A Pseudodifferential Analytic Perspective on Getzler's Rescaling
Inspired by Gilkey's invariance theory, Getzler's rescaling method, and Scott's approach to the index via Wodzicki residues, we give a localisation formula for the ℤ₂ -graded Wodzicki residue of the logarithm of a class of differential operators acting on sections of a spinor bundle o...
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| description | Inspired by Gilkey's invariance theory, Getzler's rescaling method, and Scott's approach to the index via Wodzicki residues, we give a localisation formula for the ℤ₂ -graded Wodzicki residue of the logarithm of a class of differential operators acting on sections of a spinor bundle over an even-dimensional manifold. This formula is expressed in terms of another local density built from the symbol of the logarithm of a limit of rescaled differential operators acting on differential forms. When applied to complex powers of the square of a Dirac operator, it amounts to expressing the index of a Dirac operator in terms of a local density involving the logarithm of the Getzler rescaled limit of its square.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 20 (2024), 010, 34 pages
A Pseudodifferential Analytic Perspective
on Getzler’s Rescaling
Georges HABIB ab and Sylvie PAYCHA c
a) Department of Mathematics, Faculty of Sciences II, Lebanese University,
P.O. Box, 90656 Fanar-Matn, Lebanon
E-mail: ghabib@ul.edu.lb
b) Université de Lorraine, CNRS, IECL, France
URL: https://iecl.univ-lorraine.fr/membre-iecl/habib-georges/
c) Institut für Mathematik, Universität Potsdam, Campus Golm,
Haus 9, Karl-Liebknecht-Str. 24-25, 14476 Potsdam, Germany
E-mail: paycha@math.uni-potsdam.de
URL: https://www.math.uni-potsdam.de/~paycha/paycha/Home.html
Received March 08, 2023, in final form January 11, 2024; Published online January 30, 2024
https://doi.org/10.3842/SIGMA.2024.010
Abstract. Inspired by Gilkey’s invariance theory, Getzler’s rescaling method and Scott’s
approach to the index via Wodzicki residues, we give a localisation formula for the Z2-graded
Wodzicki residue of the logarithm of a class of differential operators acting on sections of
a spinor bundle over an even-dimensional manifold. This formula is expressed in terms of
another local density built from the symbol of the logarithm of a limit of rescaled differential
operators acting on differential forms. When applied to complex powers of the square of
a Dirac operator, it amounts to expressing the index of a Dirac operator in terms of a local
density involving the logarithm of the Getzler rescaled limit of its square.
Key words: index; Dirac operator; Wodzicki residue; spinor bundle
2020 Mathematics Subject Classification: 58J40; 47A53; 15A66
Dedicated to J.P. Bourguignon for his 75th birthday
1 Introduction
On a closed Riemannian manifold (Mn, g), the algebra Ψcl(M,E) of classical pseudodifferential
operators acting on the smooth sections of a finite rank vector bundle E overM , admits a unique
(up to a multiplicative factor) trace, called the Wodzicki [19] or the noncommutative residue,
built from a residue density defined as follows. Given Q in Ψcl(M,E), the residue of Q is the
integral over M of the residue density ωRes
Q (x) := res(σ(Q)(x, ·))dx1 ∧ · · · ∧ dxn defined in (2.1)
with
res(σ(Q)(x, ·)) := 1
(2π)n
∫
|ξ|=1
trE(σ−n(Q)(x, ξ))dSξ.
Here, n is the dimension of M , trE stands for the fibrewise trace on End(E), (x, ξ) is an
element in T ∗M , and σ−n(Q)(x, ξ) is the (−n)-th homogeneous part of the symbol at (x, ξ). The
Wodzicki residue extends beyond classical pseudodifferential operators to the logarithm logθ(Q)
This paper is a contribution to the Special Issue on Differential Geometry Inspired by Mathemati-
cal Physics in honor of Jean-Pierre Bourguignon for his 75th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Bourguignon.html
ghabib@ul.edu.lb
https://iecl.univ-lorraine.fr/membre-iecl/habib-georges/
paycha@math.uni-potsdam.de
https://www.math.uni-potsdam.de/~paycha/paycha/Home.html
https://doi.org/10.3842/SIGMA.2024.010
https://www.emis.de/journals/SIGMA/Bourguignon.html
2 G. Habib and S. Paycha
(see Appendix B for the precise definition) of a pseudodifferential operatorQ with Agmon angle θ
(see (2.11)), giving rise to the logarithmic residue ωRes
logθ(Q)(x). The Wodzicki residue is local in
so far as it is expressed as the integral on M of a volume form involving the (−n)-homogeneous
component of the symbol. So it comes as no surprise that the index of the Dirac operator can
be expressed in terms of the residue. For a Z2-graded vector bundle E = E+ ⊕ E−, the index
of an elliptic odd operator /D
+
: C∞(M,E+) → C∞(M,E−) with formal adjoint /D
−
=
(
/D
+)∗
can be written [16]
Index
(
/D
+)
= −1
2
sres
(
logθ
(
/D
2))
= −1
2
∫
M
ωsRes
logθ( /D
2
)
(x),
where /D :=
[
0 /D
−
/D
+
0
]
acting on E+⊕E−, so that /D
2
= /D
− /D
+⊕ /D
+ /D
−
and θ = π. The graded
residue “sres” is defined in the same way as the residue with the fibrewise trace on End(E)
replaced by the Z2-graded trace and ωsRes
logθ( /D
2
)
(x) := sres
(
σ
(
logθ
(
/D
2))
(x, ·)
)
dx1 ∧ · · · ∧ dxn.
Inspired by the approach adopted in [16], we revisit Geztler’s rescaling in the context of in-
dex theory in the light of the logarithmic Wodzicki residue. For a class of differential operators
acting on spinors which includes /D
2
, we express the logarithmic residue density evaluated at
a point p in M in terms of another local density ω̃sres
logθ(P̃lim)
(x) (see formula (1.1)) involving
a limit P̃lim as the parameter λ goes to zero of a family of operators P̃Ge
λ built from the origi-
nal one by rescaling it at the point p (see equation (6.1)). In this sense, equation (1.3) at the limit
as λ tends to zero, can be viewed as a localisation formula of the logarithmic residue at point p.
For this purpose, we first single out a class of differential operators acting on smooth sections of
a vector bundle E, which we call geometric with respect to a metric g, whose coefficients written
in some local trivialisation are geometric sections (Definition 5.8). We consider polynomials in
the jets of the vielbeins for the metric g (Definition 5.3) and inspired by Gilkey [11], we define
their Gilkey order (at a point p) (see equation (5.5)) to be the order of those jets. We call a differ-
ential operator geometric if its coefficients are geometric polynomials whose Gilkey order obeys
a compatibility condition involving the order of the operator, see equation (5.8). Geometric dif-
ferential operators enjoy nice transformation properties under local contractions (fλ)λ∈[0,1] along
local geodesics defined by means of exponential geodesic normal coordinates (see equation (2.5)
for the definition). Indeed a geometric differential operator with respect to g transforms to
one with respect to gλ (Proposition 5.15), where gλ = λ−2f∗λg. This transformed metric can
be viewed as the pull-back metric under the canonical projection π̂ : M →M of the deformed
manifold M via a deformation to the normal cone to p, see equation (4.3).
We first consider the bundle E = ΛT ∗M . From a differential operator P in Ψcl(M,ΛT ∗M)
acting on differential forms, we define a family of operators P̃Ge
λ := λord(P )U ♯
λf
♯
λP (see (6.1))
using notations borrowed from [18], which are obtained under the combined action of the con-
tractions fλ mentioned previously and the so-called Getzler map Uλ that acts on tensors, see
Definition 3.1. We call a geometric differential operator P rescalable if P̃Ge
λ admits a limit P̃lim
when λ→ 0 (Definition 6.1). In Proposition 6.6, we give a necessary and sufficient condition for
the rescalability of a geometric differential operator in Ψcl(M,ΛT ∗M) and show that the coeffi-
cients of the limit P̃lim are polynomial expressions in the jets of the Riemannian curvature tensor.
A first result is the localisation formula (1.2) for a differential operator P acting on differential
forms. It involves a local n-degree form ω̃Res
Q (x), inspired by Scott’s proof of the index theorem
[16, Section 3.5.3] and defined for operators Q in Ψcl(M,ΛT ∗M) as (see equation (3.5)):
ω̃Res
Q (x) :=
1
(2π)n
∫
S∗
xUp
[σ−n(Q)(x, ξ)1x][n] dSξ. (1.1)
Here Up is a local exponential neighborhood of a point p in M , S∗
xUp is the unit sphere in the
cotangent space T ∗
xUp at point x and the integrand is the degree n-part of the differential form
A Pseudodifferential Analytic Perspective on Getzler’s Rescaling 3
σ−n(Q)(x, ξ)1x. When restricted to operators in the range of a Clifford map, the local form ω̃Res
Q
is proportional to the Wodzicki residue density (Corollary 3.10) and therefore becomes a global
form. It further extends to logarithmic pseudodifferential operatorsQ = logθ(P ) for a differential
operator P with Agmon angle θ. In Proposition 3.6, we show that the local n-form ω̃Res
logθ(P ) at
the point fλ(x) is the local n-form associated to the pull-back operator U ♯
λf
♯
λ(logθ(P )) at the
point x. If moreover P is rescalable, taking the limit as λ tends to zero yields the localisation
formula
ω̃Res
logθ(P )(p) = ω̃Res
logθ(P̃lim)
(x), ∀x ∈ Up. (1.2)
This formula can be applied to the Hodge Laplacian which is a geometric and rescalable operator
(see Example 6.8).
We then consider the case of a spinor module E = ΣM when M is a spin manifold of even
dimension. To define rescalability of geometric differential operators in Ψcl(M,ΣM) we use the
identification Cℓ(TM) ⊗ C ≃ End(ΣM), and the Clifford map cg : Cℓ(TM) −→ End(ΛT ∗M)
(see (C.1)) which sends an element of the Clifford algebra Cℓ(TM) on the tangent bundle
to an endomorphism of ΛT ∗M . We call a geometric differential operator P in Ψcl(M,ΣM)
rescalable if cg(P ) defined in formula (3.11) by applying cg to the coefficients of the differential
operator P , is rescalable in Ψcl(M,ΛT ∗M). We further give a necessary and sufficient condition
for the rescalability of geometric differential operators in Ψcl(M,ΣM), see Proposition 6.9.1
It follows from Proposition 3.8 that for a differential operator P ∈ Ψcl(M,ΣM) with Agmon
angle θ, the form ω̃Res
logθ(c
g(P )) defines a global density. In Corollary 6.10, we infer from the above
localisation formula (1.2) a second localisation formula for operators in Diff(M,ΣM):
ωsRes
logθ(P )(p) = (−2i)n/2ω̃Res
logθ(P̃lim)
(x), (1.3)
where P is a rescalable geometric differential operator in Diff(M,ΣM) of Agmon angle θ which
is even for the Z2-grading ΣM = Σ+M ⊕ Σ−M . This formula expresses the residue den-
sity ωsRes
logθ(P )(p) at a point p in terms of a local density ω̃Res
logθ(P̃lim)
(x) of the limit P̃lim of the
rescaled operators P̃Ge
λ .
The localisation formula (1.3) applied to the square of the Dirac operator (see Proposi-
tion 7.1), which is proven to be a rescalable geometric differential operator, confirms the results
of [16, Section 3.5.3.3] (identification of (3.5.3.12) and (3.5.3.40)). Although the limit oper-
ator P̃lim is expected to have a simpler form than the original operator as in the case of the
Dirac operator, computing ω̃Res
logθ(P̃lim)
(x) nevertheless remains a challenge since it involves its
(−n)-th homogeneous symbol.
2 The Wodzicki residue density
for classical pseudodifferential operators
In this section, we review the definition of the Wodzicki residue for classical pseudodifferential
operators acting on sections of a given vector bundle. We recall the covariance property of the
Wodzicki residue under local contractions (see Proposition 2.3). We also recall how the Wodzicki
residue extends to logarithms of classical pseudodifferential operators with appropriate spectral
properties and refer to this extension as logarithmic Wodzicki residue. Specialising to the trivial
1In this part of the work, we consider even-dimensional spin manifolds, however our study extends to manifolds
with a spinc structure. Indeed the construction relies on the identification Cℓ(TM) ⊗ C ≃ End(ΣM) which can
be extended to manifolds with a spinc structure, in which case we have the identification Cℓ(TM)⊗C ≃ End(E)
[4, Theorem 2.13] with E a vector bundle isomorphic to the spinor bundle of the spinc bundle. For simplicity, we
restrict ourselves to spin manifolds.
4 G. Habib and S. Paycha
vector bundle, we show a localisation formula for the logarithmic residue of scalar differential
operators. It identifies the logarithmic residue density at the point p of a differential operator P
with the logarithmic residue density at any point x in a small neighborhood of p of the same
operator localised at p (see Proposition 2.5).
