B Iochum
email: bruno.iochum@cpt.univ-mrs.fr

T Masson

Heat trace for Laplace type operators with non-scalar symbols

Keywords: Heat kernel, non minimal operator, asymptotic heat trace, Laplace type operator PACS: 11.15.-q, 04.62.+v 2000 MSC: 58J35, 35J47, 81T13, 46L87

For an elliptic selfadjoint operator

acting on a fiber bundle over a compact Riemannian manifold, where u µν , v µ , w are N × N -matrices, we develop a method to compute the heat-trace coefficients a r which allows to get them by a pure computational machinery. It is exemplified in any even dimension by the value of a 1 written both in terms of u µν = g µν u, v µ , w or diffeomorphic and gauge invariants. We also address the question: when is it possible to get explicit formulae for a r ?

Introduction

We consider a compact Riemannian manifold (M, g) without boundary and of dimension d together with the nonminimal differential operator

P := -[u µν (x)∂ µ ∂ ν + v ν (x)∂ ν + w(x)]. (1.1)
which is a differential operator on a smooth vector bundle V over M of fiber C N where u µν , v ν , w are N × N -matrices valued functions. This bundle is endowed with a hermitean metric. We work in a local trivialization of V over an open subset of M which is also a chart on M with coordinates (x µ ). In this trivialization, the adjoint for the hermitean metric corresponds to the adjoint of matrices and the trace on endomorphisms on V becomes the usual trace tr on matrices. Since we want P to be a selfadjoint and elliptic operator on L 2 (M, V ), we first assume that u µν (x) ξ µ ξ ν is a positive definite matrix in M N :

u µν (x) ξ µ ξ ν has only strictly positive eigenvalues for any ξ = 0.

(1.2)

We may assume without loss of generality that u µν = u νµ . In particular u µµ is a positive matrix for each µ and each u µν is selfadjoint.

The asymptotics of the heat-trace

Tr e -tP ∼ t↓0 + ∞ r=0 a r (P ) t r-d/ 2 (1.3) exists by standard techniques (see [START_REF] Gilkey | Invariance theory, the heat equation and the Atiyah-Singer index theorem[END_REF]Section 1.8.1]), so we want to compute these coefficients a r (P ).

While the spectrum of P is a priori inaccessible, the computation of few coefficients of this asymptotics is eventually possible. The related physical context is quite large: the operators P appeared in gauge field theories, string theory or the so-called non-commutative gravity theory (see for instance the references quoted in [START_REF] Avramidi | Gauged gravity via spectral asymptotics of non-Laplace type operators[END_REF][START_REF] Avramidi | Non-Laplace type operators on manifolds with boundary, in: Analysis, geometry and topology of elliptic operators[END_REF][START_REF] Avramidi | Heat kernel method and its applications[END_REF]). The knowledge of the coefficients a r are important in physics. For instance, the one-loop renormalization in dimension four requires a 1 and a 2 . When the principal symbol of P is scalar (u µν = g µν 1 N ), there are essentially two main roads towards the calculation of heat coefficients (with numerous variants): the first is analytical and based on elliptic pseudodifferential operators while the second is more concerned by the geometry of the Riemannian manifold M itself with the search for invariants or conformal covariance. Compared with the flourishing literature existing when the principal symbol is scalar, there are only few works when it is not. One can quote for instance the case of operators acting on differential forms [START_REF] Gilkey | Heat equation asymptotics of "nonminimal" operators on differential forms[END_REF][START_REF] Branson | Heat Equation Asymptotics of Elliptic Operators with Non-scalar Leading Symbol[END_REF][START_REF] Alexandrov | Heat kernel for nonminimal operators on a Kähler manifold[END_REF][START_REF] Wang | Nonminimal operators and non-commutative residue[END_REF]. The first general results are in [START_REF] Avramidi | Heat kernel asymptotics of operators with non-Laplace principal part[END_REF] or in the context of spin geometry using the Dirac operators or Stein-Weiss operators [START_REF] Avramidi | A discrete leading symbol and spectral asymptotics for natural differential operators[END_REF] also motivated by physics [START_REF] Avramidi | Gauged gravity via spectral asymptotics of non-Laplace type operators[END_REF]. See also the approach in [START_REF] Fulling | Kernel asymptotics of exotic second-order operators[END_REF][START_REF] Gusynin | Local heat kernel asymptotics for nonminimal differential operators[END_REF][START_REF] Gusynin | Complete Computation of DeWitt-Seeley-Gilkey Coefficient E4 for Nonminimal Operator on Curved Manifolds[END_REF][START_REF] Gusynin | Heat kernel expansion for nonminimal differential operations and manifolds with torsion[END_REF][START_REF] Kornyak | Heat Invariant E2 for Nonminimal Operator on Manifolds with Torsion[END_REF][START_REF] Ananthanarayan | A note on the heat kernel coefficients for nonminimal operators[END_REF][START_REF] Guendelman | On the heat kernel in covariant background gauge[END_REF][START_REF] Moss | Invariants of the heat equation for non-minimal operators[END_REF][START_REF] Toms | Local momentum space and the vector field[END_REF]. The present work has a natural algebraic flavor inherited from the framework of operators on Hilbert space comprising its own standard analytical part, so is related with the first road. In particular, it gives all ingredients to produce mechanically the heat coefficients. It is also inspired by the geometry à la Connes where P = D 2 for a commutative spectral triple (A, H, D), thus has a deep motivation for noncommutative geometry.

Let us now enter into few technical difficulties. While the formula for a 0 (P ) is easily obtained, the computation of a 1 (P ) is much more involved. To locate some difficulties, we first recall the parametrix approach, namely the use of momentum space coordinates (x, ξ) ∈ T *

x M :

d 2 (x, ξ) = u µν (x) ξ µ ξ ν , d 1 (x, ξ) = -iv µ (x) ξ µ , d 0 (x) = -w(x).
Then we can try to use the generic formula (see [START_REF] Gilkey | Heat equation asymptotics of "nonminimal" operators on differential forms[END_REF])

(-i) |α| α! (∂ α ξ b j )(∂ α x d k ) b 0 .
The functions b 2r , even for r = 1, generate typically terms of the form

tr[A 1 (λ)B 1 A 2 (λ)B 2 A 3 (λ) • • • ]
where all matrices A i (λ) = (d 2 (x, ξ) -λ) -n i commute but do not commute a priori with B i , so that the integral in λ is quite difficult to evaluate in an efficient way. Of course, one can use the spectral decomposition d 2 = i λ i π i (depending on x and ξ) to get,

i 1 ,i 2 ,i 3 ,... λ∈C dλ e -λ (λ i 1 -λ) -n i 1 (λ i 2 -λ) -n i 2 (λ i 3 -λ) -n i 3 • • • tr(π i 1 B 1 π i 2 B 2 π i 3 • • • ). (1.5)
While the λ-integral is easy via residue calculus, the difficulty is then to recombine the sum. This approach is conceptually based on an approximation of the resolvent (P -λ) -1 .

Because of previous difficulties, we are going to follow another strategy, using a purely functional approach for the kernel of e -tP which is based on the Volterra series (see [20, p. 78], [START_REF] Avramidi | Heat kernel method and its applications[END_REF]Section 1.17.2]). This approach is not new and has been used for the same purpose in [START_REF] Avramidi | Heat kernel asymptotics of operators with non-Laplace principal part[END_REF][START_REF] Avramidi | Gauged gravity via spectral asymptotics of non-Laplace type operators[END_REF][START_REF] Avramidi | Non-Laplace type operators on manifolds with boundary, in: Analysis, geometry and topology of elliptic operators[END_REF]. However our strategy is more algebraic and more in the spirit of rearrangement lemmas worked out in [START_REF] Connes | Modular curvature for noncommutative two-tori[END_REF][START_REF] Lesch | Divided differences in noncommutative geometry: rearrangement lemma, functional calculus and expansional formula[END_REF]. In particular we do not go through the spectral decomposition of u µν crucially used in [START_REF] Avramidi | Heat kernel asymptotics of operators with non-Laplace principal part[END_REF] (although in a slightly more general case than the one of Section 4). To explain this strategy, we need first to fix a few notation points.

Let K(t, x, x ) be the kernel of e -tP where P is as in (1.1) and satisfies (1.2). Then

Tr[e -tP ] = dx tr[K(t, x, x)], K(t, x, x) = 1 (2π) d dξ e -ix.ξ (e -tP e ix.ξ ).

When f is a matrix-valued function on M , we get -P (e ix.ξ f )(x) = e ix.ξ [-u µν ξ µ ξ ν + 2iu µν ξ µ ∂ ν + iv µ ξ µ + w(x)]f (x)

= -e ix.ξ [H + K + P ]f (x)

where we used

H(x, ξ) := u µν (x) ξ µ ξ ν , (1.6) K(x, ξ) := -iξ µ [v µ (x) + 2u µν (x) ∂ ν ].
(1.7)

Thus H is the principal symbol of P and it is non-scalar for non-trivial matrices u µν . If 1(x) = 1 is the unit matrix valued function, we get e -tP e ix.ξ = e ix.ξ e -t(H+K+P ) 1, so that, after the change of variables ξ → t 1/2 ξ, the heat kernel can be rewritten as

K(t, x, x) = 1 (2π) d dξ e -t(H+K+P ) 1 = 1 t d/2 1 (2π) d dξ e -H- √ tK-tP 1.
(1.8)

A repetitive application of Duhamel formula (or Lagrange's variation of constant formula) gives the Volterra series (also known to physicists as Born series):

e A+B = e A + ∞ k=1 1 0 ds 1 s 1 0 ds 2 • • • s k-1 0
ds k e (1-s 1 )A B e (s 1 -s 2 )A • • • e (s k-1 -s k )A B e s k A .

Since this series does not necessarily converge for unbounded operators, we use only its first terms to generate the asymptotics (1.3) from (1.8). When A = -H and B = -√ tK -tP , it yields for the integrand of (1.8)

e -H- √ tK-tP 1 = e -H - √ t 1 0
ds 1 e (s 1 -1)H K e -s 1 H + t ds 2 e (s 1 -1)H K e (s 2 -s 1 )H K e -s 2 H -1 0 ds 1 e (s 1 -1)H P e -s 1 H

+ O(t 2 ) 1.

(1.9)

After integration in ξ, the term in √ t is zero since K is linear in ξ while H is quadratic in ξ, so that tr K(t, x, x)

t↓0 1 t d/2 [a 0 (x) + t a 1 (x) + O(t 2 )]
with the local coefficients

a 0 (x) = tr 1 (2π) d dξ e -H(x,ξ) ,
(1.10)

a 1 (x) = tr 1 (2π) d dξ 1 0 ds 1 s 1 0 ds 2 e (s 1 -1)H K e (s 2 -s 1 )H K e -s 2 H -tr 1 (2π) d dξ [ 1 0 ds 1 e (s 1 -1)H P e -s 1 H ] (1.11)
where the function 1 has been absorbed in the last e -s i H . The coefficients a 0 (P ) and a 1 (P ) are obtained after an integration in x. Since we will not perform that integration which converges when manifold M is compact, we restrict to a r (x).

We now briefly explain how we can compute a r (P ). Expanding K and P in a r (x), one shows in Section 2 that all difficulties reduce to compute algebraic expressions like (modulo the trace)

1 (2π) d dξ 1 0 ds 1 s 1 0 ds 2 • • • s k-1 0 ds k e (s 1 -1)H B 1 e (s 2 -s 1 )H B 2 • • • B k e -s k H (1.12)
where the B i are N × N -matrices equal to u µν , v µ , w or their derivatives of order two at most. Moreover, we see (1.12) as an M N -valued operator acting on the variables (B 1 , . . . , B k ) which precisely allows to focus on the integrations on ξ and s i independently of these variables.

Then we first compute the integration in ξ, followed by the iterated integrations in s i . The main result of this section is (2.18) which represents the most general operator used in the computation of a r . We end up Section 2 by a summary of the method. We show that the previously mentioned integrations are manageable in Section 3. Actually, we prove that we can reduce the computations to few universal integrals and count the exact number of them which are necessary to get a r (x) in arbitrary dimension. In Section 4, we reduce to the case u µν = g µν u where u is a positive matrix and explicitly compute the local coefficient a 1 in Theorem 4.3 in terms of (u, v µ , w). Looking after geometric invariants like for instance the scalar curvature of M , we swap the variables (u, v µ , w) with some others based on a given connection A on V . This allows to study the diffeomorphic invariance and covariance under the gauge group of the bundle V . The coefficient a 1 can then be simply written in any even dimension in terms of a covariant derivative (combining A and Christoffel symbols). In Section 5, we use our general results to address the following question: is it possible to get explicit formulae for a r avoiding the spectral decomposition like (1.5)? We show that the answer is negative when d is odd. Finally, the case u µν = g µν u + X µν is considered as an extension of u µν = g µν 1 + X µν which appeared in the literature.

Formal computation of a r (P )

This section is devoted to show that the computation of a r (x) as (1.11) reduces to the one of the terms as in (1.12). Since a point x ∈ M is fixed here, we forget to mention it, but many of the structures below are implicitly defined as functions of x.

For k ∈ N, let ∆ k be the k-simplex

∆ k := {s = (s 0 , • • • s k ) ∈ R k+1 + | 0 ≤ s k ≤ s k-1 ≤ • • • ≤ s 2 ≤ s 1 ≤ s 0 = 1}, ∆ 0 := ∅ by convention.
We use the algebra M N of N × N -complex matrices. Denote by M N [ξ, ∂] the complex vector space of polynomials both in ξ = (ξ µ ) ∈ R d and ∂ = (∂ µ ) which are M N -valued differential operators and polynomial in ξ; for instance, P, K, H ∈ M N [ξ, ∂] with P of order zero in ξ and two in ∂, K of order one in ξ and ∂, and H of order two in ξ and zero in ∂.

