{p}.$

And finally for $X_{m}\in\mathcal{N}$ define the map

$\varepsilon_{X_{m}}:\mathcal{S}_{0}(E,\nabla)\to\bigoplus_{p=1}^{q}F_{m}^{p}/F_{m}^{p-1},$
$\varepsilon_{X_{m}}s=\varepsilon_{m}\big{(}\exp(t\nabla_{X})s\big{)},$

where $X$ is an arbitrary extension of $X_{m}$. The map $\varepsilon_{X_{m}}$
is the evaluation map over the zero fiber.

###  4.7. The smooth structure on $\mathbb{E}$

We now define a smooth structure on $\mathbb{E}$ by constructing a locally
free sheaf on $\textnormal{DNC}(M,V)$ generated by $\mathcal{S}(E,\nabla)$.
This sheaf is the sheaf of smooth functions on $\mathbb{E}$.

######  Definition 4.18.

Define the sheaf $\mathcal{E}$ on $\textnormal{DNC}(V,M)$ that consists of
maps

$\textnormal{DNC}(V,M)\to\Pi_{\mu\in\textnormal{DNC}(V,M)}\mathbb{E}_{\mu},$
$\mu\mapsto\tau(\mu)$

such that for some $f_{i}\in C^{\infty}(\textnormal{DNC}(V,M))$ and
$s_{i}\in\mathcal{S}(E,\nabla)$

$\tau(\mu)=\sum_{i=1}^{N}f_{i}(\mu)\varepsilon_{\mu}({s}_{i})$

for every $\mu\in\textnormal{DNC}(V,M)$.

The elements of $\mathcal{E}$ are naturally identified with sections over
$\mathbb{E}$.

######  Theorem 4.19.

The sheaf $\mathcal{E}$ is a locally free sheaf of constant rank of modules
over algebra $C^{\infty}(\textnormal{DNC}(V,M))$. Its rank is equal to the
rank of the vector bundle $E\to V$. The evaluation map of the previous
subsections defines a smooth structure on $\mathbb{E}$ and identifies
$\mathcal{E}$ with the space of smooth section of $\mathbb{E}$. This gives
$\mathbb{E}$ thee structure of a smooth bundle over $\textnormal{DNC}(V,M)$.
Fiber-wise $\mathbb{E}$ is given as follows

${\pi^{*}\textup{gr}(F)}$${E}$${\mathcal{N}\times\\{0\\}}$${\bigsqcup_{t\neq
0}}$${V\times\\{t\\}.}$

###### Proof.

In some neighborhood of every $\mu\in\textnormal{DNC}(V,M)$, we show the sheaf
is locally free of constant rank. Fix a trivializing neighborhood (14) around
$m\in M$ consider the associated Euler-like vector field $\mathcal{R}$. Take a
local frame $\\{e_{i}\\}_{i=1}^{\textup{rank}(E)}$ for the restricted bundle
$E|_{M}\to M$ such that each $e_{i}$ gives a local homogeneous section of
degree $q_{i}=o^{f}(e_{i})$ of the associated graded bundle

$\textup{gr}(F):=\bigoplus_{p=1}^{q}F^{p}/F^{p-1}.$

There are a unique local $\mathcal{R}$-synchronous sections $\tilde{e}_{i}$ of
$E$, so that $\tilde{e}_{i}|_{M}=e_{i}$. We show the set
$\\{\tilde{e}_{i}t^{q_{i}}\\}_{i=1}^{\textup{rank}(E)}$ generate the sheaf
locally. Clearly in a neighborhood $\mu=(m,t)\in\textnormal{DNC}(V,M)$ these
sections generate the sheaf. For $\mu=X_{m}\in\mathcal{N}_{m}$, we have

$\displaystyle\varepsilon_{X_{m}}(\tilde{e}_{i}t^{q_{i}})$
$\displaystyle=\varepsilon_{m}\exp(t\nabla_{X})\tilde{e}_{i}t^{q_{i}}$
$\displaystyle=\varepsilon_{m}\sum_{k}\frac{t^{q_{i}+k}}{k!}\nabla_{X}^{k}\tilde{e}_{i}$
$\displaystyle=\langle e_{i}\rangle_{q_{i}}+\textup{higher degree terms}.$

Hence the sheaf is locally free of constant rank equal $\textup{rank}(E)$.

Now it remains to show the same for $\mu=(v,t)\in\textnormal{DNC}(V,M)$, with
$v\notin M$. We may take a a family of sections $\\{s_{i}\\}_{i}$ of $E$
forming a frame near $v$ and with supports away from $M$. In this case, the
sections $s_{i}$ have scaling order $-\infty.$ So the set $\\{s_{i}\\}_{i}$
generates the sheaf near $(v,t)$. ∎

######  Definition 4.20.

The smooth vector bundle $\mathbb{E}\to\textnormal{DNC}(V,M)$ is called the
_rescaled bundle_.

In next subsection, we give examples of rescaled bundles.