2.1 The Wodzicki residue for classical pseudodifferential operators
Let (E, π,M) be a vector bundle over M , an n-dimensional smooth manifold, of finite rank
and let Ψcl(M,E) denote the algebra of classical polyhomogeneous pseudodifferential operators
acting on the space C∞(M,E) of smooth sections of E. These are linear maps Q : C∞(M,E) →
C∞(M,E), which read Q =
∑
i∈I QUi + SQ, where given a partition of unity (χi, i ∈ I) of M
subordinated to a finite open covering (Ui, i ∈ I) of M , the operators QUi := ψiQχi are locali-
sations of Q in open subsets Ui of M , and SQ is a smoothing operator – it maps any Sobolev
section to a smooth section. Here (ψi, i ∈ I) are smooth functions compactly supported with
support in Ui and which are identically equal to one on the support of χi for any i ∈ I. Since
we are interested in singular linear forms which vanish on smoothing operators, we reduce our
study to localised operators QU . As these choices will not influence our results, we drop the
explicit mention of the localisation and simply write Q. A pseudodifferential operator (localised
on some open subset U of M) acting on C∞(U,E) is called classical or polyhomogeneous if it is
a linear combination of pseudodifferential operators Q whose (local) symbol σ(Q) – which lies in
C∞(T ∗U,End(V )), in any local trivialisation of E over U – has a polyhomogeneous expansion
of the form
σ(Q) ∼
∞∑
j=0
σm−j(Q),
with m in C, the order of Q. Explicitly, for any N in N, the difference σ(Q)−
∑N
j=0 χσm−j(Q)
is a smooth pseudodifferential symbol of order no larger than Re(m) − N , with χ a smooth
function which vanishes in a neighborhood of zero, and σα(Q) positively homogeneous of degree
α ∈ C, that is,
σα(Q)(x, λξ) = λασα(Q)(x, ξ)
for any (x, ξ) ∈ T ∗U and λ > 0. For further details, we refer to classical books on the subject
such as [17], see also [16, Example 1.1.8]. We also consider the class of logarithmic pseudodif-
ferential operators, namely those whose symbols have a log-polyhomogeneous expansion of the
form
σ(Q)(x, ξ) = m log(|ξ|)Id + σcl(Q)(x, ξ),
where σcl(Q) is a classical symbol of nonpositive order. We define the local residue density2
ωRes
Q (x) := res(σ(Q)(x, ·))dx1 ∧ · · · ∧ dxn, (2.1)
where dx1∧· · ·∧dxn is the flat volume form in local coordinates on the (oriented) n-dimensional
manifold M and
res(σ(Q)(x, ·)) := 1
(2π)n
∫
|ξ|=1
trE(σ−n(Q)(x, ξ))dSξ,
2s-densities on an n-dimensional real vector space V are functions µ : V n → R such that µ(Av1, Av2, . . . , Avn) =
| det(A)|sµ(v1, . . . , vn) for any linear isomorphism A of V and form a one-dimensional vector space |Λ|s(V ). An
s-density on a manifold M is a section of the s-density bundle |Λ|s(TM) over M whose fibre over x consists of s-
densities on the tangent space TxM . On an n-dimensional oriented manifold M , 1-densities, also called densities,
can be canonically identified with the n-forms on M .
A Pseudodifferential Analytic Perspective on Getzler’s Rescaling 5
where trE stands for the fibrewise trace on End(E), dSξ for the standard density on the unit
sphere Sn−1 obtained as the interior product of the flat volume form dξ1∧· · ·∧dξn by the radial
vector field R :=
∑n
i=1 ξ
i ∂
∂ξi
, namely
dSξ := R⌟
(
dξ1 ∧ · · · ∧ · · · ∧ dξn
)
=
n∑
j=1
(−1)j−1ξjdξ1 ∧ · · · ∧ d̂ξj ∧ · · · ∧ dξn.
A priori, ωRes
Q (x), which is defined using a localisation of the operator Q around x, depends on
the choice of local coordinates in a neighborhood of x. M. Wodzicki [19] showed that it actually
defines a global n-form, which can be integrated to define the linear form Res on Ψcl(M,E),
called the Wodzicki or noncommutative residue:
Res(Q) :=
∫
M
ωRes
Q (x).
Remark 2.1.
1. If (M, g) is an n-dimensional smooth manifold Riemannian manifold, we can equivalently
define
res(σ(Q)(x, ·)) := 1
(2π)n
∫
S∗
xM
trE(σ−n(Q)(x, ξ))νx(ξ),
as an integral over the cotangent unit sphere S∗
xM := {ξ ∈ T ∗
xM, |ξ| = 1} endowed with
the induced Riemannian volume form νx.
2. The Wodzicki residue easily extends to a Z2-graded vector bundle E = E+⊕E− replacing
the fibrewise trace trE by a graded trace strE := trE
+ − trE
−
, in which case we set
sres(σ(Q)(x, ·)) := 1
(2π)n
∫
|ξ|=1
strE(σ−n(Q)(x, ξ))dSξ,
and
ωsRes
Q (x) := sres(σ(Q)(x, ·))dx1 ∧ · · · ∧ dxn.
2.2 Local contractions
Throughout the paper, (M, g) denotes a smooth Riemannian manifold of dimension n and p a
point in M . The local identification uses the exponential map
expp : TpM ⊃ Br −→ Up ⊂M (2.2)
around p which yields a local diffeomorphism from a ball Br of radius r > 0 centered at 0
to a local geodesic neighborhood Up of p. This exponential map is combined with a rescaling
leading to the map (this is the map exp ◦Tϵ in [9, formula (4.4.7)])
expp ◦hλ : TpM ⊃ Br/λ −→ Up ⊂M,
where
hλ : TpM ⊃ Br/λ −→ Br ⊂ TpM,
x 7−→ λx.
In the sequel, we use the following notations. From a given orthonormal basis e1(p), . . . , en(p)
of TpM at p ∈M , we build:
6 G. Habib and S. Paycha
� normal geodesic coordinates at any point x ∈ Up, by means of the map Br ⊂ Rn → Up;(
x1, . . . , xn
)
7→ x
x = expp
(
n∑
i=1
xiei(p)
)
∈ expp(Br) (2.3)
defined via the local exponential map expp in (2.2);
� a local orthonormal frame
Op(x, g) := {e1(x, g), . . . , en(x, g)}, x ∈ Up (2.4)
of TxM by the parallel transport τc : TpM → TxM along the geodesic c(t) = expp(tx),
with x in Br ⊂ TpM, which takes p to x = c(1) so that ej(x, g) = τc(ej(p)).
Unless specified otherwise, we use normal geodesic coordinates. As usual, we identify any
point x ∈ Up with its coordinates X :=
(
x1, . . . , xn
)
. Let 1 > λ > 0. By means of the map hλ,
we define a rescaled coordinate system
Y :=
(
y1 := λx1, . . . , yn := λxn
)
at any point in Up. Since we have the inclusion Br ⊂ Br/λ, the map hλ induces a diffeomorphism
fp,λ : Up −→ Uλ
p := expp(Bλr) ⊂ Up,
expp(x) 7−→ expp ◦hλ(x) = expp(λx), (2.5)
which we shall denote by fλ for simplicity. As a consequence of the above constructions, we have
f∗λ
(
∂
∂xi
)
=
∂
∂yi
= λ−1 ∂
∂xi
◦ fλ and f∗λdx
i = dyi = λdxi ◦ fλ. (2.6)
2.3 The behaviour of the Wodzicki residue under local contractions
Let us now recall the general fact on pull-back of operators. Any local diffeomorphism f : U → V
induces a local transformation on a localised pseudodifferential operator as follows: Given any Q
in Ψcl(V,E), where (E, π,M) is a vector bundle over M , we define f♯Q ∈ Ψcl(U, f
∗E) by(
f♯Q
)
s := f∗
(
Q
((
f∗
)−1
(s)
))
= Q
(
s ◦ f−1
)
◦ f, (2.7)
where s is any local section in f∗E above U . Here, f∗E is the pull-back bundle over U of the
bundle E given by
f∗E = {(x, y) ∈ U × E | f(x) = π(y)}.
The following lemma is an easy consequence of the transformation property of symbols under
the local diffeomorphism fλ. We nevertheless provide an explicit proof.
Lemma 2.2 (compare with [16, p. 381]). Given any Q in Ψcl(M,E), we have for small enough
positive λ,
σ
(
f♯λQ
)
(x, ξ) = σ(Q)
(
fλ(x),
((
(fλ)∗
)t)−1
(ξ)
)
= σ(Q)
(
fλ(x),
(
f∗λ
)t
(ξ)
)
, (2.8)
at any given point x in Up.
A Pseudodifferential Analytic Perspective on Getzler’s Rescaling 7
Proof. A local diffeomorphism f : U → V between two open subsets U and V of M , induces
a map (f∗)tx : T
∗
xU → T ∗
f(x)V and the symbol σ(Q) of Q transforms as (see, e.g., [17, equa-
tion (4.2.1)])
σ
(
f♯Q
)
(x, ξ) = σ(Q)
(
f(x),
(
(f∗)
t
)−1
(ξ)
)
+ lower order terms. (2.9)
Here “lower order terms” stands for the push forward by f of the sum
∑
|α|>0
1
α!Φα(x, ξ)∂
α
ξ σ (x, ξ)
with
Φα(x, ξ) := Dα
z e
i⟨φf
x(z),ξ⟩∣∣
z=x
and φf
x(z) := f(z)− f(x)− df(x)(z − x),
which is a polynomial in ξ of degree ≤ |α|
2 whose coefficients are linear combinations of products
of derivatives
∏
γ ∂
γf(x) of f at x with
∑
|γ| = |α| and |γ| ≥ 2. For f = fλ with U = Up and
V = Uλ
p , we have ∂i∂jf ≡ 0 for any indices i, j running from 1 to n, so that the lower order
terms vanish leading to (2.8). ■
Proposition 2.3. Let E be a vector bundle overM of finite rank. For any given Q in Ψcl(M,E),
the n-form ωRes
Q transforms covariantly under contractions fλ as
ωRes
Q ◦ fλ = ωRes
f♯λQ
,
for any λ > 0 small enough.
Proof. Applying the local residue density (2.1) at the point fλ(x) with x ∈ Up, we have
(2π)nωRes
Q (fλ(x)) =
(∫
|ξ|=1
trE(σ−n(Q)(fλ(x), ξ))dSξ
)
dy1 ∧ · · · ∧ dyn
=
(∫
|ξ|=1
trE
(
σ−n(Q)
(
fλ(x), λ
−1ξ
))
dSξ
)
dx1 ∧ · · · ∧ dxn
=
(∫
|ξ|=1
trE
(
σ−n(Q)
(
fλ(x), (f
∗
λ)
t(ξ)
))
dSξ
)
dx1 ∧ · · · ∧ dxn
(2.8)
= (2π)nωRes
f♯λ(Q)
(x).
This finishes the proof of the lemma. ■
2.4 Logarithmic residue density
For later purposes, we review here how the Wodzicki residue can be extended to logarithms of
pseudodifferential operators as defined in Appendix B. As before, we consider a vector bundle E
over M of finite rank. We say that an operator Q in Ψcl(M,E) of positive real3 order m has
a principal angle θ ∈ [0, 2π) (see [16, Section 1.5.7.1]) if the leading symbol matrix σL(Q)(x, ξ) :=
σm(Q)(x, ξ) has no eigenvalue on the ray Lθ := {reiθ, r ≥ 0} for every (x, ξ) ∈ T ∗U \ U × {0}.
In particular, the operator is elliptic and, therefore, has a purely discrete spectrum. A principal
angle θ of an operator Q is said to be an Agmon angle4 if there exists a solid angle of the ray
Λε,θ =
{
reiα, r ≥ 0, θ − ε ≤ α ≤ θ + ε
}
,
for some ε > 0, that contains no eigenvalue of Q. In that case, the operator Q is invertible.
3The order is assumed to be real so as to ensure that does not intersect all rays.
4One can actually build an Agmon angle from a small perturbation of any principal angle.
8 G. Habib and S. Paycha
Remark 2.4. We shall drop the explicit mention of the principal angle when we can choose
θ = π.
For such an operator Q, we can define the complex power Qz
θ for z ∈ C and the loga-
rithm logθ(Q) as in Appendix B. It is “nearly” classical in so far as its local symbol differs from
a classical symbol by a logarithm term. Indeed, it is shown in [16, formula (2.6.1.11)] that the
symbol of the logarithm reads
σ(logθ(Q))(x, ξ) = m log(|ξ|)Id + σcl(logθ(Q))(x, ξ), (2.10)
where σcl(logθ(Q)) is a classical symbol of order zero with homogeneous components
σ−j(logθ(Q)) of degree −j, j ≥ 0 given by (this follows from the formula above (2.6.1.11) on
p. 219 in [16])
σ−j(logθ(Q))(x, ξ) = |ξ|−j
(
∂z
(
σmz−j(Q
z
θ)
(
x,
ξ
|ξ|
)))
z=0
.
The fact that the logarithmic part of the symbol vanishes on the cotangent unit sphere
underlies the extendibility of the Wodzicki residue to logarithmic pseudodifferential operators
(for a detailed discussion, we refer the reader to [16, Section 2.7.1]). In analogy with (2.1), we set
res(σ(logθ(Q))(x, ·)) := 1
(2π)n
∫
S∗
xM
trE(σ−n(logθ(Q))(x, ξ))dSξ,
ωRes
logθ(Q)(x) := res(σ(logθ(Q))(x, ·))dx1 ∧ · · · ∧ dxn, (2.11)
which we call the logarithmic residue density of P . Given a local diffeomorphism f : U → V
and an operator Q ∈ Ψcl(M,E) with Agmon angle θ, the operator f♯Q defined in (2.7) lies in
Ψcl(M, f∗E) with the same Agmon angle θ, since equation (2.9) implies that
σL
(
f♯Q
)
(x, ξ) = σL(Q)
(
f(x), (f∗)t(ξ)
)
.
Furthermore, the relation f♯(Q− λ)−1 =
(
f♯Q− λ
)−1
gives that
f♯(Qz
θ) =
(
f♯Qθ
)z
(2.12)
for any complex number z with negative real part. Since f♯(Qk) =
(
f♯Q
)k
for any positive
integer k, it follows from the construction of the extension Qz
θ to any complex number z, that
Property (2.12) extends to z ∈ C. Similarly, one shows that
f♯(logθ(Q)) = logθ
(
f♯Q
)
,
in other words, f♯ commutes with the functional calculus. On the grounds of formula (2.10),
σ−n(logθ(Q))) = (σcl)−n(logθ(Q))) so that one can easily adapt the proof of Proposition 2.3 to
show the covariance of the logarithmic residue:
ωRes
logθ(Q) ◦ fλ = ωRes
logθ(f
♯
λQ)
, (2.13)
where we have used the fact that f♯λ and logθ commute.
A Pseudodifferential Analytic Perspective on Getzler’s Rescaling 9
2.5 A localisation formula for the logarithmic residue density
We now focus on logarithms of scalar differential operators, for which we prove a localisation
formula for the Wodzicki residue density. In the sequel, we use the following notations. For any
multiindex γ = {i1, . . . , is}, we set
Dγ
X :=
∂
∂xi1
· · · ∂
∂xis
, (2.14)
in the local normal geodesic coordinatesX =
(
x1, . . . , xn
)
at point x with the usual identification
x ↔ X. To simplify notations, unless this gives rise to an ambiguity, we henceforth write Dγ
instead of Dγ
X .