For

any k ∈ N, define a map f k (ξ) : M N [ξ, ∂] ⊗ k → M N [ξ, ∂],
evidently related to (1.12), by

f k (ξ)[B 1 ⊗ • • • ⊗ B k ] := ∆ k ds e (s 1 -1)H(ξ) B 1 e (s 2 -s 1 )H(ξ) B 2 • • • B k e -s k H(ξ) , ( 2.1)
f 0 (ξ)[a] := a e -H(ξ) , for a ∈ C =: M ⊗ 0 N . (2.2)
Here, by convention, each ∂ µ in B i ∈ M N [ξ, ∂] acts on all its right remaining terms. Remark that the map ξ → f k (ξ) is even. We first rewrite (1.9) in these notations (omitting the ξ-dependence):

e -H- √ tK-tP 1 = e -H + ∞ k=1 (-1) k f k [( √ tK + tP ) ⊗ • • • ⊗ ( √ tK + tP )] (2.3) = e -H -t 1/2 f 1 [K] + t(f 2 [K ⊗ K] -f 1 [P ]) + t 3/2 (f 2 [K ⊗ P ] + f 2 [P ⊗ K] -f 3 [K ⊗ K ⊗ K]) + t 2 (f 2 [P ⊗ P ] -f 3 [K ⊗ K ⊗ P ] -f 3 [K ⊗ P ⊗ K] -f 3 [P ⊗ K ⊗ K]) + O(t 2 ).
Since all powers of t in (2n + 1)/2 have odd powers of ξ µ 1 • • • ξ µp (with odd p), the ξ-integrals in (1.12) will be zero since f k is even in ξ, so only

a 0 (x) = tr 1 (2π) d dξ f 0 [1], (2.4
)

a 1 (x) = tr 1 (2π) d dξ (f 2 [K ⊗ K] -f 1 [P ]), (2.5
)

a 2 (x) = tr 1 (2π) d dξ (f 2 [P ⊗ P ] -f 3 [K ⊗ K ⊗ P ] -f 3 [K ⊗ P ⊗ K] -f 3 [P ⊗ K ⊗ K])
etc survive.

Our first (important) step is to erase the differential operator aspect of K and P as variables of f k to obtain variables in the space

M N [ξ] of M N -valued polynomials in ξ: because a ∂ contained in B i will apply on e (s i+1 -s i )H B i+1 • • • B k e -s k H
, by a direct use of Leibniz rule and the fact that

∂ e -sH = - s 0 ds 1 e (s 1 -s)H (∂H) e -s 1 H , ( 2.6)
we obtain the following

Lemma 2.1 When all B j are in M N [ξ, ∂], the functions f k for k ∈ N * satisfy f k (ξ)[B 1 ⊗ • • • ⊗ B i ∂ ⊗ • • • ⊗ B k ] = k j=i+1 f k (ξ)[B 1 ⊗ • • • ⊗ (∂B j ) ⊗ • • • ⊗ B k ] - k j=i f k+1 (ξ)[B 1 ⊗ • • • ⊗ B j ⊗ (∂H) ⊗ B j+1 ⊗ • • • ⊗ B k ]. (2.7)
Proof By definition (omitting the ξ-dependence)

f k [B 1 ⊗ • • • ⊗ B i ∂ ⊗ • • • ⊗ B k ] = ∆ k ds e (s 1 -1)H B 1 e (s 2 -s 1 )H B 2 • • • B i ∂(e (s i+1 -s i )H B i+1 • • • B k e -s k H ).
The derivation ∂ acts on each factor in the parenthesis: -On the argument B j , j ≥ i + 1, which gives the first term of (2.7).

-On a factor e (s j+1 -s j )H for i ≤ j ≤ k -1, we use (2.6) ∂ e (s j+1 -s j )H = -s j -s j+1 0 ds e s +s j+1 -s j )H (∂H) e -s H = -

s j s j+1
ds e (s-s j )H (∂H) e (s j+1 -s)H with s = s + s j+1 , so that in the integral, one obtains the term

- 1 0 ds 1 s 1 0 ds 2 • • • s k-1 0 s j s j+1 ds e (s 1 -1)H B 1 e (s 2 -s 1 )H B 2 • • • • • • B j e (s-s j )H (∂H) e (s j+1 -s)H B j+1 • • • B k e -s k H ).
Since, as directly checked,

1 0 ds 1 • • • s k-1 0 ds k s j s j +1 ds = ∆ k+1 ds with s j = s j for j ≤ i -1, s i = s and s j = s j-1 for j ≥ i + 1, this term is -f k+1 [B 1 ⊗ • • • ⊗ B j ⊗ (∂H) ⊗ B j+1 ⊗ • • • ⊗ B k ].
-Finally, on the factor e -s k H , one has ∂ e -s k H = -s k 0 e (s-s k )H (∂H) e -sH which gives the last term:

-f k+1 [B 1 ⊗ • • • ⊗ B k ⊗ (∂H)]. Thus (2.3) reduces to compute f k [B 1 ⊗ • • • ⊗ B k ] where B i ∈ M N [ξ].
Our second step is now to take care of the ξ-dependence: by hypothesis, each B i in the tensor product

B 1 ⊗ • • • ⊗ B k has the form B µ 1 ...µ i ξ µ 1 • • • ξ µ i with B µ 1 ...µ i ∈ M N , so that B 1 ⊗ • • • ⊗ B k is a sum of terms like B µ 1 ...µ k ξ µ 1 • • • ξ µ where B µ 1 ...µ k ∈ M ⊗ k N .
As a consequence, by linearity of f k in each variable, computation of a r requires only to evaluate terms like

1 (2π) d dξ ξ µ 1 • • • ξ µ f k (ξ)[ B µ 1 ...µ k ] ∈ M N with B µ 1 ...µ k ∈ M ⊗ k N , (2.8)
and we may assume that = 2p, p ∈ N. The next step in our strategy is now to rewrite the f k appearing in (2.8) in a way which is independent of the variables B µ 1 ...µ k , a rewriting obtained in (2.11). Then the driving idea is to show firstly that such f k can be computed and secondly that its repeated action on all variables which pop up by linearity (repeat that K has two terms while P has three terms augmented by the action of derivatives, see for instance (1.11)) is therefore a direct computational machinery. For such rewriting we need a few definitions justified on the way.

For k ∈ N, define the (finite-dimensional) Hilbert spaces

H k := M ⊗ k N , H 0 := C,
endowed with the scalar product

A 1 ⊗ • • • ⊗ A k , B 1 ⊗ • • • ⊗ B k H k := tr(A * 1 B 1 ) • • • tr(A * k B k ), a 0 , b 0 H 0 := a 0 b 0 ,
so each M N is seen with its Hilbert-Schmidt norm and

A 1 ⊗ • • • ⊗ A k 2 H k = k j=1 tr(A * j A j )
. We look at (2.8) as the action of the operator

1 (2π) d dξ ξ µ 1 • • • ξ µ l f k (ξ) acting on the finite dimensional Hilbert space H k .
Denote by B(E, F ) the set of bounded linear operators between the vector spaces E and F and let B(E) := B(E, E). For k ∈ N, let

H k := H k+1 , so H 0 = M N , m : H k → M N , m(B 0 ⊗ • • • ⊗ B k ) := B 0 • • • B k (multiplication of matrices), κ : H k → H k , κ(B 1 ⊗ • • • ⊗ B k ) := 1 ⊗ B 1 ⊗ • • • ⊗ B k , ι : M ⊗ k+1 N → B(H k , M N ), ι(A 0 ⊗ • • • ⊗ A k )[B 1 ⊗ • • • ⊗ B k ] := A 0 B 1 A 1 • • • B k A k , ι : M N → B(C, M N ), ι(A 0 )[a] := aA 0 , ρ : M ⊗ k+1 N → B( H k ), ρ(A 0 ⊗ • • • ⊗ A k )[B 0 ⊗ • • • ⊗ B k ] := B 0 A 0 ⊗ • • • ⊗ B k A k .
For A ∈ M N and k ∈ N, define the operators

R i (A) : H k → H k for i = 0, . . . , k R i (A)[B 0 ⊗ • • • ⊗ B k ] := B 0 ⊗ • • • ⊗ B i A ⊗ • • • ⊗ B k . Thus ρ(A 0 ⊗ • • • ⊗ A k ) = R 0 (A 0 ) • • • R k (A k ). (2.9)
As shown in Proposition A.1, ι is an isomorphism. The links between the three spaces

M ⊗ k+1 N , B( H k ) and B(H k , M N ) are summarized in the following commutative diagram where (m • κ * )(C)[B 1 ⊗ • • • ⊗ B k ] = m(C[1 ⊗ B 1 ⊗ • • • ⊗ B k ]): B( H k ) m • κ * M ⊗k+1 N ρ 4 4 ι * * B(H k , M N ) (2.10) For any matrix A ∈ M N and s ∈ ∆ k , define c k (s, A) := (1 -s 1 ) A ⊗ 1 ⊗ • • • ⊗ 1 + (s 1 -s 2 ) 1 ⊗ A ⊗ 1 ⊗ • • • ⊗ 1 + • • • + (s k-1 -s k ) 1 ⊗ • • • ⊗ A ⊗ 1 + s k 1 ⊗ • • • ⊗ 1 ⊗ A, c 0 (s, A) := A
where the tensor products have k + 1 terms, so that c k (s, A) ∈ M ⊗ k+1 N . This allows a compact notation since now

f k (ξ) = ∆ k ds ι[e -ξαξ β c k (s,u αβ ) ] ∈ B(H k , M N ), with f 0 (ξ) = ι(e -ξαξ β u αβ ), (2.11)
and these integrals converge because the integrand is continuous and the domain ∆ k is compact.

Since we want to use operator algebra techniques, with the help of

c k (s, A) ∈ M ⊗ k+1 N
, it is useful to lift the computation of (2.11) to the (finite dimensional C * -algebra) B( H k ) as viewed in diagram (2.10). Thus, we define

C k (s, A) := ρ(c k (s, A)) ∈ B( H k ),
and then, by (2.9)

C k (s, A) = (1 -s 1 ) R 0 (A) + (s 1 -s 2 ) R 1 (A) + • • • + (s k-1 -s k ) R k-1 (A) + s k R k (A).
Remark 2.2 All these distinctions between H k and H k or c k (s, A) and C k (s, A) seem innocent so tedious. But we will see later on that the distinctions between the different spaces in (2.10) play a conceptual role in the difficulty to compute the coefficients a r . Essentially, the computations and results take place in B( H k ) and not necessarily in the subspace M ⊗k+1 N ⊂ B( H k ) (see (2.18) for instance).

Given a diagonalizable matrix

A = C diag(λ 1 , . . . , λ n ) C -1 ∈ M N , let C ij := CE ij C -1 for i, j = 1, . . . , n
where the E ij form the elementary basis of M N defined by [E ij ] kl := δ ik δ jl . We have the easily proved result:

Lemma 2.3 We have i) R i (A 1 A 2 ) = R i (A 2 )R i (A 1 ) and [R i (A 1 ), R j (A 2 )] = 0 when i = j. ii) R i (A) * = R i (A * ). iii) When A is diagonalizable, AC ij = λ i C ij and C ij A = λ j C ij . Thus, all operators R i (A) on H k for any k ∈ N have common eigenvectors R i (A)[C i 0 j 0 ⊗ • • • ⊗ C i k j k ] = λ j i C i 0 j 0 ⊗ • • • ⊗ C i k j k ,
and same spectra as A.

In particular, there are strictly positive operators if A is a strictly positive matrix. This means that C k (s, A) ≥ 0 if A ≥ 0 and s ∈ ∆ k , and this justifies the previous lift. Now, evaluating (2.11) amounts to compute the following operators in B( H k ):

T k,p (x) := 1 (2π) d ∆ k ds dξ ξ µ 1 • • • ξ µ 2p e -ξαξ β C k (s,u αβ (x)) : H k → H k , p ∈ N, k ∈ N. (2.12) T 0,0 (x) := 1 (2π) d dξ e -ξαξ β u αβ (x) ∈ M N ρ(M N ) ⊂ B( H 0 ), (2.13)
where T k,p depends on x through u αβ only. Their interest stems from the fact they are independent of arguments B 0 ⊗ • • • ⊗ B k ∈ H k on which they are applied, so are the corner stones of this work. Using (2.10), the precise link between T k,p and f k (ξ) is

m • κ * • T k,p = 1 (2π) d dξ ξ µ 1 • • • ξ µ 2p f k (ξ). (2.14)
The fact that T k,p is a bounded operator is justified by the following Lemma 2. [START_REF] Avramidi | Heat kernel method and its applications[END_REF] The above integrals (2.12) and (2.13) converge and T k,p ∈ B( H k ).

Proof We may assume k ∈ N * since for k = 0, same arguments apply.

For any strictly positive matrix A with minimal eigenvalues λ min (A) > 0, Lemma 2.3 shows that, for any s ∈ ∆ k ,

C k (s, A) ≥ [(1 -s 1 )λ min (A) + (s 1 -s 2 )λ min (A) + • • • + s k λ min (A)] 1 H k = λ min (A) 1 H k .
We claim that the map

ξ ∈ R d → λ min (ξ α ξ β u αβ ) is continuous: the maps ξ ∈ R d → ξ α ξ β u αβ and 0 < a ∈ B( H k ) → inf(spectrum(a)) = a -1 are continuous (the set of invertible matrices is a Lie group). We use spherical coordinates ξ ∈ R d → (|ξ|, σ) ∈ R + × S d-1
, where σ := |ξ| -1 ξ is in the Euclidean sphere S d-1 endowed with its volume form dΩ.

Then λ min (ξ α ξ β u αβ ) = |ξ| 2 λ min (σ α σ β u αβ ) > 0 (remark that σ α σ β u αβ is a strictly positive matrix). Thus c := inf{λ min (σ α σ β u αβ ) | σ ∈ S d-1
} > 0 by compactness of the sphere. The usual operator-norm of H k applied on the above integral T k,p , satisfies

T k,p ≤ ∆ k ds σ∈S d-1 dΩ g (σ) ∞ 0 dr r d-1 r 2p σ µ 1 • • • σ µ 2p e -r 2 c 1 H k ≤ ∆ k ds σ∈S d-1 dΩ g (σ) ∞ 0 dr r d-1+2p e -r 2 c = vol(∆ k ) vol(S d-1 g ) Γ(d/2+p) 2 c -d/2-p .
For the ξ-integration of (2.12), we use again spherical coordinates, but now ξ = r σ with r = (g µν ξ µ ξ ν ) 1/2 , σ = r -1 ξ ∈ S d-1 g (this sphere depends on x ∈ M through g(x)) and define u[σ] := u µν σ µ σ ν which is a positive definite matrix for any σ ∈ S d-1 g . Thus we get

T k,p = 1 (2π) d ∆ k ds S d-1 g dΩ g (σ) σ µ 1 • • • σ µ 2p ∞ 0 dr r d-1+2p e -r 2 C k (s,u[σ]) (2.15) = Γ(d/2+p) 2(2π) d S d-1 g dΩ g (σ) σ µ 1 • • • σ µ 2p ∆ k ds C k (s, u[σ]) -(d/2+p) .
Thus, we have to compute the s-integration

∆ k ds C k (s, u[σ]) -α for α ∈ 1 2 N * .
We do that via functional calculus, using Lemma 2.3 iii), by considering the following integrals

I α,k (r 0 , r 1 , . . . , r k ) := ∆ k ds [(1 -s 1 )r 0 + (s 1 -s 2 )r 1 + • • • + s k r k ] -α = ∆ k ds [r 0 + s 1 (r 1 -r 0 ) + • • • + s k (r k -r k-1 )] -α
(2.16)

I α,0 (r 0 ) := r -α 0 , for α = 0, (2.17)
where 0 = r i ∈ R + corresponds, in the functional calculus, to positive operator R i (u[σ]). Such integrals converge for any α ∈ R and any k ∈ N * , even if it is applied above only to

α = d/2 + k -r ∈ 1 2 N.
Nevertheless for technical reasons explained below, it is best to define I α,k for an arbitrary α ∈ R. In short, the operator T k,p is nothing else than the operator in B( H k )

T k,p = Γ(d/2+p) 2(2π) d S d-1 g dΩ g (σ) σ µ 1 • • • σ µ 2p I d/2+p,k R 0 (u[σ]), R 1 (u[σ]), . . . , R k (u[σ]) . (2.18)
Remark that T k,p depends on x via u[σ] and the metric g.