###  4.8. Examples of rescaled bundles

In the absence of any filtration, one obtains a rescaled bundle over a the
deformation to the normal cone:

######  Example 4.21.

Consider an embedding $M\hookrightarrow V$. For every vector bundle $E\to V$
with trivial filtrations on $E|_{M}$ and $\textup{End}(E)$ and every
connection $\nabla$ on $E\to V$ we obtain a rescaled bundle

$\mathbb{E}\to\textup{DNC}(V,M).$

One can show that this bundle is independent of the choice of the connection
$\nabla$. Indeed the rescaled bundle $\mathbb{E}$ is isomorphic to the
pullback bundle $\pi_{V}^{*}E$ where $\pi_{V}:\textnormal{DNC}(V,M)\to V$ is
the canonical projection.

One interesting case, is when one considers a Riemannian manifold $V$ with the
exterior bundle $E=\Lambda^{*}T^{*}V\to V$ with the induced Levi-Civita
connection. Let $f:V\to\mathbb{R}$ be a Morse-Bott function, with the critical
submanifold $M\hookrightarrow V$. Now considering trivial filtration on
$E|_{M}$ and trivial filtration on E, one obtains a rescaled bundle rescaled
bundle $\mathbb{E}\to\textup{DNC}(V,M)$. One can show (this will be done in a
subsequent paper) the “rescaled Witten deformation” of the de Rham-Dirac
operator

$t\,\big{(}\,e^{\frac{1}{t^{2}}f}de^{-\frac{1}{t^{2}}f}+e^{-\frac{1}{t^{2}}f}d^{*}e^{\frac{1}{t^{2}}f}\,\big{)}=t\,\big{(}\,d+d^{*}+\frac{1}{t^{2}}c(df)\,\big{)}$

extends to a smooth differential operator acting on the section of the
rescaled bundle $\Gamma(\mathbb{E}).$ One can also generalize it to concede
the Novikov deformation, defined by a closed differential form $\omega$:

$t\,\big{(}\,d+d^{*}+\frac{1}{t^{2}}c(\omega)\,\big{)},$

cf. [17, 5, 8]. One can also get an equivariant version of this construction,
[6, 7].

The spinor rescaled bundle of Higson and Yi, [13], was the main inspiration of
this paper. Their setup is briefly explained in the next example:

######  Example 4.22.

Assume $V=M\times M$ and and $M\hookrightarrow M\times M$ is the diagonal
embedding. If $\mathscr{E}\to M$ is a Clifford module with a Clifford
connection $\nabla$, then $E=\mathscr{E}\boxtimes\mathscr{E}^{*}$ carries the
connection $\nabla^{E}=\nabla\boxtimes 1+1\boxtimes\nabla$. Note that $E|_{M}$
is isomorphic to

$\mathbb{C}\textup{l}(TM)\otimes\textup{End}_{\textup{Cl}}(\mathscr{E})$ (16)

and it carries the Clifford filtration which is compatible with the connection
$\nabla^{E}$. Thus we are in situation of the Subsection 4.3, where the
filtration is given by the Clifford filtration on the first factor of (16)
(see Section 5 for more details on different filtrations in this case).

The curvature of this connection satisfies

$K^{E}=c\circ q\circ\gamma(R)+{F}^{\mathscr{E}/S}\boxtimes
1+1\boxtimes{F}^{\mathscr{E}^{*}/S}.$

Here $R$ is the curvature of the Levi-Civita connection,
$\gamma:\mathfrak{so}(\mathscr{E})\xrightarrow[]{\simeq}\Lambda^{2}\mathscr{E}$
is the canonical isomorphism [16, Section 2.2.10],
$q:\Lambda^{*}TM\to\textup{Cl}(TM)$ is the quantization [2, Proposition 3.5]
map and $c:\mathbb{C}\textup{l}(TM)\to\textup{End}(\mathscr{E})$ is the
Clifford action. Thus the first summand in the above formula has Clifford
filtration order 2, and, hence, when viewed as a differential operator has
Getzler filtration order 2. Also ${F}^{\mathscr{E}/S}$ and
${F}^{\mathscr{E}^{*}/S}$ are twisting curvatures of the Clifford module
$\mathscr{E}$ and $\mathscr{E}^{*}$, respectively, cf. [2, Prop. 3.43]. Recall
that the twisting curvatures they commute with Clifford actions on
$\mathscr{E}$ and $\mathscr{E}^{*}$. So, as a differential operator of order
zero it has Getzler filtration order 0. We conclude that $K^{E}$ has Getzler
order $2$. Therefore we obtain the rescaled bundle

$\mathbb{E}\to\textnormal{DNC}(M\times M,M)=\mathbb{T}M.$

This rescaled bundle recovers the Getzler calculus [13].

Another example of the rescaled bundle arises in the context of the
equivariant index formula which will appear in a joint paper of the second
author with Yiannis Loizides, Jesus Sanchez and Shiqi Liu:

######  Example 4.23.

Consider a compact Lie group $G$ acting isometrically on $M$. For an element
$g\in G$, denote by $M^{g}$ the fixed submanifold under the action of $g$.
Associated to the diagonal embedding $M^{g}\hookrightarrow M\times M$, the
deformation space $\textnormal{DNC}(M\times M,M^{g})$ is called the _relative
tangent groupoid_ and denoted by $\mathbb{T}_{g}M$.