Given a vector bundle E →M of rank k, trivialised over an open subset U of M , Dγ acts on
a local section s|U =
∑k
i=1 αisi
∣∣
U
on U by
Dγs :=
k∑
i=1
Dγ(αi)si. (2.15)
Here {si}i=1,...,k is a basis of the bundle E|U in the local trivialisation E|U ≃ U × Rk. A dif-
ferential operator of order m ∈ Z≥0 reads P =
∑
|γ|≤m PγD
γ , which means that in the local
trivialisation E|U ≃ U × Rk of E, it acts as
P
(
k∑
j=1
αjsj
)
=
∑
|γ|≤m
k∑
i,j=1
(Pγ)ijD
γ(αj)si, (2.16)
where we have used equation (2.15). Differential operators form an algebra Diff(M,E) and we
have the following isomorphism of C∞-modules:
Diff(M,E) ≃ Diff(M)⊗C∞(M) C
∞(M,End(E)),
where we have set Diff(M) := Diff(M,M × R). Following [18], we define a family of rescaled
differential operators for any P ∈ Diff(M,M × R) by
P := λmP, λ > 0, (2.17)
and set for any small positive λ
P̃λ := λmf♯λP. (2.18)
In local normal geodesic coordinates, we have f♯λD
γ = λ−|γ|Dγ so that the family of rescaled
operators built from a differential operator P =
∑
|γ|≤m PγD
γ , locally reads (these and the
above notations Pλ are borrowed from [18])
P̃λ =
∑
|γ|≤m
λm−|γ|(Pγ ◦ fλ)Dγ .
As λ tends to zero, P̃λ converges to the operator P evaluated at the limit point p
lim
λ→0
P̃λ
∣∣
Up
=
∑
|γ|=m
Pγ(p)D
γ |p = P |p, (2.19)
where Pγ(p) corresponds to Pγ(x) evaluated at the reference point p. In the following, we
state a localisation formula for the residue of the logarithm of a differential operator (see [16,
formula (3.5.3.33), p. 382] for a similar formula).
10 G. Habib and S. Paycha
Proposition 2.5. For any differential operator P in Diff(M,M × R) with Agmon angle θ, we
have the following localisation formula:
ωRes
logθ(P )(p) = ωRes
logθ(P |p)(x)
for all x ∈ Up.
Proof. We first observe that for small positive λ
logθ
(
P̃λ
)
= logθ
(
λmf♯λP
)
= (m log λ)Id + logθ
(
f♯λP
)
,
where m is the order of P . Since the residue density vanishes on differential operators and hence
on Id, we have ωRes
logθ(P̃λ)
= ωRes
logθ(f
♯
λP )
. Equation (2.13) implies that ωRes
logθ(P̃λ)
= ωRes
logθ(P ) ◦ fλ. We
then take the limit as λ→ 0, by which P̃λ tends to P |p by (2.19). The continuity of the logarithm
combined with the continuity of the Wodzicki residue for the Fréchet topology of (log-)classical
operators of constant order then yields the statement of the proposition. ■
3 A local Berezin type n-form on Ψcl(M,ΛT ∗M)
In this section, we define a local n-form ω̃Res on Ψcl(M,ΛT ∗M) (see equation (3.5)), which
unlike the Wodzicki density, is not covariant under contractions defined in the previous section.
We give in Proposition 3.6 the behaviour of this local n-form ω̃Res under Getzler rescaling map
(see Definition 3.1) combined with the local contractions. When the manifold M is spin and for
a differential operator P acting on smooth sections of its spinor bundle, we use the expression of
the super trace in terms of a Berezin integral (see (C.2)) to relate the local n-form ω̃Res
logθ(c
g(P )) of
the logarithm (with spectral cut θ) of cg(P ) (defined in equation (3.11)) to its (super-)Wodzicki
residue ωsRes
logθ(c
g(P )). Much of this section is inspired from Simon Scott’s approach to the local
Atiyah–Singer index theorem by means of the Wodzicki residue [16, Section 3.5.3].
3.1 The Getzler rescaling map
To simplify the notation, we set ⊗q
rV := V ⊗q ⊗ (V ∗)⊗r.
Definition 3.1. The Getzler rescaling map is the tensor bundle morphism defined for any λ > 0,
by
Uλ : ⊗q
r V −→ ⊗q
rV,
t 7−→ λq−rt. (3.1)
The Getzler rescaling map Uλ restricted to ΛV induces a map
U ♯
λ : End(ΛV ) −→ End(ΛV ),
Q 7−→ U ♯
λQ : ω 7→ UλQU
−1
λ ω, (3.2)
which satisfies
U ♯
λ(v ∧ •) = λ−1v ∧ • and U ♯
λ
(
v♯g⌟•
)
= λv♯g⌟ • . (3.3)
for any v ∈ V ∗. Combining (3.3) with the Clifford map cg : Cℓ(V ) −→ End(ΛV ) defined in (C.1)
Appendix C, on the covector v ∈ V ∗ by cg(v)• = v ∧ • − v♯g⌟• yields the map
U ♯
λ ◦ cg : Cℓ(V ) −→ End(ΛV )
given by(
U ♯
λ ◦ cg
)
(v)• = λ−1v ∧ • − λv♯g⌟ • .
We have the following straightforward lemma that we will use later.
A Pseudodifferential Analytic Perspective on Getzler’s Rescaling 11
Lemma 3.2. Let eI := ei1 ·g ei2 · · · ·g eik for i1 < i2 < · · · < ik with |I| = k, it follows that
lim
λ→0
λ|I|
(
U ♯
λ ◦ cg
)
(eI) = eI∧, (3.4)
where eI∧ := ei1 ∧ · · · ∧ eik .
3.2 A local n-form on Ψcl(M,ΛT ∗M) and Getzler rescaling
In order to define the local n-form, we fix a normal geodesic neighborhood Up around a point p
in M . For (x, ξ) ∈ T ∗Up, we consider the symbol σ(Q)(x, ξ) ∈ End(ΛT ∗
xUp) of an opera-
tor Q in Ψcl(M,ΛT ∗M) in the corresponding coordinate chart. Its homogeneous component
σ−n(Q)(x, ξ) of degree −n evaluated in 1x yields a differential form σ−n(Q)(x, ξ)1x ∈ ΛT ∗
xUp.
Hence we define
ω̃Res
Q (x) :=
1
(2π)n
∫
|ξ|=1
[σ−n(Q)(x, ξ)1x][n]dSξ, (3.5)
where α[n] stands for the part of degree n of a form α in ΛT ∗
xM .
Remark 3.3.
� Note that this differs from the Wodzicki residue density. Contrarily to ωRes
Q which is
covariant with respect to the action of fλ, as we shall see shortly, ω̃Res
Q is not. Getzler’s
rescaling map will enable us to compensate this lack of covariance.
� The above constructions generalise beyond classical pseudodifferential operators, to loga-
rithmic pseudodifferential operators. For a differential operator P in Diff(M,ΛT ∗M) with
Agmon angle θ, similarly to (3.5), we define
ω̃Res
logθ(P )(x) :=
1
(2π)n
∫
|ξ|=1
[σ−n(logθ(P ))(x, ξ)1x][n]dSξ.
The maps U ♯
λ defined in (3.2) induce a transformation on differential operators as follows: for
any P =
∑
|γ|≤m PγD
γ in Diff(M,ΛT ∗M) of order m, we define
U ♯
λP :=
∑
|γ|≤m
U ♯
λ(Pγ)D
γ ∈ Diff(M,ΛT ∗M). (3.6)
A first direct consequence of (3.6) is that σL
(
U ♯
λP
)
= U ♯
λσL(P ) so that the operator U ♯
λP is also
of order m and has Agmon angle θ. We build
(
U ♯
λP
)z
θ
and logθ
(
U ♯
λP
)
following the construction
in Section 2.4.
Lemma 3.4. For any differential operator P ∈ Diff(M,ΛT ∗M) with Agmon angle θ and or-
der m, we have
σmz−j
((
U ♯
λP
)z
θ
)
(x, ξ) = U ♯
λ(σmz−j(P
z
θ )(x, ξ)),
σ−j
(
logθ
(
U ♯
λP
))
(x, ξ) = U ♯
λ(σ−j(logθ(P ))(x, ξ)) (3.7)
for any (x, ξ) ∈ T ∗U and j ≥ 0.
Proof. We prove the first identity, the second one can be shown in a similar manner. Let µ lie
on the contour Γθ. Since U
♯
λP − µ = U ♯
λ(P − µ), the same property holds on the symbolic level
σ
(
U ♯
λP
)
− µ = U ♯
λ
(
σ(P ) − µ
)
. As a result, the product formula σ
((
U ♯
λP − µ
)−1)
⋆
(
σ
(
U ♯
λP
)
−
12 G. Habib and S. Paycha
µ
)
= Id (see, e.g., [16, equation (4.8.2.2)]) which determines the homogeneous components
σ−m−j
((
U ♯
λP−µ
)−1)
of the resolvent with j ∈ Z≥0, reads σ
((
U ♯
λP−µ
)−1)
⋆
(
U ♯
λ(σ(P )−µ1)
)
= Id.
It follows that σ−m−j
((
U ♯
λP−µ
)−1)
= U ♯
λσ−m−j
(
(P − µ)−1
)
for j ∈ Z≥0. Using equation (B.3)
in Appendix B, this yields for Re(z) < 0 and j ∈ Z≥0
σmz−j(
(
U ♯
λP
)z
θ
)(x, ξ) =
i
2π
∫
Γθ
µzθσ−m−j
((
U ♯
λP − µ
)−1)
(x, ξ)dµ
=
i
2π
∫
Γθ
µzθU
♯
λσ−m−j
(
(P − µ)−1
)
(x, ξ)dµ
= U ♯
λ(σmz−j(P
z
θ )(x, ξ)).
These identities can then be extended to any complex number z. For Re(z) < k with k ∈ N, we
write
(
U ♯
λP
)z
θ
=
(
U ♯
λP
)k(
U ♯
λP
)z−k
θ
. Since σmk−j
((
U ♯
λP
)k)
= σmk−j
(
U ♯
λP
k
)
= U ♯
λσmk−j(P
k) for
any j ∈ Z≥0, it follows from (B.4) that
σmz−j
((
U ♯
λP
)z
θ
)
=
∑
a+b+|α|=j
(−i)|α|
α!
∂αξ σmk−a
((
U ♯
λP
)k)
∂αxσm(z−k)−b
((
U ♯
λP
)z−k
θ
)
∀j ∈ Z≥0
=
∑
a+b+|α|=j
(−i)|α|
α!
U ♯
λ∂
α
ξ σmk−a
(
P k
)
U ♯
λ∂
α
xσm(z−k)−b
(
P z−k
θ
)
∀j ∈ Z≥0
= U ♯
λσmz−j(P
z
θ ). ■
Lemma 3.5. Given any P ∈ Diff(M,ΛT ∗M) we have for any λ > 0(
U ♯
λ ◦ f♯
)
(P ) =
(
f♯ ◦ U ♯
λ
)
(P ) (3.8)
for any local diffeomorphism f : U → V .
Proof. First, we show that Uλ ◦ f∗ = f∗ ◦Uλ, where by definition f∗ω = ω ◦ f for any differential
form ω. Indeed, we compute
(Uλ ◦ f∗)ω = Uλ(ω ◦ f) =
n∑
i=1
λ−i(ω ◦ f)[i] =
n∑
i=1
λ−iω[i] ◦ f = (f∗ ◦ Uλ)ω.
Hence, we get for P ∈ Diff(M,ΛT ∗M)(
U ♯
λ ◦ f♯
)
(P ) = Uλ ◦ f∗ ◦ P ◦ f∗ ◦ U−1
λ = f∗ ◦ Uλ ◦ P ◦ U−1
λ ◦ f∗ =
(
f♯ ◦ U ♯
λ
)
(P ). ■
As a direct consequence of equation (3.7), we get the following
Proposition 3.6. For any differential operator P ∈ Diff(M,ΛT ∗M) of Agmon angle θ, and for
any λ > 0:
ωRes
logθ(U
♯
λP )
= ωRes
logθ P
, ω̃Res
logθ(U
♯
λP )
= λ−nω̃Res
logθ P
, ω̃Res
logθ (f
♯
λP )
= λnω̃Res
logθ P
◦ fλ. (3.9)
In particular, we get
ω̃Res
logθ(U
♯
λf
♯
λ
P )
= ω̃Res
logθ P
◦ fλ. (3.10)
A Pseudodifferential Analytic Perspective on Getzler’s Rescaling 13
Proof. For any λ > 0, we write
(2π)nωRes
logθ(U
♯
λP )
(x) =
(∫
|ξ|=1
trΛT
∗M
(
σ−n
(
logθ
(
U ♯
λP
))
(x, ξ)
)
dSξ
)
dx1 ∧ · · · ∧ dxn
(3.7)
=
(∫
|ξ|=1
trΛT
∗M
(
U ♯
λ(σ−n(logθ P )(x, ξ))
)
dSξ
)
dx1 ∧ · · · ∧ dxn
=
(∫
|ξ|=1
trΛT
∗M
(
Uλσ−n(logθ P )(x, ξ)U
−1
λ
)
dSξ
)
dx1 ∧ · · · ∧ dxn
=
(∫
|ξ|=1
trΛT
∗M (σ−n(logθ P )(x, ξ))dSξ
)
dx1 ∧ · · · ∧ dxn
= (2π)nωRes
logθ P
(x).
To prove the two other equalities, we also compute
(2π)nω̃Res
logθ(U
♯
λP )
(x) =
∫
|ξ|=1
[(
σ−n
(
logθ
(
U ♯
λP
))
(x, ξ)
)
1x
]
[n]
dSξ
(3.7)
=
∫
|ξ|=1
[(
U ♯
λ(σ−n(logθ(P ))(x, ξ))
)
1x
]
[n]
dSξ
=
∫
|ξ|=1
[
Uλ(σ−n(logθ(P ))(x, ξ)1x)
]
[n]
dSξ
=
∫
|ξ|=1
n∑
i=0
[
Uλ[σ−n(logθ(P ))(x, ·)1x][i]
]
[n]
dSξ
=
∫
|ξ|=1
n∑
i=0
[
λ−i[σ−n(logθ(P ))(x, ·)1x][i]
]
[n]
dSξ
= λ−n
∫
|ξ|=1
[σ−n(logθ(P ))(x, ·)1x][n]dSξ
= λ−n(2π)nω̃Res
logθ(P )(x).