Remark 2.5

We pause for a while to make a connection with the previous work [START_REF] Avramidi | Heat kernel asymptotics of operators with non-Laplace principal part[END_REF]. There, the main hypothesis on the matrix u µν ξ µ ξ ν is that all its eigenvalues are positive multiples of g µν ξ µ ξ ν for any ξ = 0. Under this hypothesis, we can decompose spectrally

u[σ] = i λ i π i [σ]
where the eigenprojections π i depends on σ but the associated eigenvalues λ i do not. Then, operator functional calculus gives

I d/2+p,k R 0 (u[σ]), . . . , R k (u[σ]) = i 0 ,...,i k I d/2+p,k (λ i 0 , . . . , λ i k ) R 0 (π i 0 [σ]) • • • R k (π i k [σ]) (2.19)
and

T k,p = Γ(d/2+p) 2(2π) d i 1 ,...,i k I d/2+p,k (λ i 0 , . . . , λ i k ) S d-1 g dΩ g (σ) σ µ 1 • • • σ µ 2p R 0 (π i 0 [σ]) • • • R k (π i k [σ])
where all

π i 0 [σ], . . . , π i k [σ] commute as operators in B( H k )
. However, we do not try to pursue in that direction since it is not very explicit due to the difficult last integral on the sphere; also we remind that we already gave up in the introduction the use of the eigenprojections for the same reason. Instead, we give for instance a complete computation of a 1 in Section 4 directly in terms of matrices u µν , v µ , w, avoiding this spectral splitting in the particular case where u µν = g µν u for a positive matrix u.

In conclusion, as claimed in the introduction, the above approach really reduces the computation of all a r (x) for an arbitrary integer r to the control of operators T k,p which encode all difficulties since, once known, their actions on an arbitrary variable

B 1 ⊗ • • • ⊗ B k with B i ∈ M n
are purely mechanical. For instance in any dimension of M , the calculus of a 1 (x) needs only to know, T 1,0 , T 2,1 , T 3,2 , T 4,3 . More generally, we have the following Proof As seen in (2.3), using the linearity of f k in each argument, we may assume that in

f k [B 1 ⊗• • •⊗B k ] each argument B i
is equal to K or P , so generates t 1/2 or t in the asymptotic expansion. Let n K and n P the number of K and P for such f k involved in a r (P ). Since a r (P ) is the coefficient of t r , we have 1 2 n K + n P = r and k ≥ r. In particular, n K much be even.

When B i = K = -iξ µ [v µ (x) + 2u µν (x) ∂ ν ],
again by linearity, we may assume that the argument in f (k, p)

:= f k (ξ)[B 1 (ξ) ⊗ • • • ⊗ B k (ξ)
] is a polynomial of order 2p since odd order are cancelled out after the ξ-integration. In such f (k, p), the number of ξ (in the argument) is equal to n K , so that p = 1 2 n K , and the number of derivations ∂ is n K + 2n P .

We count now all f (k, p) involved in the computation of a r (P ). We initiate the process with (k, p) = (n K + n P , 1 2 n K ), so k -p = r and after the successive propagation of ∂ as in Lemma 2.1, we end up with (k , p ) where k -p = r: in (2.7), k → k + 1 while p → p + 1 since ∂H appears as a new argument. So (k , p ) = (k , k -r) and the maximum of k is 2n K + 3n P . Here, n P = 0, . . . , r and n K = 0, . . . , 2r, so that the maximum is for k = 4r. All f (k, k -r) with r ≤ k ≤ 4r will be necessary to compute a r (P ): Let k be such that r ≤ k ≤ 3r. Then a term f (k, k -r) will be obtained by the use of Lemma 2.1 applied

on f r [u µ 1 ν 1 ∂ µ 1 ∂ ν 1 ⊗ • • • ⊗ u µrνr ∂ µr ∂ νr ]
with an action of k -r derivatives on the e sH and the reminder on the B i . The same argument, applied to

f 2r [ξ µ 1 u µ 1 ν 1 ∂ ν 1 ⊗ • • • ⊗ ξ µ 2r u µ 2r ν 2r ∂ ν 2r ], also generates a term f (k, k -r) when 2r ≤ k ≤ 4r.
Finally, remark that we can swap the ξ-dependence of f (k, p) into definition (2.12) of T k,p to end up with integrals which are advantageously independent of arguments B i .

The case r = 0 is peculiar: since k = 0 automatically, we have only to compute T 0,0 in (2.13) which gives a 0 (x) by (1.10). The link between the T 's and the I is given in (2.18). The preceding reasoning is independent of the dimension d.

Of course, in an explicit computation of a r (x), each of these 3r + 1 operators T k,k-r can be used several times since applied on different arguments [START_REF] Moss | Invariants of the heat equation for non-minimal operators[END_REF]) will be explicitly computed in Section 3.

B 1 ⊗ • • • ⊗ B k . The integral I d/2+p,k giving T k , p in (2.
We now list the terms of a 1 (x). Using the shortcuts

v := ξ µ v µ , ūν := ξ µ u µν ,
starting from (2.5) and applying Lemma 2.1, we get

-f 1 [P ] = -f 2 [u µν ⊗ ∂ µ ∂ ν H] + 2f 3 [u µν ⊗ ∂ µ H ⊗ ∂ ν H] -f 2 [v µ ⊗ ∂ µ H] + f 1 [w], (2.20) f 2 [K ⊗ K] = -f 2 [v ⊗ v] + 2f 3 [v ⊗ ūν ⊗ ∂ ν H] -2f 2 [ū µ ⊗ ∂ µ v] + 2f 3 [ū µ ⊗ ∂ µ H ⊗ v] + 2f 3 [ū µ ⊗ v ⊗ ∂ µ H] + 4f 3 [ū µ ⊗ ∂ µ ūν ⊗ ∂ ν H] + 4f 3 [ū µ ⊗ ūν ⊗ ∂ µ ∂ ν H] -4f 4 [ū µ ⊗ ∂ µ H ⊗ ūν ⊗ ∂ ν H] -4f 4 [ū µ ⊗ ūν ⊗ ∂ µ H ⊗ ∂ ν H] -4f 4 [ū µ ⊗ ūν ⊗ ∂ ν H ⊗ ∂ ν H].
(2.21)

This represents 14 terms to compute for getting a 1 (x).

Summary of the method:

We pause here to summarize the chosen method. To compute a r (x), we first expand K and P in (2.3) in terms of matrix valued differential operators which are arguments of M N -valued operators f k (ξ), and then we remove all derivative operators from the arguments using the generalized Leibniz rule (2.7). This generates a sum of terms like (2.8). Then, the method splits along two independent computational axes: the first one is to collect all the arguments B µ 1 ...µ k produced by (2.7); the second one is to compute the operators obtained by integration of f k (ξ) with respect to ξ, which, thanks to (2.10) and (2.14), requires to compute some operators T k,p . The latter operators are written in (2.18) using spherical coordinates for that ξ-integral, in terms of universal integrals I d/2+p,k defining operators depending only on u µν and the metric g. In the generic situation, the links between T k,p , I d/2+p,k , and f k are given in (2.18) and (2.14), but another link between f k and I d/2+p,k will be given in (4.4) in a particular case where the integrals (2.18) can be fully computed. The last step of the method is to collect the (matrix) traces of evaluations of operators (second axe) on the arguments (first axe): a r (x) is just a sum of such contributions. Moreover, Lemma 2.6 determines the number of integrals I d/2+p,k to compute to get a r (x).

Integral computations of I α,k

We begin with few interesting remarks on I α,k defined in (2.16) and (2.17

I α,k (r 0 , . . . , r k ) = 1 (α-1) (r k-1 -r k ) -1 [I α-1,k-1 (r 0 , . . . , r k-2 , r k ) -I α-1,k-1 (r 0 , . . . , r k-1 )]. (3.1)
(The abandoned case I 1,k is computed in Proposition 3.3.) ii) Symmetry with respect to last two variables:

I α,k (r 0 , . . . , r k-1 , r k ) = I α,k (r 0 , . . . , r k , r k-1 ).
iii) Continuities:

The functions I α,k : (R * + ) k+1 → R * + are continuous for all α ∈ R. For any (r 0 , . . . , r k-1 , r k ) ∈ (R * + ) k+1 , the map α ∈ R + → I α,k (r 0 , . . . , r k-1 , r k ) is continuous. iv) Special values: for any α ∈ R and k ∈ N, I α,k (r 0 , • • • , r 0 k+1 ) = 1 k! r -α 0 . (3.2) Proof i) In I α,k (r 0 , . . . , r k ) = 1 0 ds 1 s 1 0 ds 2 • • • s k-1 0 ds k [r 0 +s 1 (r 1 -r 0 )+• • •+s k (r k -r k-1 )] -α
, the last integral is equal to which gives the claimed relation. One checks directly from the definition (2.16) of I α+1,1 (r 0 , r 1 ) that (3.1) is satisfied for the given I α,0 .

ii

) I α,1 (r 0 , r 1 ) = 1 0 ds r 0 + s(r 1 -r 0 )] -α = 1 0 ds [r 1 + s (r 0 -r 1 )] -α = I α,1 (r 1 , r 0 )
after the change of variable s → s = 1 -s. The symmetry follows now using (3.1) by a recurrence process.

iii) The map g(s, r

) := [(1 -s 1 )r 0 + (s 1 -s 2 )r 1 + • • • + s k r k ] -α > 0 is continuous at the point r ∈ (R * + ) k+1
and uniformly bounded in a small ball B around r since then we have g(s, r) ≤ max(r -|α| min , r |α| max ) where r min or max := min or max{r i | r ∈ B} > 0. Thus, since the integration domain ∆ k is compact, by Lebesgue's dominated convergence theorem, we get the continuity of I α,k . It remains to prove the continuity of

α → I α,k (r 0 , . . . , r k-1 , r k ): If r min := min r i > 0, then (1-s 1 )r 0 +(s 1 -s 2 )r 1 +• • •+s k r k ≥ r min , so that ((1-s 1 )r 0 +(s 1 -s 2 )r 1 +• • •+s k r k ) -α ≤ r -α min and choosing a = min(1/2, r min ), r -α min ≤ a -α ≤ a -β 1 for α ∈ (β 0 , β 1 )
, we can apply again Lebesgue's dominated convergence theorem.

(iv) Using the very definition (2.16),

I α,k (r 0 , . . . , r 0 ) = ∆ k ds r -α 0 = vol(∆ k )r -α 0 = 1 k! r -α 0 .
From Lemma 2.6 and (2.18), the computation of a r (P ), for r ≥ 1 (since a 0 is known already by (4.5)) in spatial dimension d ≥ 1, requires all I d/2+k-r,k for r ≤ k ≤ 4r. The sequence I d/2,r , I d/2+1,r+1 , . . . , I d/2+3r,4r belongs to the same recursive relation (3.1), except if there is a s ∈ N such that d/2 + s = 1, which can only happen with d = 2 and s = 0 (see Case 1 below). The computation of this sequence requires then to compute I d/2,r as the root of (3.1).

The function I d/2,r can itself be computed using (3.1) when this relation is relevant. Case 1: d is even and d/2 ≤ r, recursive sequence (3.1) fails at I 1,r-d/2+1 :

I 1,r-d/2+1 → I 2,r-d/2+2 → • • • → I d/2,r → I d/2+1,r+1 → • • • → I d/2+3r,4r used to compute ar(x) . (3.3)
Case 2: d is even and r < d/2, relation (3.1) never fails and

I d/2-r,0 → I d/2-r+1,1 → I d/2-r+2,2 → • • • → I d/2,r → I d/2+1,r+1 → • • • → I d/
I d/2-r,0 → I d/2-r+1,1 → I d/2-r+2,2 → • • • → I d/2,r → I d/2+1,r+1 → • • • → I d/2+3r,4r
used to compute ar(x) .