Assume $\mathscr{E}\to M$ is a $G$-equivariant Clifford module that carries a
$G$-invariant connection $\nabla$. The bundle
$E=\mathscr{E}\boxtimes\mathscr{E}^{*}\to M\times M$ carries the induce
connection $\nabla^{E}$, Using similar argument as in Example 4.22, we obtain
a rescaled bundle

$\mathbb{E}_{g}\to\mathbb{T}_{g}M.$

Using this vector bundle one may recover the equivariant index formula.

The next section dedicated to a fourth example of the rescaled bundle, from
which we recover the Kirillov formula for the equivariant index.

##  5\. Equivariant Rescaled Bundles

Let $G$ be a compact Lie group acting on an oriented smooth manifold $M$ by
orientation-preserving isometries. Consider a $\mathbb{Z}/2$-graded Clifford
module $\mathscr{E}\to M$ which carries an even action of $G$. Let
$\nabla^{\mathscr{E}}$ be a $G$-equivariant connection on $\mathscr{E}$.

Consider the the vector bundle
$E=\mathscr{E}\boxtimes\mathscr{E}^{*}\otimes\mathbb{C}[\mathfrak{g}]_{(J)}$
with the connection

$\nabla:=\nabla^{\mathscr{E}}\boxtimes 1\otimes
1+1\boxtimes\nabla^{\mathscr{E}^{*}}\otimes 1+1\boxtimes 1\otimes d$

where $d$ is the trivial connection on the trivial bundle
$\mathbb{C}[\mathfrak{g}]_{(J)}\times M\to M.$ The restricted bundle
$F=E|_{M}$ is isomorphic to

$F\simeq\mathbb{C}\textup{l}(TM)\otimes\textup{End}_{\textup{Cl}}(\mathscr{E})\otimes\mathbb{C}[\mathfrak{g}]_{(J)}.$

It carries the filtration $F^{1}\subset F^{2}\subset\cdots\subset F^{q}$
defined by

$F^{p}:=\bigcup_{r\leq
p/2}\mathbb{C}\textup{l}^{p-2r}(TM)\otimes\textup{End}_{\textup{Cl}}(\mathscr{E})\otimes\mathbb{C}[\mathfrak{g}]^{r}_{(J)}.$

The reason for this choice of filtration is that its associated graded space
is isomorphic to

$\textup{gr}(F)\simeq\Lambda(T^{*}M)\otimes\textup{End}_{\textup{Cl}}(\mathscr{E})\otimes\mathbb{C}[\mathfrak{g}]_{(J)}$

with the grading as in (1).

The connection $\nabla$ clearly satisfies the conditions (10) and (11). Hence,
by Theorem 4.19, we obtain a rescaled bundle

$\mathbb{E}_{J}\to\mathbb{T}M.$

###  5.1. Operators on the rescaled bundles

Let $D$ be a differential operator of Getzler order $p$ (cf. Definition 4.5)
acting on $E\to M\times M$. Then we have the induced map

$t^{p}D:\mathcal{S}(E,\nabla)\to\mathcal{S}(E,\nabla)$

on the rescaled modules which gives smooth differential operator on the
rescaled bundle:

$\bm{D}\curvearrowright\Gamma(\mathbb{E}_{J}).$

######  Definition 5.1.

We define the Getzler symbol of $D$, denoted by $\sigma^{g}(D)$, as the
restriction of the operator $\bm{D}$ to the $t=0$ fiber of the tangent
groupoid:

$\sigma^{g}(D)\curvearrowright\Gamma(TM,\pi^{*}\Lambda(T^{*}M)\otimes\textup{End}_{\textup{Cl}}(\mathscr{E})\otimes\mathbb{C}[\mathfrak{g}]_{(J)})$

where $\pi:TM\to M$.

######  Example 5.2.

  * •

For a vector field $\xi\in\mathfrak{X}(M)$, consider the differential operator
$D=\nabla^{\mathscr{E}}_{\xi}$ acting on $\Gamma(M\times M,E)$ by
differentiating along the first component of $M\times M.$ This operator has
Getzler order $1$.

  * •

For a vector field $\xi\in\mathfrak{X}(M)$, consider the operator $D=c(\xi)$
acting on $\Gamma(E)$ acting by left Clifford multiplication. This operator is
of Getzler order $1$.

  * •

For a polynomial $p(X)\in\mathbb{C}[\mathfrak{g}]$ of degree $k$, the
differential operator $D=p(X)\curvearrowright\Gamma({E})$ defined by
multiplication by $p(x)$, has Getzler order $2k.$

###  5.2. A calculation of Getzler symbols of some operators

We now calculate the symbols of these operators. We need to fix some
conventions for the curvatures involved. There is an Lie algebra isomorphism

$\gamma:\mathfrak{so}(\mathscr{E})\xrightarrow[]{\simeq}\Lambda^{2}\mathscr{E}$
$T\mapsto\frac{1}{4}\sum_{i}T(e_{i})\wedge e_{i},$

where the formula is given with respect to an orthonormal basis $\\{e_{i}\\}$
(see [16, Section 2.2.10]). The curvature of the bundle $\mathscr{E}$
satisfies

$K^{\mathscr{E}}=c\circ q\circ\gamma(R)+{F}^{\mathscr{E}/S}$

where $q:\Lambda^{*}TM\to\textup{Cl}(TM)$ is the quantization (see [2,
Proposition 3.5]) map and
$c:\mathbb{C}\textup{l}(TM)\to\textup{End}(\mathscr{E})$ is the Clifford
action. We will use the notation

$\mathsf{K}=\gamma(R)+\mathsf{F}^{\mathscr{E}/S}\in\mathcal{A}^{2}(M,\Lambda^{2}\mathscr{E}),$

where $\mathsf{F}^{\mathscr{E}/S}:=q^{-1}\circ c^{-1}({F}^{\mathscr{E}/S})$.