To prove the last equality in (3.9), we use equality (2.8) to write
(2π)nω̃Res
logθ(f
♯
λP )
(x) =
∫
|ξ|=1
[
σ−n
(
f♯λ logθ P
)
(x, ξ)1x
]
[n]
dSξ
=
∫
|ξ|=1
[(
σ−n(logθ P )
(
fλ(x), (f
∗
λ)
t(ξ)
))
1x
]
[n]
dSξ
=
∫
|ξ|=1
[(
σ−n(logθ P )
(
fλ(x), λ
−1ξ
))
1x
]
[n]
dSξ
= λn(2π)nω̃Res
logθ P
(fλ(x)).
Finally, equality (3.10) is obtained by combining the last two identities in (3.9). This gives the
statement. ■
3.3 The Wodzicki residue density versus a local Berezin type density
In this paragraph, we enhance the well-known algebraic identity (C.2) to a lesser known identity
of local densities on spin manifolds. Let now (M, g) be a spin manifold of even dimension n
and let ΣM be its spinor bundle. The morphism cg defined in (C.1) induces on a differential
operator P =
∑
|γ|≤m PγD
γ in Diff(M,ΣM) of order m, the operator cg(P ) given by
cg(P ) :=
∑
|γ|≤m
cg(Pγ)D
γ ∈ Diff(M,ΛT ∗M), (3.11)
14 G. Habib and S. Paycha
where we have used the identification Cℓ(TM)⊗C ≃ End(ΣM) as in Proposition C.1. Clearly,
the operator cg(P ) has the same order as P . In order to find the relation between ω̃Res and ωRes,
we need the following lemma:
Lemma 3.7. For any differential operator P ∈ Diff(M,ΣM) of Agmon angle θ and order m,
the operator cg(P ) has also an Agmon angle θ. Also, we have that
σ−j(logθ(c
g(P )))(x, ξ) = cg(σ−j(logθ(P ))(x, ξ)), ∀j ∈ Z≥0.
Proof. From the injectivity of cg we easily deduce that the set of eigenvalues of cg(σL(P )(x, ξ))
(resp. cg(P )) is a subset of the one of σL(P )(x, ξ) (resp. P ). Thus, an Agmon angle θ for P is
also one for cg(P ). The second part of the assertion can be proved along the same lines as the
proof of equation (3.7) with cg playing the role of U ♯
λ. ■
By choosing a = σ−n(logθ(P ))(x, ξ) in (C.2), for any differential operator P ∈ Diff(M,ΣM)
which is Z2-grading, we get that
Proposition 3.8. For any differential operator P ∈ Diff(M,ΣM) with Agmon angle θ, which
is even for the Z2-grading ΣM = Σ+M ⊕ Σ−M , we have
ω̃Res
logθ(c
g(P ))(x) = jg(x)(−2i)−n/2ωsRes
logθ(P )(x),
where jg(x) =
√
det(gij(x)).
Proof. Using Lemma 3.7 for j = n, we compute
ω̃Res
logθ(c
g(P ))(x) =
1
(2π)n
∫
|ξ|=1
[σ−n(logθ(c
g(P )))(x, ξ)1x][n]dSξ
=
1
(2π)n
∫
|ξ|=1
[cg(σ−n(logθ(P ))(x, ξ))1x][n]dSξ
=
1
(2π)n
∫
|ξ|=1
[sg(σ−n(logθ(P ))(x, ξ))][n]dSξ
=
1
(2π)n
∫
|ξ|=1
(T ◦ sg)(σ−n(logθ(P ))(x, ξ))e
1 ∧ · · · ∧ endSξ
(C.2)
=
jg(x)(−2i)−n/2
(2π)n
(∫
|ξ|=1
str(σ−n(logθ(P ))(x, ξ))dSξ
)
dx1 ∧ · · · ∧ dxn
= jg(x)(−2i)−n/2ωsRes
logθ(P )(x).
Here, we use the fact that P is Z2-graded, meaning that σ−j(logθ(P ))(x, ξ) is in End(Σ±
xM) ≃
Cℓ(TxM)+ ⊗ C and, thus, equation (C.2) is applied. ■
Remark 3.9. As a consequence of Proposition 3.8, for a differential operator P ∈ Ψcl(M,ΣM)
with Agmon angle θ, ω̃Res
logθ(c
g(P )) does define a global density since ωsRes
logθ(P ) does.
Corollary 3.10. For any differential operator P ∈ Diff(M,ΣM) of Agmon angle θ and order m
which is even for the Z2-grading ΣM = Σ+M ⊕ Σ−M , we have
ω̃Res
logθ(U
♯
λf
♯
λ(c
g(P )))
= (jg ◦ fλ)(−2i)−n/2ωsRes
logθ(P ) ◦ fλ.
Proof. By (3.10) applied to the differential operator cg(P ), we write
ω̃Res
logθ(c
g(P )) ◦ fλ = ω̃Res
logθ(U
♯
λf
♯
λ
(cg(P )))
.
The statement then follows from Proposition 3.8 at the point fλ(·). ■
A Pseudodifferential Analytic Perspective on Getzler’s Rescaling 15
4 The geometric set-up
In this section, we review the geometric set up underlying Getzler rescaling. Specifically, in the
language of [6, Section 1.1], we deform the manifold M to a manifold M via a deformation to
the normal cone to a given point p, and pull back the Riemannian metric g on the manifold
under the canonical projection π̂ : M → M to a family {gλ}λ>0 of dilated metrics (see (4.4)).
This family will play a crucial role when deforming operators.
4.1 Deformation to the normal cone to a point
For an embedding M0 ↪→M of two manifolds, the deformation to the normal cone is defined as
D(M0,M) := (M × R+) ∪ (NM0 × {0}),
where NM0 is the total space of the normal bundle to M0 in M . The deformation to the
normal cone extended to the embedding of the base of a groupoid into the groupoid gives rise to
the tangent groupoid introduced by Connes [5] which proves useful in the context of manifolds
with singularities. Here, choosing a reference point p ∈M fixed throughout the paper, we take
M0 = {p} so that NM0 = TpM , in which case the deformation amounts to replacing M by the
deformed manifold around p defined as
M := (M × R+) ∪ (TpM × {0}).
The gluing of the two parts, namely M ×R+ and TpM × {0} is carried out via the local diffeo-
morphism fλ described in (2.5) as follows. We build the map (denoted by Θ in [7, Section 3.1],
but here we adopt the notations of [18])
Expp : TpM × R ⊃ Bp,r −→ M
defined on
Bp,r := ({(x, λ) ∈ TpM × R+,x ∈ Br/λ}) ∪ (TpM × {0})
by the identity map on TpM × {0} and on the remaining part of Bp,r as follows:
Expp : {(x, λ) ∈ TpM × R+,x ∈ Br/λ} −→ Up × R+ ⊂ M,
(x, λ) 7−→ (expp(λx), λ), for λ > 0. (4.1)
We consider the open set in M [6, Section 1.1]
Wp := (Up × R+) ∪ (TpM × {0}) ⊂ M.
The deformed manifold M is endowed with the smooth structure for which Expp is a dif-
feomorphism, and which restricts to the standard smooth structure on M × R+ (we refer the
reader to [12, above Lemma 4.3] for further details). Via Expp the point (x = expp(x), λ) is
identified with the point fλ(x) = expp(λx) and the point p is identified with x. We refer to the
coordinates given by (4.1) as the λ-rescaled exponential coordinates. To recover the manifold M
from the deformed manifold M, we consider the projection map
π̂ : M p1−→M × R≥0
π−→M,
(x, λ) 7−→ (x, λ) 7−→ x if λ > 0,
(x, 0) 7−→ (p, 0) 7−→ p.
16 G. Habib and S. Paycha
With M endowed with the smooth structure described above, the map π̂ is smooth allowing
to pull-back the geometry on M to M. For any section s of a vector bundle E over M , its
pull-back is a section of the pull-back bundle E := π̂∗E ⊂ M× E over M given by
(π̂∗s)(x, λ) = s(x), ∀λ > 0, ∀x ∈M
and
(π̂∗s)(x, 0) = s(p), ∀x ∈ TpM.
In particular, the tangent bundle TM →M is pulled back to
π̂∗TM = {(x, λ, y, u) ∈M × R+ × TM | x = y}
∪ {(x, 0, y, u) ∈ TpM × {0} × TM | p = y}.
Also, the local diffeomorphism (4.1) induces the isomorphism of vector bundles (see [7, Re-
mark 3.4 (e)] and [9, pp. 67–68])
(TM × R+) ∪ (TpM × {0}) −→ TM,
(x, u, λ) 7−→ (x, λu = hλ(u), λ) if λ > 0,
(x, 0) 7−→ x ∈ TpM if λ = 0.
Now, in the local exponential chart (4.1) of M, the pull-back of a section s on E is the map
π̂∗s ◦Expp : Bp,r −→ E that can be read as
(π̂∗s ◦Expp)(x, λ) = (s ◦ π̂)(expp(λx), λ) = s(expp(λx))
for any λ > 0 and x ∈ Br/λ. Also, on the remaining part of Bp,r, we have
(π̂∗s ◦Expp)(x, 0) = (s ◦ π̂)(x, 0) = s(p).
Therefore, by taking x ∈ Br ⊂ Br/λ for λ > 0 small enough and identifying it with the point
x := exppx ∈ Up, we write that
(π̂∗s)(x, λ)
Expp
= (s ◦ fλ)(x),
for x ∈ Up.
4.2 Tensor bundles pulled back by π̂
Coming back to the deformation to the normal cone, the tensor bundle T q
rM := TM⊗q⊗T ∗M⊗r
is pulled back to π̂∗T q
rM −→ M and a tensor field t written in a normal geodesic coordinates
chart with coordinates X at a point x ∈ Up as
t(x) =
∑
t
i1...iq
j1...jr
∂
∂xi1
⊗ · · · ⊗ ∂
∂xiq
⊗ dxj1 ⊗ · · · ⊗ dxjr
∣∣∣∣
x
(4.2)
is pulled back to
(π̂∗t)(x, λ)
Expp
= t ◦ fλ(x)
=
∑(
t
i1...iq
j1...jr
◦ fλ
)( ∂
∂xi1
◦ fλ
)
⊗ · · · ⊗
(
∂
∂xiq
◦ fλ
)
⊗
(
dxj1 ◦ fλ
)
⊗ · · · ⊗
(
dxjr ◦ fλ
)∣∣∣∣
x
,
A Pseudodifferential Analytic Perspective on Getzler’s Rescaling 17
for small enough λ ≥ 0. Combining (2.6) with (3.1), we deduce that
(π̂∗t)(x, λ)
Expp
= λq−r(f∗λt)(x),
for any small enough λ > 0 and (π̂∗t)(x, 0)
Expp
= t(p), for λ = 0. Specialising to q = 0 and r = 2,
yields that the local description of the pull-back of the metric g on M , viewed as a covariant
two tensor is
(π̂∗g)(x, λ)
Expp
= λ−2(f∗λg)(x) (4.3)
for small enough λ > 0 and it is g(p) for λ = 0. It is therefore natural to introduce
gλ := λ−2f∗λg, (4.4)
so that at any point in Up, we have (gλ)ij (x) = gij(fλ(x)). As a consequence of the last identity
and with the help of the Koszul formula, the Christoffel symbols Γk
ij(·, g) := g
(
∇ ∂
∂xi
∂
∂xj ,
∂
∂xk
)
satisfy for any small enough positive λ
Γk
ij(·, gλ) = λΓk
ij(fλ(·), g).
Similarly, the Christoffel symbols Γ̃t
ls(·, g) := g(∇eles, et) read in an orthonormal frame obtained
by parallel transport along the geodesic curves, satisfy
Γ̃t
ls(·, gλ) = λΓ̃t
ls(fλ(·), g). (4.5)
5 Geometric differential operators
In this section, we define the notion of geometric polynomials with respect to a given met-
ric, as smooth sections (Definition 5.3) of a given vector bundle in terms of the corresponding
vielbeins (see Appendix A). To these polynomials, we assign an order called Gilkey order, in-
spired by Gilkey’s “order of jets” in the context of his invariance theory [11, Section 2.4], see
also [15, Section 3], both of which use jets of metrics. Whereas geometric polynomials are de-
fined in terms of the jets of the vielbeins and hence of the metric tensor, the Gilkey order does
not depend on the choice of metric. We call a differential operator geometric if its coefficients
written in a local trivialisation are geometric polynomials (Definition 5.8). In Proposition 5.14,
we show that a geometric polynomial with respect to g transforms under a contraction fλ to
a geometric polynomial with respect to gλ. In Proposition 5.15, we show a similar property for
a geometric differential operator.
5.1 Valuation of local sections
Let us recall some basic facts on the jets of a vector bundle. Given any vector bundle (E, π,M)
where π : E →M is the orthogonal projection, we let Γ(E) := C∞(M,E) be the vector space of
sections of E and Γp(E) be the stalk5 of local sections at a point p. Two local sections s and s′
in Γp(E) have the same r-jet (r ∈ Z+) at p if
(Dγs)|p = (Dγs′)|p
5Let SU denote the set of local sections of E defined on an open subset U of M containing p. The stalk of
local sections at the point p is the set of ∼ equivalence classes where for two elements sU in SU and sV in SV ,
sU ∼ sV if and only sU and sV coincide in some neighborhood of p.
18 G. Habib and S. Paycha
for any multiindex γ such that 0 ≤ |γ| ≤ r. The relation
s ∼ s′ ⇐⇒ s and s′ have the same r-jet at p
defines an equivalence relation and we denote by jrps the equivalence class of s. The integer r is
called the order of the jet. The set
Jr(E) :=
{
jrps | p ∈M, s ∈ Γp(E)
}
is a manifold, called the r-th jet manifold of π. The triple (Jr(E), πr,M) is a fiber bundle where
πr : J
r(E) →M ; jrps 7→ p and, in local coordinates,
jrps = (s(p), Dγs|p; 1 ≤ |γ| ≤ r),
which can be locally represented by the polynomial
∑
|γ|≤rD
|γ|s|pXγ (here in the variable X).
The reference vector bundle E will often be implicit only when needed shall we mention it.