(3.5)

In the latter case, the root is I α,0 with α = d/2 -r half-integer, positive or negative and both situation have to be considered separately. The recursive relation (3.1), which for I α,k follows from the integration on the k-simplex ∆ k , has a generic solution:

Proposition 3.2 Given α 0 ∈ R, k 0 ∈ N and a function F : R * + → R * + , let the function J α 0 +k 0 ,k 0 : (R * + ) k+1 → R * + be defined by J α 0 +k 0 ,k 0 (r 0 , . . . , r k ) := c α 0 +k 0 ,k 0 k 0 i=0 k 0 j=0 j =i (r i -r j ) -1 F (r i ).
i) Ascending chain: Then, all functions J α 0 +k,k obtained by applying the recurrence formula (3.1) for any k ∈ N, k ≥ k 0 have the same form:

J α 0 +k,k (r 0 , . . . , r k ) := c α 0 +k,k k i=0 k j=0 j =i (r i -r j ) -1 F (r i ) (3.6) with c α 0 +k,k = (-1) k-k 0 (α 0 +k 0 )•••(α 0 +k-1) c α 0 +k 0 ,k 0 for k > k 0 . (3.7)
ii) Descending chain: when α 0 ∈ R\{-N}, the functions J α 0 +k,k defined by (3.6) for k ∈ N * starting with k 0 = 0, the root F (r 0 ) = J α 0 ,0 (r 0 ) and

c α 0 +k,k = (-1) k α 0 (α 0 +1)•••(α 0 +k-1) (3.8) satisfy (3.1). Proof i) It is sufficient to show that X := 1 α-1 (r -1 -r ) -1 [J α-1, -1 (r 0 , . . . , r -2 , r ) -J α-1, -1 (r 0 , . . . , r -1 )].
has precisely the form (3.6) for = k 0 + 1 and α = α 0 + k 0 + 1. We have

X = c α-1, -1 α-1 (r -1 -r ) -1 -2 i=0 -2 j=0 j =i (r i -r j ) -1 (r i -r ) -1 F (r i ) + -2 j=0 (r -r i ) -1 F (r ) - -2 i=0 -2 j=0 j =i (r i -r j ) -1 (r i -r -1 ) -1 F (r i ) - -2 j=0 (r -1 -r i ) -1 F (r -1 ) . (3.9)
We can combine the two sums on i = 0, . . . , -2 as:

-2 j=0 -2 j=0 j =i (r i -r j ) -1 [(r i -r ) -1 -(r i -r -1 ) -1 ] F (r i ) = (r -r -1 ) -2 j=0 j=0 j =i (r i -r j ) -1 F (r i ).
Including (r -1 -r ) -1 , the others terms in (3.9) correspond to [ j=0 j =i

(r i -r j ) -1 ] F (r i ) for i = -1 and i = (up to a sign), so that X = - c α-1, -1 α-1 j=0 j=0 j =i (r i -r j ) -1 F (r i )
which yields c α 0 +k 0 +1,k 0 +1 = -1 α 0 +k 0 c α 0 +k 0 ,k 0 and so the claim (3.7). ii) It is the same argument with k 0 = 0 and the hypothesis α 0 / ∈ -N guarantees the existence of (3.8) and moreover c α 0 ,0 = 1. Proposition 3.2 exhibits the general solution of Cases 2 and 3 in (3.4) and (3.5) (with α 0 = d/2-r, for d even and α 0 > 0, or for d odd), with, for both, F (r 0 ) = I d/2-r,0 (r 0 ) = r -α 0 0 , so that

I d/2-r+k,k (r 0 , . . . , r k ) = (-1) k (d/2-r)(d/2-r+1)•••(d/2-r+k-1) k i=0 k j=0 j =i (r i -r j ) -1 r -(d/2-r) i . (3.10)
To control the reminder Case 1 of chain (3.3) where α 0 ∈ -N (so for d even and α 0 = d/2 -r ≤ 0), we need to compute the functions

I k-α 0 ,k for α 0 ∈ {0, 1, • • • , k -1}
. This is done below and shows surprisingly enough that the generic solution of Proposition 3.2 holds true also for a different function F (r 0 ). Actually, this is simple consequence of the fact that, despite its presentation in (3.10), the RHS has no poles as function of r.

Corollary 3.3 Case 1: d even and d/2 ≤ r (so α 0 = d/2 -r ≤ 0). For any k ∈ N * , and = r -d/2 ∈ {0, 1, • • • , k -1} I k-,k (r 0 , . . . , r k ) = (-1) k--1 (k--1)! ! k i=0 k j=0 j =i (r i -r j ) -1 r i log r i . ( 3
I k-,k (r 0 , . . . , r k ) = lim r→m+ I m-r+k,k (r 0 , . . . , r n+k ) = lim r→m+ (-1) k (m-r)•••(m-r+k-1) k i=0 k j=0 j =i (r i -r j ) -1 r -(m-r) i = [ lim r→m+ (-1) k-1 (m-r)•••(m-r+ -1) 1 (m-r+ +1)•••(m-r+k-1) ] lim r→m+ k i=0 k j=0 j =i (r i -r j ) -1 r i 1 r-(m+ ) r r-(m+ ) i = (-1) k--1 (k--1)! ! k i=0 k j=0 j =i (r i -r j ) -1 r i log r i
where we used (A.1) for the second limit in last equality.

The next propositions compute explicitly Case 3 and then Case 2, and the result is not written as in (3.10) where denominators in r i -r j appear. This allows to deduce algorithmically (i.e. without any integration) the sequence

I d/2,
I +1/2,0 (r 0 ) = r --1/2 0 , I +3/2,1 (r 0 , r 1 ) = 2 2 +1 ( √ r 0 √ r 1 ) -2 -1 ( √ r 0 + √ r 1 ) -1 0≤l 1 ≤2 √ r 0 l 1 √ r 1 2 -l 1 ,
while if d/2 -r = --1/2 with ∈ N, the root and its follower are

I --1/2,0 (r 0 ) = r +1/2 0 , I -+1/2,1 (r 0 , r 1 ) = 2 2 +1 ( √ r 0 + √ r 1 ) -1 0≤l 1 ≤2 √ r 0 l 1 √ r 1 2 -l 1 .
Proof Using (3.1) and (2.17), we get when ≥ 0

I +3/2,1 (r 0 , r 1 ) = 1 +1/2 (r 0 -r 1 ) -1 [I +1/2,0 (r 1 ) -I +1/2,0 (r 0 )] = 2 2 +1 ( √ r 0 - √ r 1 ) -1 ( √ r 0 + √ r 1 ) -1 [r --1/2 1 -r --1/2 0 ]
where the term in bracket is r

--1/2 1 -r --1/2 0 = ( √ r 0 √ r 1 ) -2 -1 [r 2 +1 0 -r 2 +1 1 ] = ( √ r 0 √ r 1 ) -2 -1 ( √ r 0 - √ r 1 ) 0≤l 1 ≤2 √ r 0 l 1 √ r 1 2 -l 1 ,
which gives the result. Similar proof for the other equality.

This proposition exhibits only the two first terms of the recurrence chain in Case 3: similar formulae can be obtained at any level in which no (r i -r j ) -1 factors appear. Unfortunately, they are far more involved.

I n,k (r 0 , . . . , r k ) = (r 0 •••r k ) -1 (n-1)•••(n-k) 0≤l k ≤l k-1 ≤••• •••≤l 1 ≤n-(k+1) r l 1 -(n-(k+1)) 0 r l 2 -l 1 1 • • • r l k -l k-1 k-1 r -l k k (3.12) = (r 0 •••r k ) -(n-k) (n-1)•••(n-k) 0≤l k ≤l k-1 ≤••• •••≤l 1 ≤n-(k+1) r l 1 0 r l 2 +(n-(k+1))-l 1 1 • • • r l k +(n-(k+1))-l k-1 k-1 r (n-(k+1))-l k k . (3.13)
In (3.13), all exponents in the sum are positive while they are negative in (3.12). In particular

I n+k,k (r 0 , . . . , r k ) = (r 0 •••r k ) -n (n+k-1)•••(n+1)n 0≤l k ≤l k-1 ≤••• •••≤l 1 ≤n-1 r l 1 0 r l 2 +(n-1)-l 1 1 • • • r l k +(n-1)-l k-1 k-1 r (n-1)-l k k . ( 3.14)
Proof The first and second equalities follow directly from the third that we prove now. Equality (3.14) is true for k = 1 (the case k = 0 is just the convention (2.17)) since

I n,1 (r 0 , r 1 ) = 1 0 ds 1 (r 0 + s 1 (r 1 -r 0 ) -n = 1 n-1 (r 0 -r 1 ) -1 [r -n+1 1 -r -n+1 0 ] = 1 n-1 (r 0 r 1 ) -n+1 n-2 l 1 =0 r l 1 0 r n-2-l 1 1 .
Assuming (3.14) holds true for l = 0, . . . , k -1, formula (3.1) gives

I n+k,k (r 0 , . . . , r k ) = 1 n+k-1 (r k-1 -r k ) -1 I n+k-1,k-1 (r 0 , . . . , r k-2 , r k ) -I n+k-1,k-1 (r 0 , . . . , r k-2 , r k-1 ) .
The term in bracket is

(r 0 •••r k-2 r k ) -n (n+k-2)•••n 0≤l k-1 ≤l k-2 ≤••• •••≤l 1 ≤n-1 r l 1 0 r l 2 +(n-1)-l 1 1 • • • r l k-1 +(n-1)-l k-2 k-2 r n-1-l k-1 k -(r 0 •••r k-2 r k-1 ) -n (n+k-2)•••n 0≤l k-1 ≤l k-2 ≤••• •••≤l 1 ≤n-1 r l 1 0 r l 2 +(n-1)-l 1 1 • • • r l k-1 +(n-1)-l k-2 k-2 r n-1-l k-1 k-1
.

Thus

I n+k,k (r 0 , . . . , r k+1 ) = (r 0 •••r k-2 ) -n (n+k-1)•••n 0≤l k-1 ≤l k-2 ≤••• •••≤l 1 ≤n-1 r l 1 0 r l 2 +(n-1)-l 1 1 • • • r l k-1 +(n-1)-l k-2 k-2 (r k-1 -r k ) -1 r -n k r n-1-l k-1 k -r -n k-1 r (n-1)-l k-1 k-1 .
Since the last line is equal to

(r k-1 r k ) -1-l k-1 0≤l k ≤l k-1 r l k k-1 r l k-1 -l k k = (r k-1 r k ) -n 0≤l k ≤l k-1 r (n-1)+l k -l k-1 k-1 r (n-1)-l k k
we have proved (3.14).

The interest of (3.14) is the fact that in (2.18) we have the following: for

B 0 ⊗ • • • ⊗ B k ∈ H k , I n+k,k R 0 (u[σ]), . . . , R k (u[σ]) [B 0 ⊗ • • • ⊗ B k ] = 1 (n+k-1)•••(n+1)n 0≤l k ≤l k-1 ≤••• •••≤l 1 ≤n-1 B 0 u[σ] l 1 -n ⊗ B 1 u[σ] l 2 -l 1 -1 ⊗ • • • • • • ⊗ B k-1 u[σ] l k -l k-1 -1 ⊗ B k u[σ] -l k -1 ; (3.15)
or viewed as an operator in

B(H k , M N ) (see diagram (2.10)): m • κ * • I n+k,k R 0 (u[σ]), . . . , R k (u[σ]) [B 1 ⊗ • • • ⊗ B k ] = 1 (n+k-1)•••(n+1)n 0≤l k ≤l k-1 ≤••• •••≤l 1 ≤n-1 u[σ] l 1 -n B 1 u[σ] l 2 -l 1 -1 B 2 • • • B k-1 u[σ] l k -l k-1 -1 B k u[σ] -l k -1 .
While, if one wants to use directly (3.10) on

B 0 ⊗ • • • ⊗ B k , we face the difficulty to evaluate [R i (u) -R j (u)] -1 [B 0 ⊗ • • • ⊗ B k ] in H k .
Another defect of (3.10) shared by (3.11) is that it suggests an improper behavior of integrals I n+1,k+n when two variables r i are equal. But the continuity proved in Proposition 3.1 shows that this is just an artifact.

An example for u µν = g µν u

Here, we explicitly compute a 1 (x) assuming P satisfies (1.1) and (4.1). Given a strictly positive matrix u(x) ∈ M N where x ∈ (M, g), we satisfy Hypothesis 1.2 with

u µν (x) := g µν (x) u(x). ( 4.1)
This implies that

H(x, ξ) = |ξ| 2 g(x) u(x) where |ξ| 2 g(x) := g µν (x) ξ µ ξ ν .
Of course the fact that u[σ] = u is then independent of σ, simplifies considerably (2.18) since the integral in ξ can be performed. Thus we assume (4.1) from now on and (2.18) becomes

T k,p = g d G(g) µ 1 ...µ 2p I d/2+p,k R 0 (u), R 1 (u), . . . , R k (u) ∈ B( H k ) (4.2)
with (see [23, Section 1.1])

g d := 1 (2π) d R d dξ e -|ξ| 2 g(x) = √ |g| 2 d π d/2 , G(g) µ 1 ...µ 2p := 1 (2π) d g d dξ ξ µ 1 • • • ξ µ 2p e -g αβ ξαξ β = 1 2 2p p! ρ∈S 2p g µ ρ(1) µ ρ(2) • • • g µ ρ(2p-1) µ ρ(2p) = (2p)! 2 2p p! g (µ 1 µ 2 ...µ 2p ) (4.3)
where |g| := det(g µν ), S 2p is the symmetric group of permutations on 2p elements and the parenthesis in the index of g is the complete symmetrization over all indices. Recall from Lemma 2.3 that if u has a spectral decomposition

u = N -1 i=0 r i E i ,
each R j (u) has the same spectrum as u.

Using the shortcuts

I d/2+p,k := I d/2+p,k R 0 (u), . . . , R k (u) ,
the formula (2.8) becomes simply

1 (2π) d dξ ξ µ 1 • • • ξ µ 2p f k (ξ)[ B µ 1 ...µ 2p k ] = g d (m • κ * • I d/2+p,k )[G(g) µ 1 ...µ 2p B µ 1 ...µ 2p k ] = g d (m • I d/2+p,k )[1 ⊗ G(g) µ 1 ...µ 2p B µ 1 ...µ 2p k ]. (4.4)
In particular, it is possible to compute the dimension-free contractions G(g)

µ 1 ...µ 2p B µ 1 ...µ 2p k
before evaluating the result in the I d/2+p,k 's. For a 0 (x), we get

a 0 (x) = tr 1 (2π) d dξ e -u(x)|ξ| 2 g(x) = g d (x) tr[u(x) -d/2 ]. (4.5)
In the sequel, we use frequently the following

Lemma 4.1 For any n 1 , n 2 , n 3 ∈ N, A 1 , A 2 ∈ M N , α ∈ R + , and k = n 1 + n 2 + n 3 + 2, define X(A 1 , A 2 ) := tr(m • I α,k [1 ⊗ u ⊗ • • • ⊗ u n 1 ⊗A 1 u ⊗ • • • ⊗ u n 2 ⊗A 2 ⊗ u ⊗ • • • ⊗ u n 3 ]).
we have

X(A 1 , A 2 ) = r 0 ,r 1 r n 1 +n 3 0 r n 2 1 I α,k (r 0 , • • • , r 0 n 1 +1 , r 1 , • • • , r 1 n 2 +1 , r 0 , • • • , r 0 n 3 +1 ) tr (E 0 A 1 E 1 A 2 ) (4.6)
where E i is the eigenprojection associated to eigenvalues r i of u.

In particular,

if [A 1 , u] = 0, X(A 1 , A 2 ) = 1 k! tr(u k-α-2 A 1 A 2 ). (4.7)
Proof The number X(A 1 , A 2 ) is equal to

r i I α,k (r 0 , . . . , r k ) tr E 0 uE 1 u • • • E n 1 n 1 +1 A 1 E n 1 +1 uE 1 u • • • E n 1 +n 2 +1 n 2 +1 A 2 E n 1 +n 2 +2 u • • • uE n 1 +n 2 +n 3 +2 n 3 +1 = r i I α,k (r 0 , . . . , r k ) tr E n 1 +n 2 +2 u • • • uE n 1 +n 2 +n 3 +2 E 0 uE 1 u • • • E n 1 A 1 E n 1 +1 uE 1 u • • • E n 1 +n 2 +1 A 2 = r i r n 1 +n 3 0 r n 2 1 I α,k (r 0 , • • • , r 0 n 1 +1 , r 1 , • • • , r 1 n 2 +1 , r 0 , • • • , r 0 n 3 +1
) tr(u

n 1 +n 3 E 0 A 1 u n 2 E 1 A 2 )
yielding (4.6).