######  Proposition 5.3.

For $\xi\in\mathfrak{X}(M)$ and $p(X)\in\mathbb{C}_{J}$ of order $k\leq J$ we
have

  * •

the symbol $\sigma^{g}(\nabla^{\mathscr{E}}_{\xi})$ is given by

$s\mapsto\Big{[}\eta\mapsto\partial_{\xi}s(\eta)+\frac{1}{2}\mathsf{K}(\eta,\xi)\wedge
s(\eta)\Big{]},$

  * •

the symbol $\sigma^{g}(c({\xi}))$ is given by

$s\mapsto\Big{[}\eta\mapsto\xi\wedge s(\eta)\Big{]},$

  * •

the symbol $\sigma^{g}(p(X))$ is given by

$s\mapsto\Big{[}\eta\mapsto p(X)\cdot s(\eta)\Big{]}.$

Here these symbols are considered as operators acting on
$\Gamma(T_{m}M,\Lambda
T^{*}_{m}M\otimes\textup{End}_{\textup{Cl}}(\mathscr{E}_{m})\otimes\mathbb{C}[\mathfrak{g}]_{(J)})$
for $m\in M.$

###### Proof.

The first bullet point is quite verbatim as the proof of [13, Lemma 3.6.3].
The second bullet point has the same proof as [13, Lemma 3.6.2].

To see the third bullet point, note that the operators
${\nabla^{\mathscr{E}}_{Y}}$ and ${p(X)}$ commute, therefore

$\exp(t{\nabla^{\mathscr{E}}_{Y}}){t^{2k}p(X)}=t^{2k}{p(X)}\exp(t{\nabla^{\mathscr{E}}_{Y}})$

and hence

$\varepsilon_{Y_{m}}\Big{(}{t^{2k}p(X)}s\Big{)}=t^{2k}p(X)\varepsilon_{(X_{m},0)}(s)$

from which the statement follows.

∎

###  5.3. A conjugate connection

By Proposition 3.4, up to lower order terms the Bismut Laplacian $H(X)$ equals
$\Big{(}\nabla^{\mathscr{E},\frac{1}{4}X}\Big{)}^{*}\nabla^{\mathscr{E},\frac{1}{4}X}$.
Thus it is related to the connection $\nabla^{\mathscr{E},\frac{1}{4}X}$. But
the rescaled bundle was defined using the connection

$\nabla^{\mathscr{E}}=\nabla^{\mathscr{E},\frac{1}{4}X}-\frac{1}{4}\theta_{X}.$

In particular, the computations of Proposition 5.3 help to compute the symbol
of operators expressed in terms of the connection $\nabla^{\mathscr{E}}$. Thus
it is convenient to express $H(X)$ in terms of this connection. In other
words, we want to express it in coordinate system obtained by local
trivializing $\mathscr{E}$ using the parallel transport of
$\nabla^{\mathscr{E}}$ along geodesics (while naturally it is expressed in
trivialization obtained by parallel transport along
$\nabla^{\mathscr{E},\frac{1}{4}X}$). One way of doing it is by conjugating
$H(X)$ with the bundle map $\rho:\mathscr{E}\to\mathscr{E}$ which intertwines
those two trivializations. This bundle map is given by the parallel transport
along geodesics of the trivial line bundle with respect to the connection
$d-\frac{1}{4}\theta_{X}$. This parallel transform formula is given as
$\rho=e^{\alpha}$ where $\alpha:M\times M\to\mathfrak{g}^{*}$ is a smooth map
on $M\times M$ with values in $\mathfrak{g}^{*}.$

Specifically, consider a smooth function $\alpha:M\times
M\to\mathfrak{g}^{*}\subset\mathbb{C}[\mathfrak{g}]$ defined on the
neighborhood of the diagonal $\Delta M\hookrightarrow M\times M$ by the
formula

$\alpha_{X}(\exp_{m}(\xi),m)=-\frac{1}{4}\int_{0}^{1}(\iota(\mathcal{R})\theta_{X})\big{(}\exp(t\xi)\big{)}t^{-1}dt,$

where $\mathcal{R}$ is the local Euler-like vector field on $M\times M$
defined by

$\mathcal{R}_{(\exp_{m}(\xi),m)}=\frac{d}{dt}|_{t=1}(\exp_{m}(t\xi),m).$ (17)

Away from a neighborhood of the diagonal, we can extend the function by zero
using a cut-off function. We have
$\mathcal{R}.\alpha_{X}(\exp_{m}(\xi),m)=-\frac{1}{4}\theta_{X}(\mathcal{R}_{(\exp_{m}(\xi),m)})$.
Consider

$\omega_{X}:=-d_{s}\alpha_{X}-\frac{1}{4}\theta_{X}\in\Gamma\Big{(}\Lambda^{1}T_{s}^{*}(M\times
M)\Big{)}\otimes\mathfrak{g}^{*}$

that is a sourcewise one-form and depends on $X$ linearly, and $d_{s}$ is the
sourcewise de Rham differential. We have the equality