Definition 5.1. Given a normal geodesic coordinate system X =
(
x1, . . . , xn
)
at a point p and
for a non negative integer r, the r-valuation of a local section s ∈ Γp(E) is defined by
valrX,p(s) = min
{
|γ| ≤ r,Dγ
Xs
∣∣
p
̸= 0
}
with the notation of (2.14) provided such a minimum exists. Otherwise, following the usual
convention we set valrX,p(s) = +∞. Correspondingly, we define the valuation of s as being
valX,p(s) = min
r∈Z≥0
valrX,p(s) = min
{
|γ| | Dγs
∣∣
p
̸= 0
}
∈ [0,+∞]. (5.1)
Example 5.2. We choose E = T ∗M ⊗ T ∗M , and view g as a section of E trivialised above Up
by means of normal geodesic coordinates on Up. In that trivialisation, the expansion of the
metric around a point p is given by
gij(x) = δij −
1
3
Riklj
∣∣∣∣
p
xkxl − 1
6
Riklj;m
∣∣∣∣
p
xkxlxm +O
(
|x|4
)
, (5.2)
where Riklj;m = (∇R)miklj . Therefore, in this trivialisation, the valuation of g− Id is at least 2.
In contrast, in the trivialisation of E obtained by parallel transport, the valuation of g − Id
is +∞.
Similarly the expansion of the inverse is given by
gij(x) = δij +
1
3
Riklj
∣∣∣∣
p
xkxl +
1
6
Rg
iklj;t
∣∣∣∣
p
xkxlxt +O
(
|x|4
)
, (5.3)
so that, in these coordinates and with a slight abuse of notation, the valuation of g−1 − Id is
at least 2. Combining equations (A.3) with (5.2) (resp. (A.2) with (5.3)) yields the following
expansions [1, equation (11)]
ali(x, g) = δil −
1
6
Rijkl(p)x
jxk − 1
12
∇tRijklx
jxkxt +O
(
|x|4
)
and
bil(x, g) = δil +
1
6
Rljki(p)x
jxk +
1
12
∇tRljkix
jxkxt +O
(
|x|4
)
.
Hence, in the same way as before, the valuation of A− Id (resp. B − Id, see Appendix A) is at
least 2 as well.
A Pseudodifferential Analytic Perspective on Getzler’s Rescaling 19
Finally, using the Koszul formula combining with (5.2) and the properties of the curvature
operator, the Christoffel symbols Γk
ij(·, g) = g
(
∇ ∂
∂xi
∂
∂xj ,
∂
∂xk
)
have the following Taylor expansion
at point p in the normal geodesic coordinates
Γk
ij(x, g) =
1
3
(Riklj(p) +Rilkj(p))x
l +O
(
|x|2
)
.
Also, the Christoffel symbols in an orthonormal frame Γ̃t
ls(·, g) = g(∇eles, et) have a similar
Taylor expansion
Γ̃t
ls(x, g) = −1
2
Rlist(p)x
i +O
(
|x|2
)
, (5.4)
which shows that both Christoffel symbols have valuation at least 1.
5.2 Polynomial expressions in the jets of the vielbeins
We consider a rank k vector bundle E → M equipped with an affine connection. We trivialise
the bundle E over an exponential neighborhood Up of p using geodesic normal coordinates(
x1, . . . , xn
)
at a point x in Up by identifying the fibre Ex above x = expp(x) ∈ Up with the
fibre Ep at point p via the parallel transport along geodesics c(t) = expp(tx), x ∈ TpM . We
fix a basis (s1(p), . . . , sk(p)) of Ep, which is then transported to (s1(x, g), . . . , sk(x, g)). In this
trivialisation, sections of E may be viewed as smooth functions on Up with valued in the fixed
fibre Ep.
Definition 5.3. We call a local section s of E over Up a geometric monomial (resp. polynomial)
with respect to some metric g, if when s =
∑k
j=1 αjsj is written in the local trivialisation sj(·, g),
j = 1, . . . , k of E above Up, the coordinates αj(·, g) are monomials (resp. polynomials) in the
jets of vielbeins Ap(·, g) and Bp(·, g) for the metric g (resp. linear combinations of monomials),
namely if they are (resp. linear combinations of) expressions of the form
Sj∏
q=1
Dβj
q
(
(aj)
tq
iq
(·, g)
)
Dγj
q
(
(bj)
lq
nq(·, g)
)
, (5.5)
such that
∑Sj
q=1
∣∣βjq ∣∣+∣∣γjq ∣∣ is independent of j. In this case, we shall write s(·, g) for an expression
of the type (5.5) and call ordGil(s) :=
∑Sj
q=1
∣∣βjq ∣∣+ ∣∣γjq ∣∣ its Gilkey order.
One observes that the notion of geometric polynomial and its Gilkey order is invariant under
transformations g 7→ f∗(eφg) of the metric g, where f is a diffeomorphism on M and φ is
a smooth function on M .
Remark 5.4. When E is a subbundle of the tensor bundle, we can alternatively trivialise it
over the exponential neighborhood Up of p using geodesic normal coordinates
(
x1, . . . , xn
)
at
a point in Up. By (A.1), we have ∂
∂xi =
∑n
ℓ=1 a
ℓ
i(·, g)eℓ(·, g) and dxi =
∑n
m=1 b
i
l(·, g)el(·, g),
where (e1(·, g), . . . , en(·, g)) is the basis of TM obtained by parallel transport of some (fixed)
orthonormal basis of TpM . Inserting these relations in (4.2) yields an expression of
t(·) =
∑
t
i1...iq
j1...jr
P
ℓ1...ℓq
i1...iq
Q
j1...jq
m1...mreℓ1(·, g)⊗ · · · ⊗ eℓq(·, g)⊗ em1(·, g)⊗ · · · ⊗ emr(·, g),
where P
ℓ1...ℓq
i1...iq
and Q
j1...jq
m1...mr are linear combinations of expressions of the form (5.5). Thus, the
coordinates t
i1...iq
j1...jr
of t in (4.2) are linear combinations of expressions of the form (5.5) if and
only if its coordinates t̃
ℓ1...ℓq
m1...mr in the basis eℓ1(·, g) ⊗ · · · ⊗ eℓq(·, g) ⊗ em1(·, g) ⊗ · · · ⊗ emr(·, g)
are also linear combinations of expressions of the form (5.5). Consequently, we can use either
trivialisation in this case.
20 G. Habib and S. Paycha
Here are first examples of geometric polynomials.
Example 5.5.
1. Take E = T ∗M ⊗s T
∗M . The metric g, which is a local section of E, is a geomet-
ric monomial with respect to the metric g of Gilkey order zero since its coordinates in
the basis dxi ⊗ dxj induced by the normal geodesic coordinates x1, . . . , xn read gij(·) =∑n
l=1 a
l
i(·, g)alj(·, g) (see equation (A.2)). So is its inverse g−1 a geometric monomial of
Gilkey order zero since gij(·) =
∑n
l=1 b
i
l(·, g)b
j
l (·, g) (see equation (A.3)).
2. Take E = T ∗M ⊗ T ∗M ⊗ TM . The Christoffel symbol which is a local section of E,
is a geometric polynomial with respect to the metric g since its coordinates Γk
ij(·, g) in
the basis dxi ⊗ dxj ⊗ d
dxk induced by the normal geodesic coordinates x1, . . . , xn read by
Koszul’s formula
Γk
ij(·, g) =
n∑
l=1
gkl(·) (∂xi(gjl(·)) + ∂xj (gil(·))− ∂xl(gij(·))) , (5.6)
is a polynomial in the jets of vielbeins of Gilkey order one.
3. Similarly, the Christoffel symbols Γ̃t
ls(·, g) = g(∇eles, et) written in the orthonormal frame
(e1(·, g), . . . , ek(·, g)) obtained by parallel transport as in Remark 5.4 read as (use Einstein
convention)
Γ̃t
ls(·, g) = bil(·, g)b
j
t (·, g)bks(·, g)Γ
j
ik(·, g) + bil(·, g)b
j
t (·, g)∂xi
(
bkt (·, g)
)
gkj(·), (5.7)
are polynomials in the jets of the vielbeins of Gilkey order one.
Remark 5.6. Since jets are compatible with composition and differentiation, geometric mono-
mials form an algebra stable under differentiation.
Remark 5.7. Due to equations (A.2) and (A.3) which relate the metric to the vielbeins, the
class of polynomials we single out in Example 5.5, is consistent with the classes of polynomials
in the jets of the metric considered in [2], [8, Theorem 1.2] and [11, equation (2.4.3)]. There,
the polynomials depend on the metric tensor, its inverse – or its inverse determinant, see [2,
formula, item 1, p. 282] – and the derivatives of the metric tensor.
5.3 Geometric operators
In this subsection, we define geometric differential operators on vector bundles based on the
definition of geometric polynomials.
Definition 5.8. Let E be a vector bundle overM of finite rank equipped with an affine connec-
tion. We call a differential operator P =
∑
|γ|≤m PγD
γ in Diff(M,E) of order m geometric with
respect to a metric g if its coefficients Pγ(x) ∈ End(Ex), written in the basis obtained by parallel
transport of some fixed basis of Ep, are geometric polynomials in the jets at x of vielbeins such
that for all γ
ordGi(Pγ(x)) = ord(P )− |γ|. (5.8)
Remark 5.9. As for geometric sections, when E is a subbundle of the tensor bundle, we
can alternatively trivialise it over the exponential neighborhood Up of p using geodesic normal
coordinates
(
x1, . . . , xn
)
at a point in Up.
Example 5.10. For E = ΛT ∗M , resp. E = ΣM , for any X ∈ TM , the covariant differentia-
tion ∇X defines a geometric operator with respect to g of order 1.
A Pseudodifferential Analytic Perspective on Getzler’s Rescaling 21
1. When E = ΛT ∗M , we express the covariant derivative on a differential form α =
∑
I αIdx
I
of degree k in normal geodesic coordinates
(
x1, . . . , xn
)
around p ∈ M as follows (here
I = {i1 < · · · < ik})
∇ ∂
∂xi
α =
∑
I
(
∂
∂xi
αI
)
dxI
+
∑
t,I
αI
(
n∑
s,l=1
g
(
∇ ∂
∂xi
dxis , dxl
)
gtl(·)
)
dxi1 ∧ · · · ∧ dxt︸︷︷︸
sth-slot
∧ · · · ∧ dxik .
This shows that ∇ ∂
∂xi
is a geometric differential operator with respect to the metric g of
order 1 whose zero-th order part
∑n
m,l=1 g
(
∇ ∂
∂xi
dxis ,dxl
)
gtl(·) has coefficients given by
linear combinations of monomials (5.5) with Gilkey order 1.
2. When E = ΣM is equipped with the spin connection induced by the Levi-Civita connec-
tion, the corresponding End(ΣpM)-valued functions
ei·g : σ 7→ ei ·g σ, i = 1, . . . , n
on Up are constant along the geodesics and hence in the trivialisation induced by parallel
transport [3, Lemma 4.14]. In the normal geodesic coordinates x1, . . . , xn on Up, the
Clifford multiplication operators dxi·g are geometric operators since by (A.1) they read
dxi·g =
∑n
l=1 b
i
l(·, g)el·g.
3. When E = ΣM , the spinorial connection ∇ ∂
∂xi
acting on smooth functions from Up to Ep
reads
∇ ∂
∂xi
=
∂
∂xi
+
1
4
∑
s,t
g
(
∇ ∂
∂xi
es, et
)
︸ ︷︷ ︸
Γ̃t
is(·,g)
es ·g et·g (5.9)
and therefore defines a geometric differential operator with respect to g of order 1. Indeed,
the coefficients Γ̃t
is(·, g) are smooth real functions from Up and, by (5.7), are polynomials
in the jets of the vielbeins of Gilkey order 1.
Proposition 5.11. The product of geometric differential operators with respect to the metric g
is a geometric differential operator.
Proof. Indeed, let
P :=
∑
|γ|≤ord(P )
PγD
γ and Q :=
∑
|δ|≤ord(Q)
QδD
δ
be two geometric operators. Their composition reads
PQ =
∑
δ
|γ1|+|γ2|=|γ|
Pγ(D
γ1Qδ)D
γ2+δ.
We easily check that
ordGi(Dγ1Pδ) = ordGi(Pδ) + |γ1|,
so
ordGi(Pγ(D
γ1Qδ)) + |γ2 + δ| = ordGi(Pγ) + ordGi(Qδ) + |γ1|+ |γ2|+ |δ|
= ordGi(Pγ) + ordGi(Qδ) + |γ|+ |δ|
= ord(P ) + ord(Q) = ord(PQ).
This finishes the proof. ■
22 G. Habib and S. Paycha
Remark 5.12. In [13], the authors assign a Getzler order to linear partial differential opera-
tors D acting on the smooth sections of the spinor bundle ΣM over M which can be expressed
as a finite sum of operators of the form f ·D1 · · ·Dp where f is a smooth function and each Dj is
either a covariant derivative ∇X , or a Clifford multiplication operator X·g, or the identity opera-
tor. In our terminology, such an operator is geometric as the product of geometric operators ∇X
and X·g (see Example 5.10). We will later see in Example 6.8 that the exterior differential d is
a geometric operator but not of the form f ·D1 · · ·Dp. Also, the notion of geometric operator
generalises to classical pseudodifferential operators in requiring a condition similar to (5.8) on
the homogeneous components of the symbol, see [15].
In the following, we consider again a vector bundle E over M of rank k equipped with
a connection ∇.
Lemma 5.13. Let E be a vector bundle over M of rank k and let s(p) ∈ Ep, for some fixed
p ∈ M . We denote by s(·, g) the section in Γ(E) obtained by parallel transport of s(p) along
the exponential curve c(t) = expp(tx) corresponding to the metric g with x ∈ Br ⊂ TpM . We
also denote by s(·, gλ) the section obtained by parallel transport of s(p) along the exponential
curve γ(t) corresponding to the metric gλ. Then, we have
s(·, gλ) = s(fλ(·), g).
Proof. First, we notice that if c(t) = expp(tx) is the exponential curve corresponding to the
metric g with x ∈ Br ⊂ TpM , then the curve γ : I → Up given by
γ(t) := fλ ◦ c(t) = expp(tλx)
is the exponential curve associated with the metric f∗λg as well for the metric gλ. The sec-
tion s(·, g)◦ fλ is parallel along the curve γ(t) as a direct consequence from the fact that s(·, g) is
parallel on E along the curve c(t). Now the initial condition and the uniqueness of the parallel
transport allow to deduce the result. ■
Proposition 5.14. Let E be a vector bundle over M of rank k. Let s be a local section of E
which is a geometric monomial (resp. polynomial) with respect to the metric g of Gilkey order
ordGi(s). The local section f∗λs is a geometric monomial (resp. polynomial) with respect to the
metric gλ of the same Gilkey order ordGi(s).