The particular case follows from tr(E

0 A 1 E 1 A 2 ) = δ 0,1 tr(E 0 A 1 A 2 ) and I α,k (r 0 , • • • , r 0 k+1 ) = 1 k! r -α 0 . (4.8)
We also quote for further references the elementary Corollary 4.2 For any symmetric tensor S ab = S ba , any A a , A b ∈ M N and any function g : (r 0 , r 1 )

∈ (R * + ) 2 → R, r i g(r 0 , r 1 )S ab tr(π r 0 A a π r 1 A b ) = r i 1 2 [g(r 0 , r 1 ) + g(r 1 , r 0 )] S ab tr(π r 0 A a π r 1 A b ). (4.9)
We now divide the computation of a 1 (x) into several steps.

Collecting all the arguments

As a first step, we begin to collect all terms B µ 1 ...µ 2p k of (4.4) due to the different variables appearing in (2.20) and (2.21), including their signs.

Variable in f 1 : w.

Variables in f 2 without the common factor ξ µ 1 ξ µ 2 and summation over µ 1 , µ 2 :

-u µν ⊗ ∂ µ ∂ ν H → -g µν (∂ µ ∂ ν g µ 1 µ 2 ) u ⊗ u -2g µν (∂ µ g µ 1 µ 2 ) u ⊗ ∂ ν u -g µν g µ 1 µ 2 u ⊗ ∂ µ ∂ ν u -v µ ⊗ ∂ µ H → -(∂ µ g µ 1 µ 2 ) v µ ⊗ u -g µ 1 µ 2 v µ ⊗ ∂ µ u -v ⊗ v → -v µ 1 ⊗ v µ 2 -2ū µ ⊗ ∂ µ v → -2g µµ 1 u ⊗ ∂ µ v µ 2 .
Variables in f 3 without the commun factor Π 4 i=1 ξ µ i and summation over the µ i :

2u µν ⊗ ∂ µ H ⊗ ∂ ν H → +2g µν (∂ µ g µ 1 µ 2 )(∂ ν g µ 3 µ 4 ) u ⊗ u ⊗ u +2g µν (∂ µ g µ 1 µ 2 )g µ 3 µ 4 u ⊗ u ⊗ ∂ ν u +2g µν g µ 1 µ 2 (∂ ν g µ 3 µ 4 ) u ⊗ ∂ µ u ⊗ u +2g µν g µ 1 µ 2 g µ 3 µ 4 u ⊗ ∂ µ u ⊗ ∂ ν u 2v ⊗ ūµ ⊗ ∂ µ H → +2g µµ 2 (∂ µ g µ 3 µ 4 ) v µ 1 ⊗ u ⊗ u + 2g µµ 2 g µ 3 µ 4 v µ 1 ⊗ u ⊗ ∂ µ u 2ū µ ⊗ ∂ µ H ⊗ v → +2g µµ 1 (∂ µ g µ 2 µ 3 ) u ⊗ u ⊗ v µ 4 + 2g µµ 1 g µ 2 µ 3 u ⊗ ∂ µ u ⊗ v µ 4 2ū µ ⊗ v ⊗ ∂ µ H → +2g µµ 1 (∂ µ g µ 3 µ 4 ) u ⊗ v µ 2 ⊗ u + 2g µµ 1 g µ 3 µ 4 u ⊗ v µ 2 ⊗ ∂ µ u 4ū µ ⊗ ∂ µ ūν ⊗ ∂ ν H → +4g µµ 1 (∂ µ g νµ 2 )(∂ ν g µ 3 µ 4 ) u ⊗ u ⊗ u +4g µµ 1 (∂ µ g νµ 2 )g µ 3 µ 4 u ⊗ u ⊗ ∂ ν u +4g µµ 1 g νµ 2 (∂ ν g µ 3 µ 4 ) u ⊗ ∂ µ u ⊗ u +4g µµ 1 g νµ 2 g µ 3 µ 4 u ⊗ ∂ µ u ⊗ ∂ ν u 4ū µ ⊗ ūν ⊗ ∂ µ ∂ ν H → +4g µµ 1 g νµ 2 (∂ µ ∂ ν g µ 3 µ 4 ) u ⊗ u ⊗ u +4g µµ 1 g νµ 2 (∂ µ g µ 3 µ 4 ) u ⊗ u ⊗ ∂ ν u +4g µµ 1 g νµ 2 (∂ ν g µ 3 µ 4 ) u ⊗ u ⊗ ∂ µ u +4g µµ 1 g νµ 2 g µ 3 µ 4 u ⊗ u ⊗ ∂ µ ∂ ν u
Variables in f 4 without the commun factor Π 6 i=1 ξ µ i and summation over the µ i :

-4ū µ ⊗ ∂ µ H ⊗ ūν ⊗ ∂ ν H → -4g µµ 1 (∂ µ g µ 2 µ 3 )g νµ 4 (∂ ν g µ 5 µ 6 ) u ⊗ u ⊗ u ⊗ u -4g µµ 1 (∂ µ g µ 2 µ 3 )g νµ 4 g µ 5 µ 6 u ⊗ u ⊗ u ⊗ ∂ ν u -4g µµ 1 g µ 2 µ 3 g νµ 4 (∂ ν g µ 5 µ 6 ) u ⊗ ∂ µ u ⊗ u ⊗ u -4g µµ 1 g µ 2 µ 3 g νµ 4 g µ 5 µ 6 u ⊗ ∂ µ u ⊗ u ⊗ ∂ ν u -4ū µ ⊗ ūν ⊗ ∂ µ H ⊗ ∂ ν H → -4g µµ 1 g νµ 2 (∂ µ g µ 3 µ 4 )(∂ ν g µ 5 µ 6 ) u ⊗ u ⊗ u ⊗ u -4g µµ 1 g νµ 2 (∂ µ g µ 3 µ 4 )g µ 5 µ 6 u ⊗ u ⊗ u ⊗ ∂ ν u -4g µµ 1 g νµ 2 g µ 3 µ 4 (∂ ν g µ 5 µ 6 ) u ⊗ u ⊗ ∂ µ u ⊗ u -4g µµ 1 g νµ 2 g µ 3 µ 4 g µ 5 µ 6 u ⊗ u ⊗ ∂ µ u ⊗ ∂ ν u -4ū µ ⊗ ūν ⊗ ∂ ν H ⊗ ∂ µ H → -4g µµ 1 g νµ 2 (∂ ν g µ 3 µ 4 )(∂ µ g µ 5 µ 6 ) u ⊗ u ⊗ u ⊗ u -4g µµ 1 g νµ 2 (∂ ν g µ 3 µ 4 )g µ 5 µ 6 u ⊗ u ⊗ u ⊗ ∂ µ u -4g µµ 1 g νµ 2 g µ 3 µ 4 (∂ µ g µ 5 µ 6 ) u ⊗ u ⊗ ∂ ν u ⊗ u -4g µµ 1 g νµ 2 g µ 3 µ 4 g µ 5 µ 6 u ⊗ u ⊗ ∂ ν u ⊗ ∂ µ u.
A second and tedious step is now to do in (4.4) the metric contractions G(g)

µ 1 ...µ 2p B µ 1 ...µ 2p k
for previous terms where the G(g) µ 1 ...µ 2p are given by:

G(g) µ 1 µ 2 = 1 2 g µ 1 µ 2 , (4.10) G(g) µ 1 µ 2 µ 3 µ 4 = 1 4 (g µ 1 µ 2 g µ 3 µ 4 + g µ 1 µ 3 g µ 2 µ 4 + g µ 1 µ 4 g µ 2 µ 3 ) , ( 4.11)
G(g) µ 1 µ 2 µ 3 µ 4 µ 5 µ 6 = 1 8 + g µ 1 µ 2 g µ 3 µ 4 g µ 5 µ 6 + g µ 1 µ 2 g µ 3 µ 5 g µ 4 µ 6 + g µ 1 µ 2 g µ 3 µ 6 g µ 4 µ 5 + g µ 1 µ 3 g µ 2 µ 4 g µ 5 µ 6 + g µ 1 µ 3 g µ 2 µ 5 g µ 4 µ 6 + g µ 1 µ 3 g µ 2 µ 6 g µ 4 µ 5 + g µ 1 µ 4 g µ 2 µ 3 g µ 5 µ 6 + g µ 1 µ 4 g µ 2 µ 5 g µ 3 µ 6 + g µ 1 µ 4 g µ 2 µ 6 g µ 3 µ 5 + g µ 1 µ 5 g µ 2 µ 3 g µ 4 µ 6 + g µ 1 µ 5 g µ 2 µ 4 g µ 3 µ 6 + g µ 1 µ 5 g µ 2 µ 6 g µ 3 µ 4 + g µ 1 µ 6 g µ 2 µ 3 g µ 4 µ 5 + g µ 1 µ 6 g µ 2 µ 4 g µ 3 µ 5 + g µ 1 µ 6 g µ 2 µ 5 g µ 3 µ 4 . (4.12)
Keeping the same order already obtained in the first step, we get after the contactions: Contribution of f 1 variable: w (no contraction since p = 0). Contribution of f 2 variables:

-1 2 g µν g ρσ (∂ µ ∂ ν g ρσ ) u ⊗ u -g µν g ρσ (∂ ν g ρσ ) u ⊗ ∂ µ u -d 2 g µν u ⊗ ∂ µ ∂ ν u -1 2 g ρσ (∂ µ g ρσ ) v µ ⊗ u -d 2 v µ ⊗ ∂ µ u -1 2 g µν v µ ⊗ v ν -u ⊗ ∂ µ v µ .
Contribution of f 3 variables:

1 2 g µν g ρσ g αβ (∂ µ g ρσ )(∂ ν g αβ ) + g µν g ρσ g αβ (∂ µ g ρα )(∂ ν g σβ ) u ⊗ u ⊗ u + d+2 2 g µν g ρσ (∂ ν g ρσ ) u ⊗ u ⊗ ∂ µ u + d+2 2 g µν g ρσ (∂ ν g ρσ ) u ⊗ ∂ µ u ⊗ u + d(d+2) 2 g µν u ⊗ ∂ µ u ⊗ ∂ ν u + 1 2 g ρσ (∂ µ g ρσ ) v µ ⊗ u ⊗ u + g µν (∂ ρ g ρν ) v µ ⊗ u ⊗ u + d+2 2 v µ ⊗ u ⊗ ∂ µ u + 1 2 g ρσ (∂ µ g ρσ ) u ⊗ u ⊗ v µ + g µν (∂ ρ g ρν ) u ⊗ u ⊗ v µ + d+2 2 u ⊗ ∂ µ u ⊗ v µ + 1 2 g ρσ (∂ µ g ρσ ) u ⊗ v µ ⊗ u + g µν (∂ ρ g ρν ) u ⊗ v µ ⊗ u + d+2 2 u ⊗ v µ ⊗ ∂ µ u + g ρσ (∂ µ g µν )(∂ ν g ρσ ) + 2g ρσ (∂ µ g νρ )(∂ ν g µσ ) u ⊗ u ⊗ u + (d + 2)(∂ ν g µν ) u ⊗ u ⊗ ∂ µ u + [g µν g ρσ (∂ ν g ρσ ) + 2(∂ ν g µν )] u ⊗ ∂ µ u ⊗ u + (d + 2)g µν u ⊗ ∂ µ u ⊗ ∂ ν u + [g µν g ρσ (∂ µ ∂ ν g ρσ ) + 2(∂ µ ∂ ν g µν )] u ⊗ u ⊗ u + [g µν g ρσ (∂ ν g ρσ ) + 2(∂ ν g µν )] u ⊗ u ⊗ ∂ µ u + [g µν g ρσ (∂ ν g ρσ ) + 2(∂ ν g µν )] u ⊗ u ⊗ ∂ µ u + (d + 2)g µν u ⊗ u ⊗ ∂ µ ∂ ν u.
which, once collected, gives

g µν g ρσ (∂ µ ∂ ν g ρσ ) + 2(∂ µ ∂ ν g µν ) + g ρσ (∂ µ g µν )(∂ ν g ρσ ) + 2g ρσ (∂ µ g νρ )(∂ ν g µσ ) 1 2 g µν g ρσ g αβ (∂ µ g ρσ )(∂ ν g αβ ) + g µν g ρσ g αβ (∂ µ g ρα )(∂ ν g σβ ) u ⊗ u ⊗ u + (d + 6) 1 2 g µν g ρσ (∂ ν g ρσ ) + (∂ ν g µν ) u ⊗ u ⊗ ∂ µ u + d+4 2 g µν g ρσ (∂ ν g ρσ ) + 2(∂ ν g µν ) u ⊗ ∂ µ u ⊗ u + (d+2) 2 2 g µν u ⊗ ∂ µ u ⊗ ∂ ν u + (d + 2)g µν u ⊗ u ⊗ ∂ µ ∂ ν u + 1 2 g ρσ (∂ µ g ρσ ) + g µν (∂ ρ g ρν ) (v µ ⊗ u ⊗ u + u ⊗ v µ ⊗ u + u ⊗ u ⊗ v µ ) + d+2 2 (v µ ⊗ u ⊗ ∂ µ u + u ⊗ ∂ µ u ⊗ v µ + u ⊗ v µ ⊗ ∂ µ u)
Contribution of f 4 variables: We use the following symmetry: in previous three terms of f 4 , one goes from the first to the second right terms by the change (µ 2 , µ 3 , µ 4 ) → (µ 3 , µ 4 , µ 2 ) and from the second to the third terms via (µ, ν) → (ν, µ) and (µ 1 , µ 2 ) → (µ 2 , µ 1 ). So after the contraction of the first term and using that symmetry (which explains the factors 3 and 2), we get

3 -1 2 g µν g ρσ g αβ (∂ µ g ρσ )(∂ ν g αβ ) -g µν g ρσ g αβ (∂ µ g ρα )(∂ ν g σβ ) -2g ρσ (∂ µ g µν )(∂ ν g ρσ ) -2g ρσ (∂ µ g µρ )(∂ ν g νσ ) -2g ρσ (∂ µ g νρ )(∂ ν g µσ ) u ⊗ u ⊗ u ⊗ u -(d + 4) (∂ µ g µν ) + 1 2 g µν g ρσ (∂ µ g ρσ ) 3 u ⊗ u ⊗ u ⊗ ∂ ν u + 2 u ⊗ u ⊗ ∂ ν u ⊗ u + u ⊗ ∂ ν u ⊗ u ⊗ u -1 2 (d + 4)(d + 2)g µν 2 u ⊗ u ⊗ ∂ µ u ⊗ ∂ ν u + u ⊗ ∂ µ u ⊗ u ⊗ ∂ ν u .
It worth to mention that all results of this section 4.1 are valid in arbitrary dimension d of the manifold.