  * •

$\omega_{X}|_{\Delta M}=0$ and

  * •

in a neighborhood of the diagonal

$\iota(\mathcal{R})\omega_{X}=0.$ (18)

From (18), we obtain

$\mathscr{L}(\mathcal{R})\omega_{X}=-\frac{1}{4}\iota(\mathcal{R})d\theta_{X}$

which means the first order terms of Taylor expansion of $\omega$ is the same
as $-\frac{1}{4}\iota(\mathcal{R})d\theta_{X}.$

######  Lemma 5.4.

For $\xi,\eta\in\mathfrak{X}(M)$ we have

$d\theta_{X}(\xi,\eta)=-2(\mu^{M}(X)\xi,\eta).$

So in particular, $-\frac{1}{4}\iota(\mathcal{R})d\theta_{X}$ is the one-form
dual to the vector field $\frac{1}{2}\mu(X)\mathcal{R}.$ ∎

So we have

$\omega_{X}=\frac{1}{4}\sum_{j}(\mu(X)\mathcal{R},\partial_{j})dx^{j}+\mathcal{O}(2)$
(19)

where $\mathcal{O}(2)$ is a linear in $X$.

######  Lemma 5.5.

For $\eta\in\mathfrak{X}(M)$ and $\sigma\in\Gamma(M\times
M,\mathscr{E}\boxtimes\mathscr{E}^{*})\otimes\mathbb{C}[\mathfrak{g}]_{(J)},$we
have

$o^{\textup{sc}}(\omega_{X}(\eta)\sigma)\geq o^{\textup{sc}}(\sigma)-1.$

###### Proof.

Since $\omega_{X}$ vanishes along the the diagonal $\Delta M\hookrightarrow
M\times M$, the scaling order increase by at least $1$ and since it is a
linear in $X$ it increase the polynomial order by at most $1$. So the
equivariant scaling (scaling order minus two times the polynomial order) order
decreases by at most $1$. ∎

Therefore $t\omega_{X}(\eta)$ acts on the rescaled bundle and it gives
corresponding operators on the quotient module and the rescaled bundle:

$\bm{\omega_{X}(\eta)}:\mathcal{{S}}_{J,0}(E,\nabla)\to\mathcal{S}_{J,0}(E,\nabla)$

and

$\bm{\omega_{X}(\eta)}:\Gamma(\mathbb{T}M,\mathbb{E}_{J})\to\Gamma(\mathbb{T}M,\mathbb{E}_{J}).$

######  Lemma 5.6.

As an operator on $\mathcal{S}_{J,0}(E)$, we have

$\bm{\omega_{X}(\eta)}\sigma=\frac{1}{4}(\mu(X)\mathcal{R},\eta)\cdot\sigma$

###### Proof.

This follows immediately from (19). ∎

Consider the map

$\rho:M\times M\to C^{\infty}(\mathfrak{g})$ (20)

$\rho_{X}(m_{1},m_{2})=e^{\alpha_{X}(m_{1},m_{2})}.$

Define the conjugate connection

$\rho_{X}\nabla^{\mathscr{E},X}\rho_{X}^{-1}$

and denote it by

$\widehat{\nabla}^{X}.$

Note that $\widehat{\nabla}^{X}=\nabla^{\mathscr{E}}+\omega_{X}$. By Lemma
5.6, the scaled operator $t\widehat{\nabla}^{X}_{\eta}$ acts on the rescaled
module and hence on the rescaled bundle. From Proposition 5.3 it follows:

######  Proposition 5.7.

The Getzler’s symbol of $\widehat{\nabla}^{X}_{\xi}$ is given for $m\in M$ by

$\sigma^{g}(\widehat{\nabla}^{X}_{\xi})\curvearrowright\Gamma(T_{m}M,\Lambda
T^{*}_{m}M\otimes\textup{End}_{\textup{Cl}}(\mathscr{E}_{m})\otimes\mathbb{C}[\mathfrak{g}]_{(J)})$
$s\mapsto\Big{[}\eta\mapsto\partial_{\xi}s(\eta)+\frac{1}{2}\mathsf{K}(\eta,\xi)\wedge
s(\mu)+\frac{1}{4}(\mu(X)\eta,\xi)s(\xi)\Big{]}.$

∎

Denote by $\widehat{H}(X)$ the conjugate operator $\rho_{X}H(X)\rho_{X}^{-1}$.
From Propositions 3.4 and 5.7 it follows that the Getzler symbol of the
operator $\widehat{H}(X)$, in a local orthonormal frame $\\{e_{i}\\}_{i}$, is
given by the formula

$\sigma^{g}(\widehat{H}(X))=-\sum_{i}\Big{(}\partial_{i}-\frac{1}{4}\sum_{j}\big{(}R_{ij}+\mu^{M}_{ij}(X)\big{)}x^{j}\Big{)}^{2}+\mathsf{F}^{\mathscr{E}/S}+\mu^{\mathscr{E}/S}(X).$
(21)

where $R_{ij}=(R\partial_{i},\partial_{j})$. We will use the notation
$\widehat{\mathcal{H}}(X):=\sigma^{g}(\widehat{H}(X))$ for the symbol. So in
particular, the family of operators $t^{2}\widehat{H}(X)$ acting on the
rescaled bundle $\Gamma(\mathbb{E}_{J})$ over the $t$-fibers of the tangent
groupoid along with the operator $\widehat{\mathcal{H}}(X)$ acting on the
rescaled bundle over the $0$-fibers gives a smooth differential operator
acting on $\Gamma(\mathbb{E}_{J})$.