Proof. Since the local section s is a geometric monomial with respect to the metric g, it can be
written as s(·, g) =
∑k
j=1 αj(·, g)sj(·, g) where αj(·, g) is a monomial in the jets of the vielbeins.
The section f∗λs is equal to f∗λs =
∑k
j=1 αj(fλ(·), g)sj(fλ(·), g). In order to express f∗λs in terms
of the metric gλ, we first differentiate the relations from Lemma A.1 in Appendix A,
ali(·, gλ) = ali(fλ(·), g) and bil(·, gλ) = bil(fλ(·), g),
to get that αj(·, gλ) = λord
Gi(s)αj(fλ(·), g). On the other hand, by Lemma 5.13, we have that
sj(·, gλ) = sj(fλ(·), g) for all j. Therefore, we deduce that
f∗λs = λ−ordGi(s)
k∑
j=1
αj(·, gλ)sj(·, gλ).
That means f∗λs is a geometric monomial with respect to the metric gλ and that s and f∗λs have
the same Gilkey order. ■
As a direct consequence, we get the following result on geometric differential operators.
A Pseudodifferential Analytic Perspective on Getzler’s Rescaling 23
Proposition 5.15. Let E be a vector bundle over M of finite rank equipped with an affine
connection. Let P be a differential operator in Diff(M,E) of order m geometric with respect to
the metric g. The differential operator f♯λP in Diff(M,E) of order m is geometric with respect
to the metric gλ.
Proof. If P =
∑
|γ|≤m PγD
γ , then f♯λP =
∑
|γ|≤m f∗λPγf
♯
λD
γ . Since by assumption, the coeffi-
cients Pγ are geometric polynomials with respect to the metric g, it follows from the previous
proposition that their pull-backs f∗λPγ are geometric polynomials with respect to the metric gλ
with the same Gilkey order as those of the Pγ ’s. Since the order of f
♯
λP coincides with that of P ,
for any multi-index γ we have
ordGi(f∗λPγ) = ordGi(Pγ) = ord(P )− |γ| = ord
(
f♯λP
)
− |γ|,
so that f♯λP satisfies equation (5.8). ■
6 Getzler rescaled geometric differential operators
In this section, we focus on geometric differential operators P in Diff(M,E) for E = ΣM , resp.
E = ΛT ∗M given in Definition 5.8. We study their behaviour under the combined action of
a contraction fλ and the map Uλ defined in (3.2) as well as the limit as λ tends to zero of the
resulting operator. For that purpose we introduce the operators P̃Ge|(·,λ) in (6.1), resp. in (6.2),
where the superscript Ge stands for Getzler. We call P rescalable if P̃Ge|(·,λ) admits a limit P̃lim
when λ tends to zero. In Proposition 6.6 in the case E = ΛT ∗M , resp. in Proposition 6.9 in the
case E = ΣM , we give a necessary and sufficient condition for P to be rescalable and show that
the coefficients of the limit operator P̃lim are polynomials in the jets of the curvature tensor onM .
For P in Diff(M,ΛT ∗M), resp. in Diff(M,ΣM), we give in Theorem 6.7, resp. Corollary 6.10
(for an even order operator) a localisation formula similar to the one in Proposition 2.5. This
time instead of the local residue form ωRes
logθ(P̃lim)
, our localisation formula involves the local
n-form ω̃Res
logθ(P̃lim)
.
In the same spirit as (2.17) and (2.18), we set for P ∈ Diff(M,ΛT ∗M) of order m
PGe
∣∣
(·,λ) := λmU ♯
λP
∣∣
(·) and P̃Ge
λ := f♯λP
Ge, (6.1)
resp. for P ∈ Diff(M,ΣM)
PGe
∣∣
(·,λ) := λm
(
U ♯
λ ◦ cg(P )
)∣∣
(·) and P̃Ge
λ := f♯λP
Ge, (6.2)
where U ♯
λP and cg(P ) are given in equations (3.6) and (3.11), respectively.
Definition 6.1. We call a differential operator P in Diff(M,E) with E = ΛT ∗M , resp. E = ΣM
rescalable at a point p if and only if P̃Ge
λ
∣∣
Up
introduced in (6.1), resp. in (6.2) admits a limit P̃lim
when λ goes to zero.
Remark 6.2. Note that rescalability is a local notion valid at a point, and is defined via a local
normal geodesic coordinates.
Proposition 6.3. Rescalable operators in Diff(M,E) at point p for E = ΛT ∗M , resp. E = ΣM ,
form a subalgebra.
Proof. Let P1, P2 be two operators in Diff(M,ΛT ∗M) of order m1 and m2, respectively. Since
the order is additive on products of operators and the degree is also additive on wedge products
of forms, we have
λm1+m2f♯λU
♯
λ(P1P2) =
(
λm1f♯λU
♯
λ(P1)
)(
λm2f♯λU
♯
λ(P2)
)
.
24 G. Habib and S. Paycha
If P1 and P2 are rescalable, the limits as λ→ 0 exist on the right-hand side, and hence so do they
on the left-hand side, which shows that the product P1P2 is rescalable. Replacing U ♯
λ by U ♯
λ ◦ c
g
and using the fact that cg is an algebra morphism yields the result for E = ΣM . ■
The following technical lemma will be useful.
Lemma 6.4. In local normal geodesic coordinates X, and with the notations of (5.1), let q :=
valX,p(h) be the valuation of a local section h ∈ Γp(E) around p, where E is the trivial bundle
E :=M × R →M and p ∈M .
1. For any real number θ, as λ tends to zero, the expression λ−θDγ(h ◦ fλ)
� converges if and only if θ ≤ max(|γ|, q);
� if θ < max(|γ|, q), it converges to zero.
2. If θ = max(|γ|, q), the expression λ−θDγ(h◦ fλ) converges to the coefficient of order θ−|γ|
in the Taylor expansion of h at point p.
Proof. The proof is based on the fact that λ−θDγ(h◦ fλ) = λ|γ|−θ(Dγh)◦ fλ. By definition of q,
we have h = O(|x|q) so that for q ≥ |γ|, we have (Dγh) ◦ fλ = λq−|γ|O
(
|x|q−|γ|). If, q < |γ|, we
have (Dγh) ◦ fλ = O(1), which ends the proof of (1) and (2) observing that in the convergent
case, the limit corresponds to the θ − |γ| coefficient in the Taylor expansion. ■
We now specialise to a monomial say P in the jets of the vielbeins as in (5.5) written in
normal geodesic coordinates. We set
ΘX,p
(x,g)(P ) :=
∑
s
max
(
|βs|, valX,p
(
atsis(x, g)
))
+max
(
|γs|, valX,p
(
blsns
(x, g)
))
.
In the following, we shall often drop the explicit mention of X, p, x, g and simply write Θ(P ).
Example 6.5.
1. Recall that gij(x) =
∑
l a
l
i(x, g)a
l
j(x, g). We have Θ(gij) = 0 if i = j and it is at least 2
otherwise.
2. Using the Koszul formula (5.6), the Christoffel symbols Γk
ij(x, g) can be written Γk
ij(x, g) =∑
l P
l
i,j,k(x, g) with Θ
(
P l
i,j,k
)
= 1 for l = i = j = k and at least 2 otherwise.
3. Similarly, by relation (5.7) the Christoffel symbols Γ̃t
ls(x, g) can be written
Γ̃t
ls(x, g) =
∑
k
P k
l,s,t(x, g)
with Θ
(
P k
l,s,t
)
≥ 2. Notice here that Θ
(
P k
l,s,t
)
cannot be equal to 1, since this corresponds
to l = s = t = k which would imply Γ̃t
ls(x, g) = Γ̃s
ls(x, g) = 0 and hence would yield
a contradiction.
Proposition 6.6. Let P ∈ Diff(M,ΛT ∗M) be a geometric differential operator of order m with
respect to a metric g. In a local trivialisation around a point p of ΛT ∗M induced by normal
geodesic coordinates
(
x1, . . . , xn
)
on Up, the operator P applied to a section s =
∑
I αIdx
I(
dxI := dxi1 ∧ · · · ∧ dxi|I| for I = {i1, . . . , i|I|}
)
reads
P
(∑
I⊂N
αIdx
I
)
=
∑
|γ|≤m
∑
I,J⊂N
(Pγ)IJ(·, g)Dγ(αI)dx
J ,
A Pseudodifferential Analytic Perspective on Getzler’s Rescaling 25
where (Pγ)IJ(·, g) are polynomials as in (5.5). The operator P is rescalable at a point p if and
only if |J | − |I| ≤ Θ((Pγ)IJ). In this case, the limit rescaled operator reads
P̃lim =
∑
|γ|≤m
∑
|J |−|I|=Θ((Pγ)IJ )
(Pγ)
lim
IJ (p, g)
((
dxI
)∗ ⊗ dxJ
)
Dγ ,
where
(Pγ)
lim
IJ (p, g) = lim
λ→0
(
λ|I|−|J |(Pγ)IJ(·, gλ)
)
, ∀I, J ⊂ N
is a polynomial expression in the jets of the Riemannian curvature tensor.
Proof. The local expression of P in the theorem results from Remark 5.4, equations (2.16)
and (4.2). Hence, from the definition of U ♯
λP = UλPU
−1
λ , we get(
U ♯
λP
)(∑
I⊂N
αIdx
I
)
=
∑
|γ|≤m
∑
I,J⊂N
λ|I|−|J |(Pγ)IJ(·, g)Dγ(αI)dx
J .
Since P is a geometric differential operator, the coefficients (Pγ)IJ(·, g) are in the jets of the
vielbeins as in (5.5) with ordGi((Pγ)IJ) + |γ| = m. Now, we write
P̃Ge
λ (s ◦ fλ) = λm
(
f♯λU
♯
λP
)
(s ◦ fλ) = λm
(
U ♯
λP
)
(s) ◦ fλ
=
∑
|γ|≤m
∑
I,J⊂N
λm+|I|−|J |(Pγ)IJ(fλ(·), g)Dγ(αI)dx
J
∣∣
fλ(·)
=
∑
|γ|≤m
∑
I,J⊂N
λ|γ|+|I|−|J |(Pγ)IJ(·, gλ)λ−|γ|Dγ(αI ◦ fλ)dxJ ◦ fλ. (6.3)
In the last equality, we use the fact that (Pγ)IJ(·, gλ) = λord
Gi((Pγ)IJ )(Pγ)IJ(fλ(·), g). Hence, we
deduce that
P̃Ge
λ =
∑
|γ|≤m
∑
I,J⊂N
λ|I|−|J |(Pγ)IJ(·, gλ)
((
dxI
)∗ ⊗ dxJ
)∣∣
fλ
Dγ .
Now by Lemma A.1 in Appendix A, we write for (Pγ)IJ(·, gλ)
S∏
s=1
Dαs
(
atsis(·, gλ)
)
Dβs
(
blsns
(·, gλ)
)
=
S∏
s=1
Dαs
(
atsis(·, g) ◦ fλ
)
Dβs
(
blsns
(·, g) ◦ fλ
)
.
For convenience, we have dropped the explicit mention of the indices I and J . Applying
Lemma 6.4 to ha := atsis(·, g) and hb := blsns
(·, g) with both θa and θb non negative integers
such that θa + θb = |J | − |I|, it tells us that the expression λ−θaDγ(f∗λha) converges if and only
if θa ≤ max(|αs|, qa), with qa := valX,p
(
atsis(x, g)
)
and that the limit vanishes if we have a strict
inequality. If θa = max(|αs|, qa), the limit is a polynomial in the jets of the curvature tensor.
Similarly for hb and θb. Hence, the only non zero terms which survive in the limit of (6.3) as
λ→ 0, correspond to θa = max(|αs|, qa) and θb = max(|βs|, qb) and hence Θ((Pγ)IJ) = |J | − |I|.
This yields the statement of the theorem. ■
We prove a localisation formula for the local form ω̃Res
logθ(P ) when E = ΛT ∗M .
Theorem 6.7. Let P in Diff(M,ΛT ∗M) be a geometric differential operator with respect to the
metric g of Agmon angle θ. If P is rescalable, then
ω̃Res
logθ(P )(p) = ω̃Res
logθ(P̃lim)
(x), ∀x ∈ Up,
where P̃lim := limλ→0 P̃Ge
λ .
26 G. Habib and S. Paycha
Proof. Combining equation (6.1) with equation (3.10) yields
ω̃Res
logθ(P )(fλ(x)) = ω̃Res
logθ(U
♯
λ(f
♯
λP ))
(x) = ω̃Res
logθ(P̃Ge
λ )
(x),
where we have commuted U ♯
λ and f♯λ thanks to equation (3.8). By Proposition 6.6, the limit
P̃lim := limλ→0 P̃Ge
λ exists, from which we deduce the statement of the theorem by letting λ tend
to zero in the above identities. ■
Example 6.8. The exterior differential d : ΛT ∗M → ΛT ∗M
(
as well as its L2-adjoint δ
)
is
not a rescalable operator. Indeed, by writing d =
∑n
j=1 dx
j∧∇ ∂
∂xj
and using the local expression
of ∇ ∂
∂xj
in Example 5.10, we have for any α =
∑
I αIdx
I that
d
(∑
I
αIdx
I
)
=
∑
j,I
(
∂
∂xj
αI
)
dxj ∧ dxI
+
∑
t,I
αI
(
n∑
s,l=1
g
(
∇ ∂
∂xj
dxis , dxl
)
gtl(·)
)
dxj ∧ dxi1 ∧ · · · ∧ dxt︸︷︷︸
sth-slot
∧ · · · ∧ dxik ,
which shows that it is a geometric operator. For the j’s that do not belong to I in the first
sum of the right-hand side, we have |J | − |I| = 1 and the corresponding Θ = 0. Therefore, the
condition in Proposition 6.6 is not fulfilled and, thus, d is not rescalable. However, the Hodge
operator ∆ = dδ + δd is a geometric rescalable operator. The fact that it is geometric comes
from Proposition 5.11. To show that it is rescalable, we use the Bochner–Weitzenböck formula
on k-forms: ∆ = ∇∗∇+W [k], where ∇∗∇ is given by
∇∗∇ = −gij(x)
(
∇ ∂
∂xi
∇ ∂
∂xj
− Γk
ij(x, g)∇ ∂
∂xk
)
andW [k] =
∑n
i,j=1 e
∗
j∧(ei⌟R(ei, ej)) is the Bochner operator. Here R is the curvature operator of
the manifoldM . Indeed, by replacing∇ ∂
∂xi
by its expression and performing some computations,
one can easily see that ∆ (we use Einstein convention) has the form
∆
(
αIdx
I
)
= − gij
∂
∂xi
∂
∂xj
(αI)dx
I
− gij
∂
∂xj
(αI)Γ
k
iisgtkdx
i1 ∧ · · · ∧ dxt︸︷︷︸
sth-slot
∧ · · · ∧ dxik + · · · .