Application of operators I d/2+p,k

We can now compute in (4.4) the application of

I d/2+p,k on each previous G(g) µ 1 ...µ 2p B µ 1 ...µ 2p k
. We restrict to even dimension d = 2m since we will prove in Section 5.2 that it is impossible to get explicit formulae when d is odd (explicit in the sense that we do want to avoid ending with formulae like (1.5)).

Lemma 2.6 tells us that we have only to apply the four operators given by (3.10) for k = 1, 2, 3, 4:

I m+k-1,k (r 0 , . . . , r k ) = (-1) k (m-1)(m)•••(m+k-2) k i=0 k j=0 j =i (r i -r j ) -1 r -(m-1) i , when m = 1, I k,k (r 0 , . . . , r k ) = (-1) k+1 (k-1)! k i=0 k j=0 j =i (r i -r j ) -1 r -1 i log r i , when m = 1 = lim m→1 I m+k-1,k (r 0 , . . . , r k ) (see Proposition 3.1).
They are applied on the above f k variables which have the form

A 1 ⊗ • • • ⊗ A k where each A i is equal to u, v µ ,
B 1 = B 2 = u. Case 2: (B 1 = ∂ µ u, B 2 = u), (B 1 = ∂ µ ∂ ν u, B 2 = u), (B 1 = v µ , B 2 = u), (B 1 = ∂ µ v µ , B 2 = u). Case 3: (B 1 = v µ , B 2 = v ν ), (B 1 = v µ , B 2 = ∂ µ u), (B 1 = ∂ µ u, B 2 = ∂ ν u).
We use the shortcut

J m+k-1,k [A 1 ⊗ • • • ⊗ A k ] := tr(m • I m+k-1,k [1 ⊗ A 1 ⊗ A 2 ⊗ • • • ⊗ A k ]).
We first give two examples of such computations of J. The first one corresponds to Case 0 and is given by the variable w in f 1 . Its contribution to a 1 is:

J m,1 [w] = tr(m • I m,1 (R 0 (u), R 1 (u))[1 ⊗ w]) = r 0 ,r 1 I m,1 (r 0 , r 1 ) tr(E 0 wE 1 ) = r 0 I m,1 (r 0 , r 0 ) tr(E 0 w) = tr(u -m w) since I m,1 (r 0 , r 0 ) = r -m 0 by (3.
2). Our second example comes from Case 3 and is given by the variable -1 2 g µν v µ ⊗ v ν in f 2 : we have

J m+1,2 [v µ ⊗ v ν ] = tr(m • I m+1,2 (R 0 (u), R 1 (u), R 2 (u))[1 ⊗ v µ ⊗ v ν ]) = r 0 ,r 1 ,r 2 I m+1,2 (r 0 , r 1 , r 2 ) tr(E 0 v µ E 1 v ν E 2 ) = r 0 ,r 1 I m+1,2 (r 0 , r 1 , r 0 ) tr(E 0 v µ E 1 v ν ).
Thus, using Corollary 4.2 and (A.3), its contribution to a 1 is

-1 2 g µν J m+1,2 [v µ ⊗ v ν ] = -gµν 2 r 0 ,r 1 1 2 [I m+1,2 (r 0 , r 1 , r 0 ) + I m+1,2 (r 1 , r 0 , r 1 )] tr(E 0 v µ E 1 v ν ) (4.13) = -1 2 g µν r 0 ,r 1 1 2m m-1 =0 r --1 0 r -m 1 tr(E 0 v µ E 1 v ν ) = -1 4m g µν m-1 =0 tr(u --1 v µ u -m v ν ).
We now consider all contributions. Case 1: Thanks to (3.2)

J m+k-1,k [u ⊗ • • • ⊗ u k ] = 1 k! tr(u -m+1 )
and we get

-1 2 g µν g ρσ (∂ µ ∂ ν g ρσ ) J m+1,2 [u ⊗ u] + g µν g ρσ (∂ µ ∂ ν g ρσ ) + 2(∂ µ ∂ ν g µν ) + g ρσ (∂ µ g µν )(∂ ν g ρσ ) + 2g ρσ (∂ µ g νρ )(∂ ν g µσ ) 1 2 g µν g ρσ g αβ (∂ µ g ρσ )(∂ ν g αβ ) + g µν g ρσ g αβ (∂ µ g ρα )(∂ ν g σβ ) J m+2,3 [u ⊗ u ⊗ u] + 3 -1 2 g µν g ρσ g αβ (∂ µ g ρσ )(∂ ν g αβ ) -g µν g ρσ g αβ (∂ µ g ρα )(∂ ν g σβ ) -2g ρσ (∂ µ g µν )(∂ ν g ρσ ) -2g ρσ (∂ µ g µρ )(∂ ν g νσ ) -2g ρσ (∂ µ g νρ )(∂ ν g µσ ) J m+3,4 [u ⊗ u ⊗ u ⊗ u] = α tr(u -m+1 )
where .4). So this contribution is

α := 1 3 (∂ µ ∂ ν g µν ) -1 12 g µν g ρσ (∂ µ ∂ ν g ρσ ) + 1 48 g µν g ρσ g αβ (∂ µ g ρσ )(∂ ν g αβ ) + 1 24 g µν g ρσ g αβ (∂ µ g ρα )(∂ ν g σβ ) -1 12 g ρσ (∂ µ g µν )(∂ ν g ρσ ) + 1 12 g ρσ (∂ µ g νρ )(∂ ν g µσ ) -1 4 g ρσ (∂ µ g µρ )(∂ ν g νσ ). (4.14) Case 2: (B 1 = ∂ µ u, B 2 = u) generating terms in tr(u -m ∂ µ u) with coefficient -1 2 g µν g ρσ (∂ ν g ρσ ) + (2m+6) 3! 1 2 g µν g ρσ (∂ ν g ρσ ) + (∂ ν g µν ) + 1 3! 2m+4 2 g µν g ρσ (∂ ν g ρσ ) + 2(∂ ν g µν ) -6(2m+4) 4! (∂ ν g µν ) + 1 2 g µν g ρσ (∂ ν g ρσ ) = m-2 6 1 2 g µν g ρσ (∂ ν g ρσ ) -(∂ ν g µν ) (B 1 = ∂ µ ∂ ν u, B 2 = u) generating terms in tr(u -m ∂ µ ∂ ν u) with coefficient -2m 2 g µν 1 2! + (2m + 2)g µν 1 3! = -m-2 6 g µν (B 1 = v µ , B 2 = u) generating terms in tr(u -m v µ ) with coefficient -1 2 1 2! g ρσ (∂ µ g ρσ ) + 3 3! 1 2 g ρσ (∂ µ g ρσ ) + g µν (∂ ρ g ρν ) = 1 2 g µν (∂ ρ g ρν ). (B 1 = ∂ µ v µ , B 2 = u) generating a term in tr(u -m ∂ µ v µ ) with coefficient -1 2! = -1 2 . Case 3: (B 1 = v µ , B 2 = ∂ µ u) generating terms in tr[E 0 v µ E 1 (∂ µ u)] with coefficient r 0 ,r 1 -mI m+1,2 (r 0 , r 1 , r 0 ) + 2m+2 2 [r 1 I m+2,3 (r 0 , r 1 , r 1 , r 0 ) + r 1 I m+2,3 (r 1 , r 1 , r 0 , r 1 ) + r 0 I m+2,3 (r 0 , r 0 , r 1 , r 0 )] = 1 2m m-1 =0 (m -2 ) r --1 0 r -m 1 thanks to (A
1 2m m-1 =0 (m -2 ) tr[u --1 v µ u -m (∂ µ u)] (B 1 = ∂ µ u, B 2 = ∂ ν u) generating terms in tr[E 0 (∂ µ u)E 1 (∂ ν u)] with coefficient 2(m + 1) 2 g µν r 0 I m+2,3 (r 0 , r 0 , r 1 , r 0 ) -2(m + 2)(m + 1)g µν [2r 2 0 I m+3,4 (r 0 , r 0 , r 0 , r 1 , r 0 ) + r 0 r 1 I m+3,4 (r 0 , r 0 , r 1 , r 1 , r 0 )] = g µν g 3 (r 0 , r 1 )
with the definition of g 3 in (A.5). Thanks to (A.6), this contribution is

1 6m g µν m-1 =0 (m 2 -2m -3 (m -1 -)) tr[u --1 (∂ µ u)u -m (∂ ν u)].

Main results

The recollection of all contributions (4.4) for a 1 (x) from Cases 0-3 is now ready for the first interesting result:

Theorem 4.3 Assume that P = -(u g µν ∂ µ ∂ ν + v ν ∂ ν + w) is a selfadjoint elliptic operator acting on L 2 (M, V
) for a 2m-dimensional boundaryless Riemannian compact manifold (M, g) and a vector bundle V over M where u, v µ , w are local maps on M with values in M N , with u positive and invertible. Then, its local

a 1 (x) heat-coefficient in (1.3) for x ∈ M is a 1 =g 2m tr(u -m w) + α tr(u -m+1 ) + m-2 6 1 2 g µν g ρσ (∂ ν g ρσ ) -(∂ ν g µν ) tr(u -m ∂ µ u) -m-2 6 g µν tr(u -m ∂ µ ∂ ν u) + 1 2 g µν (∂ ρ g ρν ) tr(u -m v µ ) -1 2 tr(u -m ∂ µ v µ ) -1 4m m-1 =0 g µν tr(u --1 v µ u -m v ν ) + 1 2m m-1 =0 (m -2 ) tr[u --1 v µ u -m (∂ µ u)] + m-1 =0 m-2 6 -(m--1) 2m g µν tr[u --1 (∂ µ u)u -m (∂ ν u)] (4. 15
)
where α is given in (4.14).

Since the operator P is not written in terms of objects which have simple (homogeneous) transformations by change of coordinates and gauge transformation, this result does not make apparent any explicit Riemannian or gauge invariant expressions. This is why we have not used normal coordinates until now. Nevertheless, from Lemma A.5, one can deduce after a long computation: Lemma 4.4 Under a gauge transformation, a 1 (x) given by (4.15) is gauge invariant.

As shown in A.4, with the help of a gauge connection A µ , one can change the variables (u, v µ , w) to variables (u, p µ , q) well adapted to changes of coordinates and gauge transformations (see (A.12) and (A.13)). For u µν = g µν u, (A.14) and (A.15) becomes

v µ = -1 2 g µν g ρσ (∂ ν g ρσ ) + ∂ ν g µν u + g µν (∂ ν u) + g µν (uA ν + A ν u) + p µ (4.16) w = -1 2 g µν g ρσ (∂ ν g ρσ ) + ∂ ν g µν uA µ + g µν (∂ ν u)A µ + g µν u(∂ µ A ν ) + g µν A µ uA ν + p µ A µ + q. ( 4.17)
Relations (4.16) and (4.17) can be injected into (4.15) to get an explicitly diffeomorphism and gauge invariant expression. In order to present the result of this straightforward computation, let us introduce the following notations. Given the Christoffel symbols Γ ρ µν :=

1 2 g ρσ (∂ µ g σν + ∂ ν g σµ -∂ σ g µν ), the Riemann curvature tensor R α βµν := ∂ µ Γ α βν -∂ ν Γ α βµ + Γ α µρ Γ ρ βν -Γ α νρ Γ ρ βµ
, and the Ricci tensor R µν := R ρ µρν , the scalar curvature R := g µν R µν computed in terms of the derivatives of the inverse metric is

R = g µν g ρσ (∂ µ ∂ ν g ρσ ) -(∂ µ ∂ ν g µν ) + g ρσ (∂ µ g µν )(∂ ν g ρσ ) + 1 2 g ρσ (∂ µ g νρ )(∂ ν g µσ ) -1 4 g µν g ρσ g αβ (∂ µ g ρσ )(∂ ν g αβ ) -5 4 g µν g ρσ g αβ (∂ µ g ρα )(∂ ν g σβ ), (4.18
) and one has

g µν Γ ρ µν = 1 2 g ρσ g αβ (∂ σ g αβ ) -∂ σ g ρσ , Γ σ σρ = -1 2 g αβ (∂ ρ g αβ ).
Let ∇ µ be the (gauge) covariant derivative on V (and its related bundles):

∇ µ s := ∂ µ s + A µ s for any section s of V.
From (A.13), u, p µ and q are sections of the endomorphism vector bundle End(V ) = V * ⊗ V (while v µ and w are not), so that ∇ µ u = ∂ µ u + [A µ , u] (and the same for p µ and q). We now define ∇ µ , which combines ∇ µ and the linear connection induced by the metric g:

∇ µ u := ∂ µ u + [A µ , u] = ∇ µ u, ∇ µ p ρ := ∂ µ p ρ + [A µ , p ρ ] + Γ ρ µν p ν = ∇ µ p ρ + Γ ρ µν p ν ∇ µ ∇ ν u := ∂ µ ∇ ν u + [A µ , ∇ ν u] -Γ ρ µν ∇ ρ u = ∂ µ ∇ ν u + [A µ , ∇ ν u] -Γ ρ µν ∇ ρ u , so that, if ∆ ∇ := g µν ∇ µ ∇ ν is the connection Laplacian ∆ ∇ u = g µν ∇ µ ∇ ν u -1 2 g µν g αβ (∂ µ g αβ ) -∂ µ g µν ∇ ν u, ∇ µ p µ = ∇ µ p µ -1 2 g αβ (∂ µ g αβ )p µ .
Any relation involving u, p µ , q, g and ∇ µ inherits the homogeneous transformations by changes of coordinates and gauge transformations of these objects. Let us state now the result of the computation of a 1 (x) in terms of (u, p µ , q): Theorem 4.5 Assume that P = -(|g| -1/2 ∇ µ |g| 1/2 g µν u∇ ν + p µ ∇ µ + q) is a selfadjoint elliptic operator acting on L 2 (M, V ) for a 2m-dimensional boundaryless Riemannian compact manifold (M, g) and a vector bundle V over M where u, p µ , q are sections of endomorphisms on V with u positive and invertible. Then, its local

a 1 (x) heat-coefficient in (1.3) for x ∈ M is a 1 =g 2m 1 6 R tr[u 1-m ] + tr[u -m q] -m+1 6 tr[u -m ∆ ∇ u] -1 2 tr[u -m ∇ µ p µ ] + m-1 =0 2m 2 -4m+3 12m
-(m--1) 2m g µν tr[u --1 ( ∇ µ u)u -m ( ∇ ν u)] + 1 2m m-1 =0 (m -2 -1) tr[u --1 p µ u -m ( ∇ µ u)] -1 4m m-1 =0 g µν tr[u --1 p µ u -m p ν ] (4.19)
where

g 2m = √ |g| 4 m π m .
Proof This can be checked by an expansion of the RHS of (4.19). A more subtle method goes using normal coordinates in (4.15), (4.16), (4.17), knowing that a 1 (x) is a scalar and (u, p µ , q) are well adapted to change of coordinates. The coefficients in the sum of the second line of this expression have been symmetrized ↔ (m --1) using the trace property and the change of variable → m --1.