The operator (21) is a _generalized harmonic oscillator operator_ (cf. [2,
Definition 4.11]) and the Mehler kernel associated to the heat operator
$e^{-\tau\widehat{\mathcal{H}}(X)}$, $\tau>0$, is given by the formula

${K}_{\tau}^{\widehat{\mathcal{H}}}(X,\xi)=(4\pi\tau)^{-n/2}\textup{det}^{1/2}\Big{(}\frac{\tau(R+\mu^{M}(X))/2}{\sinh\tau(R+\mu^{M}(X))/2}\Big{)}\exp(-\tau(\mathsf{F}^{\mathscr{E}/S}+\mu^{\mathscr{E}/S}(X)))$
$\exp(-\frac{1}{4\tau}\langle\xi|\tau(R+\mu^{M}(X))/2\coth\tau(R+\mu^{M}(X))/2|\xi\rangle).$

which can be written in term of equivariant curvature terms as

$(4\pi\tau)^{-n/2}\textup{det}^{1/2}\Big{(}\frac{\tau
R_{\mathfrak{g}}(X)/2}{\sinh\tau
R_{\mathfrak{g}}(X)/2}\Big{)}\exp(-\frac{1}{4\tau}\langle\xi|\tau
R_{\mathfrak{g}}/2\coth\tau
R_{\mathfrak{g}}/2|\xi\rangle)\exp(-\mathsf{F}^{\mathscr{E}/S}_{\mathfrak{g}}(X)).$
(22)

###  5.4. The heat kernel asymptotics

We denote by $K_{\tau}(x,y,X)\in\Gamma(M\times
M,\mathscr{E}\boxtimes\mathscr{E}^{*})$ the heat kernel of
$e^{-\tau\widehat{H}(X)}$ where $\widehat{H}(X)$ is the conjugate operator
$\rho_{X}H(X)\rho_{X}^{-1}$ and $H(X)=H_{\frac{1}{4}}(X)$ is the Bismut
Laplacian (5).

######  Lemma 5.8.

[18, Theorem 7.15] Fix $X\in\mathfrak{g}$. For $J\geq 0$, there exists an
asymptotic expansion

$K_{\tau}(x,y,X)\sim(4\pi
t)^{-n/2}\exp\Big{(}-\frac{d^{2}(x,y)}{4t}\Big{)}\,\sum_{j\geq
0}\tau^{j}\Phi_{j,J}(x,y,X)$

where $\Phi_{j,J}(x,y,X)\in\Gamma(M\times
M,\mathscr{E}\boxtimes\mathscr{E}^{*})$. In a neighborhood of the diagonal, we
have the recurrence relations (cf. (7.17) in [18])

$(\nabla_{\mathcal{R}}+j+\nabla_{\mathcal{R}}\log
g^{1/4})\Phi_{j,J}=-\widehat{H}(X)\Phi_{j-1,J}$ (23)

where $\mathcal{R}$ is the Euler-like vector filed (17),
$\Phi_{0,J}(x,y,X):\mathscr{E}_{y}\to\mathscr{E}_{x}$ is the parallel
transport along the geodesic between $x$ and $y$ with respect to the
connection $\nabla^{\mathscr{E}}$. Furthermore,

$g=\textup{det}(g_{ij})$

where $g_{ij}(x,y)$ are functions definedin a neighborhood of the diagonal on
$M\times M$ whose restriction to the source fibers $M\times\\{y\\}$ are the
components of the metric in the normal coordinate at $y$. ∎

######  Remark 5.9.

From the recurrence relations (23), it follows that the coefficients
$\Phi_{j,J}(x,y,X)$ are polynomials in variable $X$. Hence we have

$\Phi_{j,J}(x,y,X)\in\Gamma(M\times
M,\mathscr{E}\boxtimes\mathscr{E}^{*}\otimes\mathbb{C}[\mathfrak{g}]_{(J)}).$

For every $N\geq 0$ we will denote by

$K_{\tau}^{N}:=(4\pi\tau)^{-n/2}e^{-\frac{d^{2}(x,y)}{4\tau}}\sum^{N}_{j\geq
0}\tau^{j}\Phi_{j,J}(x,y,X)$

the approximate heat kernel. Consider the rescaled heat kernel
$\sigma_{\tau}^{N}(x,y,X):=t^{n}\psi(d(x,y))K_{t^{2}\tau}^{N}$, where
$\psi:\mathbb{R}\to\mathbb{R}^{\geq 0}$ is the cut-off functions such that
$\psi|_{[-r/2,r/2]}=1$ and $\psi$ vanishes outside $[-r,r]$ where $r$ is the
injectivity radius of $M$.