Since the Laplacian preserves the degree, then |J | − |I| = 0 ≤ Θ is always satisfied. The limit of
the rescaled operator corresponds to polynomials with Θ = 0. Hence by the computation of Θ
in Example 6.5, we get that
P̃lim
(∑
I
αIdx
I
)
= −
∑
i,I
∂2αI
(∂xi)2
dxI .
Therefore, the localisation formula in Theorem 6.7 can be applied for the Hodge Laplacian
and we get
ω̃Res
logθ(∆)(p) = ω̃Res
logθ(P̃lim)
(x), ∀x ∈ Up.
Proposition 6.9. Let P ∈ Diff(M,ΣM) be a geometric differential operator with respect to the
metric g of order m. In the trivialisation
{
e1, . . . , en
}
induced by parallel transport, the operator
reads
P =
∑
|γ|≤m
∑
I
(Pγ)I(·, g)eI ·g Dγ , (6.4)
A Pseudodifferential Analytic Perspective on Getzler’s Rescaling 27
where (Pγ)I(·, g) are polynomials as in (5.5) and eI ·g = ei1 ·g · · · ·g eik with i1 < · · · < ik and
|I| = k. The operator P is rescalable if and only if |I| ≤ Θ((Pγ)I). In this case, the limit rescaled
operator in (6.2) reads
P̃lim =
∑
|γ|≤m
∑
|I|=Θ((Pγ)I)
(Pγ)
lim
I (p, g)eI ∧Dγ ,
where (Pγ)
lim
I (p, g) := limλ→0
(
λ−|I|(Pγ)I(·, gλ)
)
is a polynomial expression in the jets of the
Riemannian curvature tensor.
Proof. In any local trivialization of ΣM , the operator P can be written as
P =
∑
|γ|≤m
Pγ(·, g)Dγ ,
where Pγ(·, g) ∈ End(ΣM) ≃ Cℓ(TM) ⊗ C. In the local trivialisation above a normal geodesic
chart induced by parallel transport, we write
Pγ(·, g) =
∑
I
(Pγ)I(·, g)eI .
Recall here that the End(ΣpM)-valued functions ei·g : σ 7→ ei ·g σ for each i = 1, . . . , n on Up
are constant along the geodesics and hence in this trivialisation [3, Lemma 4.14]. Now, as P is
geometric, we get that (Pγ)I(·, g) are in the jets of the vielbeins as in (5.5). Hence, we get (6.4).
In particular, we deduce that
cg(P ) =
∑
|γ|≤m
∑
I
(Pγ)I(·, g)cg
(
eI
)
Dγ .
Now, we apply equation (6.2) to a section s to get
P̃Ge
λ (s ◦ fλ) = λm
(
f♯λU
♯
λc
g(P )
)
(s ◦ fλ) = λm
(
U ♯
λc
g(P )s
)
◦ fλ
=
∑
|γ|≤m
∑
I⊂N
λm(Pγ)I(fλ(·), g)U ♯
λ
(
cg
(
eI
))
(Dγs)
∣∣
fλ(·)
.
=
∑
|γ|≤m
∑
I⊂N
λm−|I|(Pγ)I(fλ(·), g)λ|I|U ♯
λ
(
cg
(
eI
))
λ−|γ|Dγ(s ◦ fλ)
=
∑
|γ|≤m
∑
I⊂N
λ−|I|(Pγ)I(·, gλ)λ|I|U ♯
λ
(
cg
(
eI
))
Dγ(s ◦ fλ).
Here, we use the fact that
(Pγ)I(·, gλ) = λord
Gi((Pγ)I)(Pγ)I(fλ(·), g) and ordGi((Pγ)I) + |γ| = m,
since (Pγ)I(·, g) are in the jets of the vielbeins. Therefore, we deduce that
P̃Ge
λ =
∑
|γ|≤m
∑
I⊂N
λ−|I|(Pγ)I(·, gλ)λ|I|U ♯
λ
(
cg
(
eI
))
Dγ .
Now, using (3.4), we have that λ|I|U ♯
λ
(
cg
(
eI
))
converges to eI∧ as λ→ 0. Also, by Lemma 6.4,
the term λ−|I|(Pγ)I(·, gλ) converges if and only if |I| ≤ Θ((Pγ)I). Thus, the operator P is
rescalable if and only if |I| ≤ Θ((Pγ)I). The limit rescaled operator follows then easily. ■
We prove a localisation formula for the local form ω̃Res
logθ(P ) when E = ΣM .
28 G. Habib and S. Paycha
Corollary 6.10. Let P in Diff(M,ΣM) be a geometric differential operator with respect to the
metric g of Agmon angle θ which is even for the Z2-grading ΣM = Σ+M ⊕ Σ−M . If P is
rescalable, then we have
ω̃Res
logθ(P̃lim)
(x) = (−2i)−n/2ωsRes
logθ(P )(p), ∀x ∈ Up,
where P̃lim := limλ→0 P̃Ge
λ with P̃Ge
λ as in (6.2).
Proof. Using the relation in Corollary 3.10, the fact that P is rescalable and that jg ◦ fλ → 1
as λ→ 0 yield the result. ■
7 The rescaled square of the Dirac operator
In this section, we show that whereas the Dirac operator /D (which is a geometric operator and
hence so its square) is not rescalable, its square is. We then apply the results of the previous
section to P = /D
2
and compute P̃lim with the help of Proposition 6.9. This allows to find the
expression of P̃lim in terms of the curvature operator of M as in [10]. We then derive from
Corollary 6.10 a localisation formula (7.1) for the graded residue of the logarithm of /D
2
.
We recall that the Dirac operator on a spin manifold (Mn, g) is the differential operator of
order one given by /D :=
∑n
i=1 e
i ·g∇ei , where ∇ei is the spinorial Levi-Civita covariant derivative
in the direction of ei. Using (5.9), it reads as
/D =
∑
i=l
ali(·, g)ei ·g ∂xm +
∑
i ̸=l
ali(·, g)ei ·g ∂xl +
1
4
n∑
i,l,s,t=1
Γ̃t
ls(·, g)ali(·, g)ei ·g es ·g et ·g .
The Dirac operator is geometric. Indeed, the above expression involves a sum of three terms,
each of which is expressed in terms of jets of vielbeins
(
see Example 5.5 for Γ̃
)
and satisfies
condition (5.8). Yet it is not rescalable. With the notations of Proposition 6.9 with P = /D,
the condition |I| ≤ Θ((Pγ)I) is not satisfied in the first of the three sums since |I| = 1 and
Θ((Pγ)I) = 0. Recall that the valuation of ali(·, g) is zero if l = i and at least 2 otherwise.
Proposition 7.1. The square of the Dirac operator /D
2
is a rescalable geometric differential
operator. Setting P := /D
2
, the operator P̃Ge
λ in (6.2), read in a local trivialisation of ΣM at
point x in Up obtained by parallel transport along geodesics, converges to
−
∑
i
(
∂xi −
1
8
∑
j,s,t
Rijst(p)x
jes ∧ et∧
)2
,
where {e1(p), . . . , en(p)} is an orthonormal frame TpM .
Proof. Since /D is geometric so is its square /D
2
by Proposition 5.11. We now show it is
rescalable. Since the action by Clifford multiplication ei·g of the vectors of an orthonormal
frame of TxM obtained from {e1(p), . . . , en(p)} by parallel transport is constant in x, in the
following we will simply write {e1, . . . , en}. We use the Schrödinger–Lichnerowicz formula [14]
to write
/D
2∣∣
x
= −
n∑
i,j=1
gij(x)
(
∇xi∇xj − Γk
ij(x, g)∇xk
)
+
1
4
Scal(x)
= −
n∑
i,j=1
n∑
k,l=1
gij(x)
(
∂xi +
1
4
Γ̃l
ik(x, g)e
k ·g el·g
) n∑
s,t=1
(
∂xj +
1
4
Γ̃t
js(x, g)e
s ·g et·g
)
A Pseudodifferential Analytic Perspective on Getzler’s Rescaling 29
+
n∑
i,j,k=1
gij(x)Γk
ij(x, g)
n∑
s,t=1
(
∂xk +
1
4
Γ̃t
ks(x, g)e
s ·g et·g
)
+
1
4
Scal(x)
= −
n∑
i,j=1
gij(x)∂2xixj︸ ︷︷ ︸
(I)
−
n∑
i,j=1
1
4
gij(x)
n∑
s,t=1
∂xi
(
Γ̃t
js(x, g)
)
es ·g et·g︸ ︷︷ ︸
(II)
−
n∑
i,j,s,t=1
1
2
gij(x)Γ̃t
js(x, g)e
s ·g et ·g ∂xi︸ ︷︷ ︸
(III)
− 1
16
n∑
i,j=1
∑
k ̸=l,s ̸=t
gij(x)Γ̃l
ik(x, g)Γ̃
t
js(x, g)e
k ·g el ·g es ·g et·g︸ ︷︷ ︸
(IV)
+
n∑
i,j,k=1
gij(x)Γk
ij(x, g)
(
∂xk +
1
4
n∑
s,t=1
Γ̃t
ks(x, g)e
s ·g et·g
)
︸ ︷︷ ︸
(V)
+
1
4
Scal(x),
where Scal is the scalar curvature of the metric g. Combining equations (A.2) and (5.7) with the
Koszul formula, we can express /D
2
in terms of the vielbeins. To avoid lengthy computations,
we only sketch the computation for the third term in the above equation to show that the
relation |I| ≤ Θ((Pγ)I) holds. Thus, the operator is rescalable. To show the inequality, we first
observe that |I| equals 2. According to Examples 6.5, the coefficient in (III) can be written
as a sum of polynomials of the vielbeins such that the corresponding Θ is at least 2
(
since the
one corresponding to Γ̃l
im(x, g) is at least 2
)
. With the help of (4.5) and (5.4), the limit of the
rescaled operator of (III) is equal to
lim
λ→0
λ−2
∑
i,j,s,t
gijλ (x)Γ̃
t
js(x, gλ)
(
es ∧ et∧
)
∂xi = −1
2
∑
i,j,k,s,t
δijRjkst(p)x
k
(
es ∧ et∧
)
∂xi .
The same thing can be done for the first term (I) which converges to −
∑
i ∂
2
xi . The second
term (II) converges to 1
8
∑
i,j,s,t δ
ijRijst(p)e
s ∧ et ∧ . The fourth term (IV) tends to
− 1
64
∑
i,j,k,l,s,t,q
δijRiqkl(p)Rjlst(p)x
qxlek ∧ el ∧ es ∧ et ∧ .
The other terms converge to 0. Therefore, we deduce that
lim
λ→0
P̃Ge
λ = −
∑
i
∂2xi +
1
8
∑
i,j,s,t
δijRijst(p)e
s ∧ et +
1
4
∑
i,j,k,s,t
δijRjkst(p)x
k
(
es ∧ et∧
)
∂xi
− 1
64
∑
i,j,k,l,s,t,q
δijRiqkl(p)Rjlst(p)x
qxlek ∧ el ∧ es ∧ et∧
= −
∑
i
(
∂xi −
1
8
∑
j,s,t
Rijst(p)x
jes ∧ et∧
)2
,
which confirms the fact, which we checked by hand in the previous tedious computations, that
the operator is rescalable. ■
30 G. Habib and S. Paycha
We can apply Corollary 6.10 to the square of the Dirac operator P = /D
2
. Since the opera-
tor /D
2
is non negative self-adjoint, it has a well defined logarithm logθ
(
/D
2)
(here θ = π). Hence,
we have that
ω̃Res
logθ(P̃lim)
(x) = (−2i)−n/2ωsRes
logθ( /D
2
)
(p), (7.1)
for any x ∈ Up. The computation of the Wodzicki residue on the right-hand side is tedious.
In [16], it is derived from the heat-kernel asymptotics of P̃lim.
A Vielbein
Let (Mn, g) be a Riemannian manifold. Let Fp(·, g) :=
{
∂
∂x1 , . . . ,
∂
∂xn
}
be the cartesian frame
on Up built from the geodesic coordinates around p defined in (2.3). From a given orthonor-
mal basis e1(p), . . . , en(p) of TpM at p ∈ M , we build a local orthonormal frame Op(·, g) :=
{e1(·, g), . . . , en(·, g)}, of TM obtained by the parallel transport along small geodesics as in (2.4).
A linear map Ap(·, g) : TM → TM (resp. its inverse Bp(·, g)) which takes the basis Fp(·, g)
to Op(·, g) (resp. Op(·, g) to Fp(·, g)) can be represented by a n× n matrix A =
(
ali(·, g)
)
(resp.
B =
(
bil(·, g)
)
) with
∂
∂xi
=
n∑
l=1
ali(·, g)el(·, g), ∀i ∈ [[1, n]],
el(·, g) =
n∑
j=1
bjl (·, g)
∂
∂xj
, ∀l ∈ [[1, n]]. (A.1)
Also, we have that
dxi =
n∑
m=1
bil(·, g)el(·, g), ∀i ∈ [[1, n]], el(·, g) =
n∑
j=1
alj(·, g)dxj , ∀l ∈ [[1, n]].
With these conventions, and dropping the explicit mention of p whenever this does not lead to
confusion, we write A(·, g) =
(
ali(·, g)
)
i,l
and we have that
n∑
l=1
ali(·, g)alj(·, g) = gij(·) or equivalently AAt = G, (A.2)
where G has entries gij(·). Similarly, we have
n∑
l=1
bil(·, g)b
j
l (·, g) = gij(·) or equivalently BtB = G−1. (A.3)
Taking the scalar product of the first equation in (A.1) with em yields after inserting the second
equation
ali(·, g) =
n∑
l=1
gij(·)bjl (·, g) and bjl (·, g) =
n∑
l=1
gij(·)ali(·, g). (A.4)
The second equation in (A.4) is also derived by multiplying by the inverse of the metric.