Corollary 4.6

The previous formula can be written in a more compact way as

a 1 =g 2m 1 6 R tr[u 1-m ] + tr[u -m q] -m+1 6 tr[u -m ∆ ∇ u] -1 2 tr[u -m ∇ µ p µ ] + 1 4m m-1 =0 g µν tr[u --1 [(2 -1) ∇ µ u -p µ ]u -m [(2 -1) ∇ ν u + p ν ]] + 1 2m m-1 =0 g µν tr[u( ∇ µ u --1 )u( ∇ ν u -m )] -m 2 -2m+3 3m m-1 =0 g µν tr[u --1 ( ∇ µ u)u -m ( ∇ ν u)] (4.20)
where p µ = g µν p ν .

Proof This relation can be obtained by expanding the second and third lines of (4.20) using the combinatorial equality

m-1 =0 g µν tr[u(∂ µ u --1 )u(∂ ν u -m )] = m-1 =0 m + (m --1) g µν tr[u --1 (∂ µ u)u -m (∂ ν u)]
that is a tedious computation.

The above formula (4.20) is not the unique way to write (4.19). Some variations are possible using for instance the relations

g µν tr[u(∂ µ ∂ ν u -m )] = -m g µν tr[u -m (∂ µ ∂ ν u)] + (m + 1) m-1 =0 g µν tr[u --1 (∂ µ u)u -m (∂ ν u)], m-1 =0 g µν tr[u(∂ µ u -)(∂ ν u -m )] = m-1 =0 m-1 2 + (m --1) g µν tr[u --1 (∂ µ u)u -m (∂ ν u)], g µν tr[∂ µ (u -m ∂ ν u)] = g µν tr[u -m (∂ µ ∂ ν u)] - m-1 =0 g µν tr[u --1 (∂ µ u)u -m (∂ ν u)],
which make all appear the expression

g µν tr[u --1 (∂ µ u)u -m (∂ ν u)] in (4.19).
Corollary 4.7 When M has dimension 4, the last two terms of (4.20) compensate and the formula simplifies as

a 1 = |g| 1/2 16 π 2 1 6 R tr[u -1 ] + tr[u -2 q] -1 2 g µν tr[u -2 ∇ µ ∇ ν u] -1 2 tr[u -2 ∇ µ p µ ] + 1 4 g µν tr[u -2 ( ∇ µ u -p µ )u -1 ( ∇ ν u + p ν )] . ( 4.21)
Remark 4.8 For the computation of a r (x) with r ≥ 2, directly in terms of variables (u, p µ , q), the strategy is to use normal coordinates from the very beginning, which simplifies the computation of terms B µ 1 ...µ 2p k of (4.4). Then an equivalent result to Theorem 4.3 would be obtained, but only valid in normal coordinates. Thus, by the change of variables (4.16), (4.17), a final result as Theorem 4.5 could be calculated. Remark 4.9 In the present method, the factor 1 6 R is explicitly and straightforwardly computed from the metric entering u µν = g µν u, as in [START_REF] Avramidi | Non-Laplace type operators on manifolds with boundary, in: Analysis, geometry and topology of elliptic operators[END_REF] for instance. Many methods introduce R using diffeomorphism invariance and compute the coefficient 1 6 using some "conformal perturbation" of P (see [START_REF] Gilkey | Asymptotic formulae in spectral geometry[END_REF]Section 3.3]).

Case of scalar symbol

: u(x) = f (x) 1 N
Let us consider now the specific case u(x) = f (x) 1 N , where f is a nowhere vanishing positive function. Then (4.16) simplifies to

v µ = [-1 2 g µν g ρσ (∂ ν g ρσ ) + ∂ ν g µν ]f 1 N + g µν (∂ ν f ) 1 N + 2f g µν A ν
+ p µ and we can always find A µ such that p µ = 0:

A µ = 1 2 g µν f -1 v ν + [ 1 2 g ρσ (∂ µ g ρσ ) -g µν (∂ ρ g ρν ) -f -1 (∂ µ f )] 1 N .
One can check, using (A.8), that A µ satisfies the correct gauge transformations. This means that P can be written as

P = -(|g| -1/2 ∇ µ |g| 1/2 g µν f ∇ ν + q)
where the only matrix-dependencies are in q and A µ . Since u is in the center, ∇ µ u = (∂ µ f ) 1 N and (4.19) simplifies as

a 1 = g 2m f -m [ N 6 f R + tr[q] -m+1 6 N g µν (∂ µ ∂ ν f ) + m+1 6 N g µν Γ ρ µν (∂ ρ f ) + m 2 -m+1 12 N g µν f -1 (∂ µ f )(∂ ν f )]
in which A µ does not appears. Now, if f is constant, we get the well-known result (see [START_REF] Gilkey | Asymptotic formulae in spectral geometry[END_REF]Section 3.3]):

a 1 = g 2m f -m N 6 R + tr[q] .

About the method

Existence

For the operator P given in (1.1), the method used here assumes only the existence of asymptotics (1.3). This is the case when P is elliptic and selfadjoint. Selfadjointness of P on L 2 (M, V ) is not really a restriction since we remark that given an arbitrary family of u µν satisfying (1.2), skewadjoint matrices ṽµ and a selfadjoint matrix w, we get a formal selfadjoint elliptic operator P defined by (1.1) where

v µ = ṽµ + (∂ ν log|g| 1/2 ) u µν + ∂ ν u µν , w = w + 1 2 [-∂ µ ṽµ + (∂ µ log|g| 1/2 ) ṽµ ].
A crucial step in our method is to be able to compute the integral (2.18) for a general u µν . The case u µν = g µν u considered in Section 4 makes that integral manageable.

On explicit formulae for u µν = g µν u

For u µν = g µν u, the proposed method is a direct computational machinery. Since the method can be computerized, this could help to get a r in Case 2 (d even and r < d/2). Recall the steps: 1) to expand the arguments of the initial f k 's, 2) to contract with the tensor G(g), 3) to apply to the corresponding operators I α,k , 4) to collect all similar terms. Further eventual steps: 5) to change variables to (u, p µ , q), 6) to identify (usual) Riemannian tensors and covariant derivatives (in terms of A µ and Christoffel symbols).

Is it possible to get explicit formulae for a r from the original ingredients (u, v µ , w) of P ? An explicit formula should look like (4.15) or (4.19). This excludes the use of spectral decomposition of u as in (1.5) which could not be recombined as

finite sum tr[h (0) (u)B 1 h (1) (u)B 2 • • • B k h (k) (u)]
where the h i are continuous functions and the B i are equal to u, v µ , w or their derivatives. The obstruction to get such formula could only come from the operators I α,k and not from the arguments B i . Thus the matter is to understand the u-dependence of I α,k .

Let us consider I α,k as a map u → I α,k (u). An operator map u → A(u) ∈ B( H k ) is called u-factorizable (w.r.t. the tensor product) if it can be written as

A(u) = finite sum R 0 (h (0) (u)) R 1 (h (1) (u)) • • • R k (h (k) (u))
where the h (i) are continuous functions on R * + . Lemma 5.1 Let u → A(u) = F (R 0 (u), . . . , R k (u)) the operator map defined by a continuous function F : (R * + ) k+1 → R * + . Then A is u-factorizable iff F is decomposed as

F (r 0 , . . . , r k ) = finite sum h (0) (r 0 )h (1) (r 1 ) • • • h (k) (r k ) (5.1)
for continuous functions h (i) .

Proof Let λ i be the eigenvalues of u and π i the associated eigenprojections. If F is decomposed, then by functional calculus, one gets

A(u) = i 0 ,...,i k F (λ i 0 , . . . , λ i k ) R 0 (π i 0 ) • • • R k (π i k ) = i 0 ,...,i k finite sum h (0) (λ i 0 ) • • • h (k) (λ i k ) R 0 (π i 0 ) • • • R k (π i k ) = finite sum R 0 (h (0) (u)) • • • R k (h (k) (u)).
If A is u-factorizable, then this computation can be seen in the other way around to show that F is decomposed.

The general solutions (3.10) and (3.11) for the operators I α,k are not manifestly u-factorizable because of the factors (r i -r j ) -1 . For Case 2, Proposition 3.5 shows that I α,k is indeed a u-factorizable operator (see also (3.15)).

The explicit expressions of the operators I α,k ∈ B( H k ) don't give a definitive answer about the final formula: for instance, when applied to an argument containing some u's, the expression can simplify a lot (see for example Case 1 of Section 4.2). Moreover, the trace and the multiplication introduce some degrees of freedom in the writing of the final expression. This leads us to consider two operators A and A as equivalent when

tr • m • κ * • A[B 1 ⊗ • • • ⊗ B k ] = tr • m • κ * • A [B 1 ⊗ • • • ⊗ B k ] for any B 1 ⊗ • • • ⊗ B k ∈ H k . The equivalence is reminiscent of the lift from B(H k , M N ) to B( H k ) (see Remark 2.
2) combined with the trace.

Lemma 5.2 With

I α,k (r 0 , . . . , r k-1 ) := I α,k (r 0 , . . . , r k-1 , r 0 ) (= lim r k →r 0 I α,k (r 0 , . . . , r k-1 , r k )), the operator I α,k := I α,k (R 0 (u), . . . , R k-1 (u)) = I α,k (R 0 (u), . . . , R k-1 (u), R 0 (u)) ∈ B( H k ) is equivalent with the original I α,k . Proof For any B 1 ⊗ • • • ⊗ B k ∈ H k , using previous notations, one has tr(• m • κ * • I α,k [B 1 ⊗ • • • ⊗ B k ]) = i 0 ,...,i k I α,k (λ i 0 , . . . , λ i k ) tr π i 0 B 1 π i 1 • • • B k π i k = i 0 ,...,i k I α,k (λ i 0 , . . . , λ i k ) tr π i k π i 0 B 1 π i 1 • • • B k = i 0 ,...,i k I α,k (λ i 0 , . . . , λ i k ) δ i 0 ,i k tr π i 0 B 1 π i 1 • • • B k = i 0 ,...,i k-1 I α,k (λ i 0 , . . . , λ i k-1 ) tr π i 0 B 1 π i 1 • • • B k = tr • m • κ * • I α,k [B 1 ⊗ • • • ⊗ B k ].
The equivalence between I α,k and I α,k seems to be the only generic one we can consider.

We have doubts on the fact that the operators I α,k can be always equivalent to some u-factorizable operators. For instance, Proposition 5. [START_REF] Avramidi | Non-Laplace type operators on manifolds with boundary, in: Analysis, geometry and topology of elliptic operators[END_REF] The contribution to a 1 of (4.13) generates always a non-explicit formula when the dimension d is odd, unless u and v µ have commutation relations.

In fact,

h d (r 1/2 0 , r 1/2 1 ) := 1 2 [I d/2+1,2 (r 0 , r 1 , r 0 ) + I d/2+1,2 (r 1 , r 0 , r 1 )] = 1 d (r 1 r 0 ) -d/2 r d/2 0 -r d/2 1 r 0 -
d (xy) d h d (x, y) = x 2m -y 2m x 2 -y 2 = m-1 =0 x 2m-2 -2 y 2 . Assume now d = 2m + 1. Then, d (xy) d h d (x, y) = x 2m+1 -y 2m+1 x 2 -y 2 = y 2m 1 x+y + m-1 =0 x 2m-1-2 y 2 ,
(for m = 0 there is no sum). This expression is not decomposable since the map (x + y) -1 is not decomposable: Suppose we have such decomposition (x + y) -1 = N =1 h 1, (x) h 2, (y) for N ∈ N * . Let (x i , y i ) 1≤i≤N be N points in (R * + ) 2 and consider the N × N -matrix c i,j := (x i + y j ) -1 . Then

det(c) = N i,j=1 (x i + y j ) -1 1≤i<j≤N (x i -x j )(y i -y j ).
This expression shows that we can choose a family (x i , y i ) 1≤i≤N such that det(c) = 0. With such a family, define the two matrices a i, := h 1, (x i ) and b i, := h 2, (y i ). Then

c i,j = (x i + y j ) -1 = N =1 h 1, (x) h 2, (y j ) = N =1 a i, b j, so that, in matrix notation, c = a • t b, which implies that det(a) = 0 and det(b) = 0. From (x + y j ) = N =1 h 1, (x) b j, , we deduce h 1, (x) = j b -1 j, (x + y j ) -1 and, similarly, h 2, (y) = i a -1 i, (x i + y) -1 . This gives (x + y) -1 = i,j, a -1 i, b -1 j, (x + y j ) -1 (x i + y) -1 .
This expression must hold true on (R * + ) 2 and when x, y → 0 + , the LHS goes to +∞ while the RHS remains bounded. This is a contradiction.

Explicit formulae of a r for scalar symbols (u

(x) = f (x) 1 N )
When u is central, the operator defined by I α,k (r 0 , . . . , r k ) is equivalent to the operator defined by

I α,k (r 0 ) := lim r j →r 0 I α,k (r 0 , . . . , r k ) = 1 k! r -α 0 .
Thus (4.4) reduces to

1 (2π) d dξ ξ µ 1 • • • ξ µ 2p f k (ξ)[ B µ 1 ...µ 2p k ] = g d m[u -d/2-p ⊗ G(g) µ 1 ...µ 2p B µ 1 ...µ 2p k ] = g d G(g) µ 1 ...µ 2p u -d/2-p m[B µ 1 ...µ 2p k
] so the tedious part of the computation of a r is to list all arguments B µ 1 ...µ 2p k and to contract them with G(g) µ 1 ...µ 2p . This can be done with the help of a computer in any dimension for an arbitrary r. All formulae are obviously explicit. An eventual other step is to translate the results in terms of diffeomorphic and gauge invariants.