######  Proposition 5.10.

For every $j\geq 0$, we have

$o^{\textup{sc}}(\Phi_{j,J})\geq-2j.$

###### Proof.

We use induction on $j$. Note that for $j=0$ the case is clear by Remark 5.9
and Theorem 4.15, since $\Phi_{0,J}$ is the parallel transport map and hence
synchronous. We show for $j>0$

$o^{\textup{sc}}(\Phi_{j,J})=o^{\textup{sc}}\big{(}(\nabla_{\mathcal{R}}+j+\nabla_{\mathcal{R}}\log
g^{1/4})\Phi_{j,J}\big{)}$ (24)

from which, we then deduce

$o^{\textup{sc}}(\Phi_{j,J})\geq
o^{\textup{sc}}(-\widehat{H}(X)\Phi_{j-1,J})\geq-2(j-1)-2=-2j.$

So we just need to prove (24). Using Theorem 4.15, we only need to show the
equality in a trivializing neighborhood associated with the Euler-like vector
field $\mathcal{R}$. So if the Taylor expansion of $\Phi_{i,J}$ is given by

$\Phi_{j,J}\sim\sum_{I}y^{I}\sigma_{I},$

then we have the Taylor expansion

$(\nabla_{\mathcal{R}}+j)\Phi_{j,J}\sim\sum_{I}(|I|+j)y^{I}\sigma_{I}$

which has the same scaling order (=Taylor order) as $\Phi_{j,J}.$ the Euler-
like vector field $\mathcal{R}$ vanishes along the diagonal, hence so does
$\nabla_{\mathcal{R}}\log g^{1/4}$. Thus this term has no effect on the
scaling order and therefore the equality (24) follows.

∎

Note that for $\tau>0$ the we may consider
$\psi(d(x,y))e^{-\frac{d^{2}(x,y)}{4\tau t^{2}}}$ as a smooth function on
$\mathbb{T}M.$ By Proposition 5.10, we have

$\sum^{N}_{j\geq 0}\tau^{j}t^{2j}\Phi_{j,J}\in\mathcal{S}_{J}(E,\nabla)$

and hence $\sigma_{\tau}^{N}(x,y,X)\in\Gamma(\mathbb{T}M,\mathbb{E}_{J}).$

The following lemma is a generalization of Lemma 2.8 in [15]. Consider a
section $s\in\Gamma(M\times
M\times\mathbb{R},\mathscr{E}\boxtimes\mathscr{E}^{*})$ and the natural
projection $\pi:\mathbb{T}M\to M\times M\times\mathbb{R}$.

######  Lemma 5.11.

For $N\geq n$, the section $t^{N+1}s$, after composing with
$\pi:\mathbb{T}M\to M\times M\times\mathbb{R},$ may be considered as a smooth
section of the rescaled bundle $\mathbb{E}_{J}\to\mathbb{T}M$ that vanishes to
order $2N-n+1$ on the zero fiber of the tangent groupoid.

###### Proof.

Since the function $t:\mathbb{T}M\to\mathbb{R}$ be the canonical projection.
This projection as a function vanishes on the zero fiber. Since we may write

$t^{2N+1}s=t^{2N-n}\cdot t^{n+1}s,$

it is enough to show that $t^{n}s$ gives a smooth section of the rescaled
bundle that vanishes along the zero fiber. Note that we may find sections
$\sigma_{1},\cdots,\sigma_{p}\in\Gamma(M\times
M,\mathscr{E}\boxtimes\mathscr{E}^{*})$ and functions $f_{1},\cdots,f_{p}\in
C^{\infty}(M\times M\times\mathbb{R},\mathscr{E}\boxtimes\mathscr{E}^{*})$
such that

$s=\sum_{j=1}^{p}f_{j}(x,y,t)\sigma_{j}(x,y).$

Since $t^{n+1}\sigma_{j}$ is an element of the rescaled module that gives zero
values on the zero fiber of the tangent groupoid and $f_{i}\circ\pi\in
C^{\infty}(\mathbb{T}M)$, we obtain an smooth section of the rescaled bundle
of the form

$\sum_{j=j}^{p}f_{i}\circ\pi\cdot t^{n+1}\sigma_{j}$

that vanishes along the zero fiber of $\mathbb{T}M$. ∎

From Lemma 5.8, we have

$K_{\tau t^{2}}-\sigma_{\tau}^{N}=\mathcal{O}(t^{2N+1})$

where $\mathcal{O}(t^{2N+1})\in\Gamma(M\times
M\times\mathbb{R},\mathscr{E}\boxtimes\mathscr{E}^{*})$ and if $N>n/2$, by
Lemma 5.11, it gives a smooth section of the rescaled bundle vanishing over
the zero fiber of the tangent groupoid. In particular, we have

######  Theorem 5.12.

The rescaled heat kernels $\bm{K}_{\tau}:=K_{\tau t^{2}}$ gives a smooth
section of the rescaled bundle $\mathbb{E}_{J}\to\mathbb{T}M$ which over the
zero fiber of the tangent groupoid agrees with values of sections
$\sigma_{\tau}^{N}$ for $N>n/2.$ ∎

The section $\bm{K}_{\tau}\in\Gamma(\mathbb{T}M,\mathbb{E}_{J})$ satisfies the
heat equation

$(\partial_{\tau}+\bm{\widehat{H}}(X))\bm{K}_{\tau}=0$

where $\bm{\widehat{H}}(X)$ is the smooth extension of the operator
$t^{2}\widehat{H}(X)$ to zero fiber of the tangent groupoid. Note that on zero
fiber of the tangent groupoid, $\bm{\widehat{H}}(X)$ acts as the harmonic
oscillator operator $\widehat{\mathcal{H}}(X)$ given in (21). Therefore we
have

$(\partial_{\tau}+\widehat{\mathcal{H}}(X))(\bm{K_{\tau}|_{t=0}})=0$

The uniqueness property of Mehler kernel then proves

$\bm{K}|_{t=0}={K}_{\tau}^{\widehat{\mathcal{H}}}$ (25)

where ${K}_{\tau}^{\widehat{\mathcal{H}}}$ is as in (22).