Lemma A.1. At any point in Up and λ > 0, we have
ali(·, gλ) = ali(fλ(·), g) and bil(·, gλ) = bil(fλ(·), g).
A Pseudodifferential Analytic Perspective on Getzler’s Rescaling 31
Proof. To prove the equality ali(·, gλ) = ali(fλ(·), g), we let {ē1(·, gλ), . . . , ēn(·, gλ)} denote the
orthonormal frame obtained by parallel transport from {e1(p), . . . , en(p)} with respect to the
metric gλ. We know from Lemma 5.13 that ēi(·, gλ) = ei(fλ(·), g). Using (A.1) with respect to
the metric gλ, we have that
∂
∂xi
◦ fλ =
n∑
l=1
ali(·, gλ)ēl(·, gλ).
Also, (A.1) applied to the point fλ(·) gives that
∂
∂xi
◦ fλ =
n∑
l=1
ali(fλ(·), g)el(fλ(·), g).
Comparing both equations yields the result. The second equality can be proven in the same
way. ■
B Complex powers and logarithms of elliptic operators
Let E →M be a vector bundle E over M of rank k. We consider an operator Q in Ψcl(M,E) of
positive real order m with angle θ ∈ [0, 2π). For 0 < δ, we define the contour along the ray Lθ
around the spectrum of Q:
Γθ = Γ1
θ ∪ Γ2
θ ∪ Γ3
θ,
where
Γ1
θ :=
{
reiθ, δ < r
}
, Γ2
θ =
{
δeit, θ − 2π < t < θ
}
, Γ3
θ =
{
rei(θ−2π), δ < r
}
.
For any operator Q ∈ Ψcl(M,E) with positive real order m and Agmon angle θ, the resol-
vent
(
Q − λ
)−1
is a bounded linear operator on L2(M,E) with operator norm O(|λ|−1) as λ
tends to infinity in a sector around the contour Γθ. For Re(z) < 0, the Cauchy integral [16,
equation (1.5.7.1)]
Qz
θ :=
i
2π
∫
Γθ
λzθ(Q− λ)−1dλ,
converges in the operator norm to a bounded linear operator on L2(M,E). Here λzθ = |λ|zeizargθλ,
with argθλ the argument of λ in ]θ, θ + 2π[. For any real s, it also defines a linear operator
Qz
θ : H
s(M,E) → Hs(M,E) on the Sobolev closure Hs(M,E) of the space C∞(M,E) of smooth
sections of E and, therefore, induces a linear operator on C∞(M,E). Complex powers of elliptic
operators Qz
θ can be extended to any z ∈ C with Re(z) < k for any k ∈ N by
Qz
θ := QkQz−k
θ , (B.1)
thus giving rise to a group of well-defined complex powers Qz
θ acting on C∞(M,E) (see [16,
Section 1.5.7.1])
Q0
θ = Id, Qz+w
θ = Qz
θQ
w
θ , ∀(z, w) ∈ C2.
For Re(z) < 0, the symbol of Qz
θ reads [16, formula (4.8.2.6)]
σ(Qz
θ) ∼
i
2π
∫
Γθ
λzθ(σ(Q)− λ)∗−1dλ,
32 G. Habib and S. Paycha
where σ(Q) is the symbol of Q, the star stands for the inverse in the symbol algebra and
(σ(Q)− λ)∗−1 = σL(Q)∗−1
(
σ(Q)σL(Q)∗−1 − λσL(Q)∗−1
)∗−1
= σL(Q)−1(1 + symbol of order < 0)∗−1
is obtained by means of an expansion in ξ. For Re(z) < 0, the complex power Qz
θ is a classical
pseudodifferential operator of order mz, whose symbol has the asymptotic expansion
σ(Qz
θ)(x, ξ) ∼
∞∑
j=0
σmz−j(Q
z
θ)(x, ξ), (B.2)
where σmz−j(Q
z
θ)(x, ξ) are the positively homogeneous functions of degree mz − j given by [16,
formula (4.8.2.7)]:
σmz−j(Q
z
θ)(x, ξ) =
i
2π
∫
Γθ
λzθσ−m−j
(
(Q− λ)−1
)
(x, ξ)dλ. (B.3)
For a complex number z with Re(z) < k and k ∈ N, the complex power Qz
θ given by (B.1) is also
a classical pseudodifferential operator of order mz as a product of classical pseudodifferential
operators, Qk of order mk and Qz−k
θ of order m(z − k). It has an asymptotic expansion as
in (B.2) whose homogeneous components are given by the product formula for symbols [17,
formula (10.16)]:
σmz−j(Q
z
θ) =
∑
k,a+b+|α|=j
(−i)|α|
α!
∂αξ σmk−a
(
Qk
)
∂αxσm(z−k)−b
(
Qz−k
θ
)
, ∀j ∈ Z≥0. (B.4)
For z0 ∈ C with Re(z0) < 0 and following the notations of [16, formula (2.6.1.6)], we consider
the Cauchy integral
Lθ(Q, z0) :=
i
2π
∫
Γθ
logθ(λ)λ
z0
θ (Q− λ)−1dλ,
which is absolutely convergent and defines an bounded linear operator Lθ(Q, z0) : H
s(M,E) −→
Hs−mRe(z0)−ϵ(M,E) for any real number s and for any positive ϵ. As we did for complex powers,
we extend Lθ(Q, z0) to Re(z0) < k for any k ∈ N by [16, formula (2.6.1.7)]
Lθ(Q, z0) := Lθ(Q, z0 − k)Qk.
thus defining Lθ(Q, z0) for any z0 ∈ C such that
Lθ(Q, z0) = Lθ(Q, 0)Q
z0
θ = logθ(Q)Qz0
θ .
Here we have set logθ(Q) := Lθ(Q, 0), which defines a bounded linear operator
Lθ(Q, z0) : Hs(M,E) → Hs−ϵ(M,E)
for any real number s and any ϵ > 0. Since logθ(λ)λ
z
θ = λzθ logθ(λ), extending instead by
Lθ(Q, z0) := QkLθ(Q, z0 − k),
gives rise to the same family of operators and logθ(Q)Qz
θ = Qz
θ logθ(Q) for any complex number z.
A Pseudodifferential Analytic Perspective on Getzler’s Rescaling 33
C The supertrace versus the Berezin integral
and the Getzler rescaling
Following [3, Section 3], we review the construction and properties of the Berezin integral to-
gether with its relation with the supertrace as well as Geztler’s rescaling on differential forms.
Let V be a real vector space. A linear map T : ΛV → R is called a Berezin integral if T vanishes
on ΛkV for k < n = dim(V ). If V is an oriented Euclidean vector space equipped with a scalar
product g, there exist a canonical Berezin integral defined as the projection of any element
in ΛV onto its component on the n-form e1 ∧ · · · ∧ en where {e1, . . . , en} is an orthonormal basis
of (V, g). We will denote this Berezin integral by T
T
(
e1 ∧ · · · ∧ en
)
= 1, T
(
eI
)
= 0 if |I| < n.
Let us now consider the Clifford algebra Cℓ(V ) of (V, g). The isometry −Id ∈ O(V, g) gives rise
to the map
Φ: Cℓ(V ) −→ Cℓ(V ),
ei1 ·g · · · ·g eik 7−→ (−1)kei1 ·g · · · ·g eik
with i1 < · · · < ik. Here “·g” denotes the Clifford multiplication with respect to the metric g.
The map Φ clearly satisfies Φ2 = Id. Therefore, we get a splitting of Cℓ(V ) into
Cℓ(V ) = Cℓ(V )+ ⊕ Cℓ(V )−,
where Cℓ(V )± := {a ∈ Cℓ(V ) | Φ(a) = ±a}. Now we have the proposition [3, Proposition 3.19].
Proposition C.1. Let V be an oriented Euclidean vector space of even dimension n. There
exists a unique Z2-graded Clifford module S = S+ ⊕ S−, such that
Cℓ(V )⊗ C ≃ End(S).
In particular, dimS± = 2
n
2
−1. Also, we have that Cℓ(V )+ · S± ⊂ S± and Cℓ(V )− · S± ⊂ S∓.
Therefore, we have the isomorphism
Cℓ(V )+ ⊗ C ≃ End
(
S±).
Notice that S± are defined as the eigenspaces of S associated with the eigenvalues ±1 of the
complex volume form ωC = i
n
2 e1 ·g · · · ·g en with n even. Now, there is a natural super trace on
Cℓ(V )⊗ C, defined by
str(a) :=
{
trS
+
(a)− trS
−
(a) if a ∈ Cℓ(V )+,
0 if a ∈ Cℓ(V )−,
where, as before, trE stands for the fibrewise trace on End(E). In order to relate this supertrace
with the Berezin integral, we assign to any vector v ∈ V ∗ the endomorphism cg(v) ∈ End(ΛV )
that uniquely extends to a morphism of algebra bundles
cg : Cℓ(V ) −→ End(ΛV ) (C.1)
defined by cg(v)• = v ∧ • − v♯g⌟•, where v♯g is the vector in V associated to its covector v
by the musical isomorphism. The symbol map sg : Cℓ(V ) → ΛV is the isomorphism given by
sg(a) := cg(a)1. Indeed, to show sg (as well as cg) is injective, we assume that sg(a) = 0 for
some a =
∑
i1<···<ik
ai1...ike
i1 ·g · · · ·g eik . Then cg(a)1 =
∑
i1<···<ik
ai1...ike
i1 ∧· · ·∧eik = 0. Hence,
ai1...ik = 0 and, thus, a = 0. The bijectivity of sg comes from the equality of the dimensions.
In [3, Proposition 3.21], it is shown that there is a unique supertrace which is related to the
Berezin integral by the following
str(a) = (−2i)
n
2 (T ◦ sg)(a). (C.2)
for any a ∈ Cℓ(V )⊗ C.
34 G. Habib and S. Paycha
Acknowledgements
The first named author would like to thank the Alfried Krupp Wissenschaftskolleg in Greifswald
for the support. We are grateful to the Humboldt Foundation for funding a Linkage Programm
between the University of Potsdam in Germany and the Lebanese University, as well as the
American University of Beirut in Lebanon. We also thank the referees for their very helpful
comments.
References
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Münster J. Math. 14 (2021), 1–40, arXiv:1705.09588.
[8] Epstein D.B.A., Natural tensors on Riemannian manifolds, J. Differential Geometry 10 (1975), 631–645.
[9] Freed D., Lectures on Dirac operators, Unpublished notes, 1987, available at https://web.ma.utexas.edu/
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1 Introduction
2 The Wodzicki residue density for classical pseudodifferential operators
2.1 The Wodzicki residue for classical pseudodifferential operators
2.2 Local contractions
2.3 The behaviour of the Wodzicki residue under local contractions
2.4 Logarithmic residue density
2.5 A localisation formula for the logarithmic residue density
3 A local Berezin type n-form on Psi_{cl} (M, Lambda T^*M)
3.1 The Getzler rescaling map
3.2 A local n-form on Psi_{cl} (M, Lambda T^*M) and Getzler rescaling
3.3 The Wodzicki residue density versus a local Berezin type density
4 The geometric set-up
4.1 Deformation to the normal cone to a point
4.2 Tensor bundles pulled back by hat pi
5 Geometric differential operators
5.1 Valuation of local sections
5.2 Polynomial expressions in the jets of the vielbeins
5.3 Geometric operators
6 Getzler rescaled geometric differential operators
7 The rescaled square of the Dirac operator
A Vielbein
B Complex powers and logarithms of elliptic operators
C The supertrace versus the Berezin integral and the Getzler rescaling
References
|
| id | nasplib_isofts_kiev_ua-123456789-212113 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T07:04:42Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Habib, Georges Paycha, Sylvie 2026-01-28T13:57:17Z 2024 A Pseudodifferential Analytic Perspective on Getzler's Rescaling. Georges Habib and Sylvie Paycha. SIGMA 20 (2024), 010, 34 pages 1815-0659 2020 Mathematics Subject Classification: 58J40; 47A53; 15A66 arXiv:2303.04013 https://nasplib.isofts.kiev.ua/handle/123456789/212113 https://doi.org/10.3842/SIGMA.2024.010 Inspired by Gilkey's invariance theory, Getzler's rescaling method, and Scott's approach to the index via Wodzicki residues, we give a localisation formula for the ℤ₂ -graded Wodzicki residue of the logarithm of a class of differential operators acting on sections of a spinor bundle over an even-dimensional manifold. This formula is expressed in terms of another local density built from the symbol of the logarithm of a limit of rescaled differential operators acting on differential forms. When applied to complex powers of the square of a Dirac operator, it amounts to expressing the index of a Dirac operator in terms of a local density involving the logarithm of the Getzler rescaled limit of its square. The first-named author would like to thank the Alfried Krupp Wissenschaftskolleg in Greifswald for the support. We are grateful to the Humboldt Foundation for funding a Linkage Programme between the University of Potsdam in Germany and the Lebanese University, as well as the American University of Beirut in Lebanon. We also thank the referees for their very helpful comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Pseudodifferential Analytic Perspective on Getzler's Rescaling Article published earlier |
| spellingShingle | A Pseudodifferential Analytic Perspective on Getzler's Rescaling Habib, Georges Paycha, Sylvie |
| title | A Pseudodifferential Analytic Perspective on Getzler's Rescaling |
| title_full | A Pseudodifferential Analytic Perspective on Getzler's Rescaling |
| title_fullStr | A Pseudodifferential Analytic Perspective on Getzler's Rescaling |
| title_full_unstemmed | A Pseudodifferential Analytic Perspective on Getzler's Rescaling |
| title_short | A Pseudodifferential Analytic Perspective on Getzler's Rescaling |
| title_sort | pseudodifferential analytic perspective on getzler's rescaling |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212113 |
| work_keys_str_mv | AT habibgeorges apseudodifferentialanalyticperspectiveongetzlersrescaling AT paychasylvie apseudodifferentialanalyticperspectiveongetzlersrescaling AT habibgeorges pseudodifferentialanalyticperspectiveongetzlersrescaling AT paychasylvie pseudodifferentialanalyticperspectiveongetzlersrescaling |