###  5.5. Supertraces on the rescaled bundle

Let $M\times\mathbb{R}\hookrightarrow\mathbb{T}M$ denote the embedding of the
unit space of the tangent groupoid.

######  Lemma 5.13.

The following map is well-defined

$\mathfrak{s}:\Gamma(\mathbb{T}M,\mathbb{E}_{J})\to
C^{\infty}(M\times\mathbb{R},\mathbb{C}[\mathfrak{g}]_{(J)})$

$\sigma\mapsto\left\\{\begin{array}[]{ll}(m,t)\mapsto
t^{-n}\textnormal{str}{(\sigma(m,m,t))}&t\neq 0\\\ (m,0)\mapsto\
\textnormal{str}{(\sigma(0_{m},0))}&t=0\end{array}\right.$

Here, $C^{\infty}(M\times\mathbb{R},\mathbb{C}[\mathfrak{g}]_{(J)})$ denotes
the $\mathbb{C}[\mathfrak{g}]_{(J)}$-valued smooth functions on
$M\times\mathbb{R}$. The supertrace $\textup{str}(\sigma(m,m,t))$ is induced
by the supertrace on $\mathscr{E}\otimes\mathscr{E}^{*}$ and the supertrace
$\textup{str}(\sigma(0_{m},0))$ is given by the Berezin integral (cf. [2, Page
40]).

###### Proof.

Since every $\sigma\in\Gamma(\mathbb{T}M,\mathbb{S})$ can be written as a sum

$\sigma=\sum_{k}f_{k}\sigma_{k}$

where $f_{k}\in C^{\infty}(\mathbb{T}M)$ and
$\sigma_{k}\in\mathcal{S}_{J}(E,\nabla),$ it is enough to prove the lemma for
$\sigma\in\mathcal{S}_{J}(E,\nabla)$. So assume $\sigma=\sum_{k}s_{k}t^{-k}$
where $s_{k}\in\Gamma(M\times
M,\mathscr{E}\boxtimes\mathscr{E}^{*}\otimes\mathbb{C}[\mathfrak{g}]_{(J)})$
is of scaling order $k$ or more. We claim $\textup{str}(s_{k}(m,m))=0$ for
$k>-n$. To see why, note that by definition

$o^{\textup{sc}}(s_{k})\leq-o^{\textup{f}}(s_{k}).$

So when $k>-n$,

$o^{\textup{f}}(s_{k})<n$

and therefore $\textup{str}(s_{k}(m,m))=0.$ Also for $k<-n$,
$t^{-n}\textup{str}(s_{k})(x,x)t^{-k}$ extends smooth to zero at $t=0$, since
$k-n>0.$ For $k=-n$ the function $t^{-n}\textup{str}(s_{-n}(m,m))t^{n}$ is
constant in $t$. Therefore

$t^{-n}\textup{str}(\sigma(m,m,t))$

extends smoothly to

$\textup{str}(s_{-n}(m,m))$

at $t=0$ which equals $\textup{str}(\sigma(0_{m},0)).$ ∎

Using Lemma 5.13, we define a family of functionals on
$\Gamma(\mathbb{T}M,\mathbb{E}_{J})$:

$\textnormal{Str}_{t}:\Gamma(\mathbb{T}M,\mathbb{E}_{J})\to\mathbb{C}$

given as

$\textnormal{Str}_{t}(\sigma)=\int_{M}\mathfrak{s}(\sigma)(m,t)dm$ (26)

and obviously $\textnormal{Str}_{t}(\sigma)\to\textnormal{Str}_{0}(\sigma).$

######  Theorem 5.14 (McKean-Singer).

For $t\neq 0$ and $\tau>0$

$\textnormal{Str}_{t}(\bm{K}_{\tau})$

is independent of $t,\tau$ and by Proposition 3.2 equals the equivariant index

$\textup{ind}(e^{-X},\not{D}).$

Since the supertraces $\textup{Str}_{t}$ (26), are defined for every $t$,
Theorem 5.14 also applies for $t=0$. So by evaluating at $\tau=1$ and $t=0$
and using (25) we obtain

######  Theorem 5.15 (Kirillov Formula).

The equivariant index is given by the integral of equivariant differential
forms as follows:

$\textup{ind}(e^{-X},\not{D})=(2\pi
i)^{-n/2}\int_{M}\widehat{\textnormal{A}}_{\mathfrak{g}}(X,M)\textnormal{Ch}_{\mathfrak{g}}(X,\mathscr{E}/S).$

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