Application to quantum field theory

Second-order differential operators which are on-minimal have a great importance in physics and have justified their related heat-trace coefficients computation (to quote but a few see almost all references given here). For instance in the interesting work [START_REF] Moss | Invariants of the heat equation for non-minimal operators[END_REF][START_REF] Toms | Local momentum space and the vector field[END_REF], the operators P given in (1.1) are investigated under the restriction

u µν = g µν 1 + ζ X µν ,
where ζ is a parameter (describing for ζ = 0 the minimal theory), under the constraints for the normalized symbol X(σ) := X µν σ µ σ ν with |σ| 2 g = g µν σ µ σ ν = 1 given by

X(σ) 2 = X(σ), for any σ ∈ S 1 g , ( 5.2)
∇ ρ X µν = 0.

(5.3)

Here, ∇ ρ is a covariant derivative involving gauge and Christoffel connections. In covariant form, the operators are

P = -(g µν ∇ µ ∇ ν + ζ X µν ∇ µ ∇ ν + Y ).
Despite the restrictions (5.2)-( 5.3) meaning that the operator X is a projector and the tensorendomorphism u is parallel, this covers the case of operators describing a quantized spin-1 vector fields like

P µ ν = -(δ µ ν ∇ 2 + ζ ∇ µ ∇ ν + Y µ ν )
, or a Yang-Mills fields like

P µ ν = -(δ µ ν D 2 -ζ D µ D ν + R µ ν -2F µ ν )
where D µ := ∇ ν + A µ and A µ , F µν are respectively the gauge and strength fields, or a perturbative gravity (see [START_REF] Moss | Invariants of the heat equation for non-minimal operators[END_REF] for details).

Remark first that

H(x, ξ) = u µν ξ µ ξ ν = |ξ| 2 g [1 + ζ X (ξ/|ξ| g )], so that e -H(x,ξ) = [e -(1+ζ)|ξ| 2 g -e -|ξ| 2 g ] X + e -|ξ| 2 g 1 V .
Thus (1.10) becomes

a 0 (x) = 1 (2π) d dξ [e -(1+ζ)|ξ| 2 g -e -|ξ| 2 g ] tr X + 1 (2π) d e -|ξ| 2 g tr 1 V = [ σ∈S 1 g dΩ(σ) tr X(σ)] [ 1 (2π) d ∞ 0 dr r d-1 (e -(1+ζ)r 2 -e -r 2 )] + g d N = Γ(d/2) 2(2π) d [(1 + ζ) -d/2 -1] [ σ∈S 1 g dΩ(σ) tr X(σ)] + g d N.
One has

g d := 1 (2π) d R d dξ e -|ξ| 2 g(x) = |g| 1/2 2 d π d/2 and σ∈S 1 g dΩ(σ) tr X(σ) = tr(X µν ) σ∈S 1 g dΩ(σ) σ µ σ ν .
Using (4.3), we can get

σ∈S 1 g dΩ(σ) σ µ 1 • • • σ µ 2p = 2 (2π) d g d Γ(d/2+p) G(g) µ 1 ...µ 2p ,
so that we recover [18, (2.34)]:

a 0 (x) = g d N + d -1 g µν tr(X µν ) (1 + ζ) -d/2 -1 .
Now, let us consider the more general case

u µν = g µν u + ζ X µν (5.4)
where u is a strictly positive matrix u(x) ∈ M N as in Section 4, X µν as before and assume [u(x), X µν (x)] = 0 for any x ∈ (M, g) and µ, ν. Previous situation was u = 1 V , the unit matrix in M N . Once Lemma 2.1 has been applied, the difficulty to compute a r (x) is to evaluate the operators T k,p defined by (2.15). Here we have u

[σ] = u + ζ X(σ)
, where the two terms commute. With the notation

X i := R i [ X(σ)],
we have

C k (s, u[σ]) = (1 -s 1 )R 0 [u + ζ X(σ)] + (s 1 -s 2 )R 1 [u + ζ X(σ)] + . . . • • • + (s k-1 -s k )R k-1 [u + ζ X(σ)] + s k R k [u + ζ X(σ)] = C k (s, u) + ζ (1 -s 1 ) X 0 + (s 1 -s 2 ) X 1 + • • • + (s k-1 -s k ) X k-1 + s k X k
so that, using the fact that each X i is a projection with eigenvalues i = 0, 1:

e -r 2 C k (s,u[σ]) = e -r 2 C k (s,u) ( i )∈{0,1} k+1 e -ζr 2 [(1-s 1 ) 0 +(s 1 -s 2 ) 1 +•••+(s k-1 -s k ) k-1 +s k k ] × × X 0 0 (1 -X 0 ) 1-0 • • • X k k (1 -X k ) 1-k . Notice that X i i (1 -X i ) 1-i = [(1 -i )g µν + (2 i -1)R i (X µν )]σ µ σ ν . With the definition I ( i ), ζ α,k (r 0 , r 1 , . . . , r k ) := I α,k (r 0 + ζ 0 , r 1 + ζ 1 , . . . , r k + ζ k ), the operators T k,p of (2.18) become T k,p = Γ(d/2+p) 2(2π) d S d-1 g dΩ g (σ) σ µ 1 • • • σ µ 2p σ α 0 σ β 0 • • • σ α k σ β k × ( i )∈{0,1} k+1
I ( i ), ζ d/2+p,k R 0 (u), R 1 (u), . . . , R k (u) × × [(1 -0 )g α 0 β 0 + (2 0 -1)R 0 (X α 0 β 0 )] • • • [(1 -k )g α k β k + (2 k -1)R k (X α k β k )]. (5.5)
The computations of these operators are attainable using the method given in Section 4, but with some more complicated combinatorial expressions requiring a computer. We will still get explicit formulae for a 1 (x) in any even dimension as in Theorem 4.5. The main combinatorial computation is to make the contractions between G(g) µ 1 ...µ 2p α 0 β 0 ...α k β k from the first line of (5.5) with the operators of the last line applied on variables B µ 1 ...µ 2p k . When u = 1 V in (5.4), the operators in the second line of (5.5) are just the multiplication by the numbers

I d/2+p,k (1 + ζ 0 , 1 + ζ 1 , . . . , 1 + ζ k ).
Thus one gets explicit formulae for all coefficients a r (x) in any dimension.

Conclusion

On the search of heat-trace coefficients for Laplace type operators P with non-scalar symbol, we develop, using functional calculus, a method where we compute some operators T k,p acting on some (finite dimensional) Hilbert space and the arguments on which there are applied. This splitting allows to get general formulae for these operators and so, after a pure computational machinery will yield all coefficients a r since there is no obstructions other than the length of calculations. The method is exemplified when the principal symbol of P has the form g µν u where u is a positive invertible matrix. It gives a 1 in any even dimension which is written both in terms of ingredients of P (analytic approach) or of diffeomorphic and gauge invariants (geometric approach). As just said, the method is yet ready for a computation of a r with r ≥ 2 for calculators patient enough, as well for the case g µν u + ζ X µν as in Section 5.4. Finally, the method answers a natural question about explicit expressions for all coefficients a r : we proved that u-factorizability is always violated when the dimension is odd and it is preserved in even dimension d when d/2 -r > 0. We conjecture this always holds true in all even dimension.

A. Appendix

A.1. Some algebraic results

Let A be a unital associative algebra over C, with unit 1. Denote by (C * (A, A) = ⊕ k≥0 C k (A, A), δ) the Hochschild complex where C k (A, A) is the space of linear maps ω :

A ⊗ k → A and (δω)[b 0 ⊗ • • • ⊗ b k ] = b 0 ω[b 1 ⊗ • • • ⊗ b k ] + k i=0 (-1) i ω[b 0 ⊗ • • • ⊗ b i-1 ⊗ b i+1 ⊗ • • • ⊗ b k ] + (-1) k+1 ω[b 0 ⊗ • • • ⊗ b k-1 ]b k for any ω ∈ C k (A, A) and b 0 ⊗ • • • ⊗ b k ∈ A ⊗ k .
Define the differential complex (T * A = ⊕ k≥0 A ⊗ k+1 , d) with

d(a 0 ⊗ • • • ⊗ a k ) = 1 ⊗ a 0 ⊗ • • • ⊗ a k + k i=0 (-1) i a 0 ⊗ • • • ⊗ a i-1 ⊗ 1 ⊗ a i+1 ⊗ • • • ⊗ a k + (-1) k+1 a 0 ⊗ • • • ⊗ a k ⊗ 1
for any a 0 ⊗ • • • ⊗ a k ∈ A ⊗ k+1 . Both C * (A, A) and T * A are graded differential algebras, the first one for the product

(ω ω )[b 1 ⊗ • • • ⊗ b k+k ] = ω[b 1 ⊗ • • • ⊗ b k ] ω [b k+1 ⊗ • • • ⊗ b k+k ]
and the second one for

(a 0 ⊗ • • • ⊗ a k )(a 0 ⊗ • • • ⊗ a k ) = a 0 ⊗ • • • ⊗ a k a 0 ⊗ • • • ⊗ a k ∈ A ⊗ k+k +1 = T k+k A.
The following result was proved in [START_REF] Masson | Géométrie non commutative et applications à la théorie des champs[END_REF]: Recall that an associative algebra A is central simple if it is simple and its center is C. Central simple algebras have the following properties, proved for instance in [START_REF] Lam | A first course in noncommutative rings[END_REF]: [START_REF] Avramidi | Gauged gravity via spectral asymptotics of non-Laplace type operators[END_REF] If B is a central simple algebra and C is a simple algebra, then B ⊗ C is a simple algebra. If moreover C is central simple, then B ⊗ C is also central simple.

Lemma A.
Proof (of Proposition A.1) A pure combinatorial argument shows that ι is a morphism of graded differential algebras for the structures given above.

Assume that A is central simple. The space A ⊗ k+1 is an associative algebra for the product

(a 0 ⊗ • • • ⊗ a k ) • (a 0 ⊗ • • • ⊗ a k ) = a 0 a 0 ⊗ • • • ⊗ a k a k , which is central simple by Lemma A.2. Let J k = Ker ι ∩ A ⊗ k+1 . Then, for any α = i a 0,i ⊗ • • • ⊗ a k,i ∈ J k , any β = b 0 ⊗ • • • ⊗ b k ∈ A ⊗ k+1 , and any c 1 ⊗ • • • ⊗ c k ∈ A ⊗ k , one has ι(α • β)[c 1 ⊗ • • • ⊗ c k ] = i a 0,i b 0 c 1 a 1,i b 1 c 2 • • • b k-1 c k a k,i b k = ι(α)[b 0 c 1 ⊗ • • • ⊗ b k-1 c k ]b k = 0, so that α • β ∈ J k .
The same argument on the left shows that J k is a two-sided ideal of the algebra A ⊗ k+1 , which is simple. Since ι is non zero (ι(1 ⊗ • • • ⊗ 1) = 0), one must have J k = 0: this proves that ι is injective.

The algebra A = M N is central simple [START_REF] Lam | A first course in noncommutative rings[END_REF], so that ι is injective, and moreover the spaces C k (A, A) and A ⊗ k+1 have the same dimensions: this shows that ι is an isomorphism.

Remark A. [START_REF] Avramidi | Non-Laplace type operators on manifolds with boundary, in: Analysis, geometry and topology of elliptic operators[END_REF] In [START_REF] Masson | Géométrie non commutative et applications à la théorie des champs[END_REF], the graded differential algebras C * (A, A) and T * A are equipped with a natural Cartan operations of the Lie algebra A (where the bracket is the commutator) and it is shown that ι intertwines these Cartan operations.

A.2. Some combinatorial results

Lemma A. [START_REF] Avramidi | Heat kernel method and its applications[END_REF] Given a family a 0 , . . . , a r of different complex numbers, we have Thus det

           1 1 • • • 1 a 0 a 1 • • • a r a 2 0 a 2 1 • • • a 2 r . . . . . . • • • . . . a r-1 0 a r-1 1 • • • a r-1 r a s 0 a s 1 • • • a s r            = β(s)
after an expansion of the determinant with respect to the last line. But this is zero since the last line coincides with the line s + 1 of the matrix when s = 0, . . . , r -1.

ii) The irreducible fraction expansion of f (z) = 1 (z-a 0 )(z-a 1 )•••(z-ar) is r n=0 Res(f )(an) z-an yielding (A.2).

A.3. Few properties of functions I d/2+k,k

We collect here some special combination of function I d/2+k-r,k . Let g 1 (r 0 , r 1 ) := I d/2+1,2 (r 0 , r 1 , r 0 ). Then

  e -λ tr [b 2r (x, ξ, λ)] (1.4) where λ belongs to a anticlockwise curve C around R + and (x, ξ) ∈ T * (M ). Here the functions b 2r are defined recursively by b 0 (x, ξ, λ) := (d 2 (x, ξ) -λ) -1 , b r (x, ξ, λ) := -r=j+|α|+2-k j<r

Lemma 2 . 6

 26 For any dimension d of the manifold M and x ∈ M , given r ∈ N, the computation of a r (P ) needs exactly to know each of the 3r + 1 operators T k,k-r where r ≤ k ≤ 4r or equivalently to know I d/2,r , I d/2+1,r+1 , . . . , I d/2+3r,4r .

): Proposition 3 . 1

 31 Properties of functions I α,k : i) Recursive formula valid for 1 = α ∈ R and k ∈ N * :

  2+3r,4r used to compute ar(x) . (3.4) Case 3: d is odd, relation (3.1) never fails and

. 11 )

 11 Proof Let d/2 = m ∈ N * . Then using the continuity of Proposition 3.1 with(3.10)

Proposition 3 . 4

 34 r , I d/2+1,r+1 , . . . , I d/2+3r,4r . Case 3: d is odd. If d/2 -r = + 1/2 with ∈ N, the root and its follower are

Proposition 3 . 5

 35 Case 2: d even and r < d/2. For k ∈ N * and N n = d/2 -r + k ≥ k + 1,

Proposition A. 1

 1 The map ι :T * A → C * (A, A) defined by ι(a 0 ⊗ • • • ⊗ a k )[b 1 ⊗ • • • ⊗ b k ] := a 0 b 1 a 1 b 2 • • • b k a k is a morphism of graded differential algebras.If A is central simple, then ι is injective, and if A = M N (algebra of N × N matrices) then ι is an isomorphism.

2 )

 2 =n (a n -a m ) -1 = 0, for any s ∈ {0, 1, . . . , r -=n (a n -a m ) -1 , ∀z ∈ C\{a 0 , . . . , a r }. (A.Proof i) If α(s) := r n=0 a s n r m=0, m =n (a n -a m ) -1 and β(s) := α(s) 0≤l<k≤r (a k -a l ) = r n=0 (-1) r-n a s n r 0≤l<k≤r k =n, l =n (a k -a l )then it is sufficient to show that β(s) = 0 for s = 0, . . . , r-1. Recall first that the determinant of a Vandermonde p × p-matrix is det -b j ).

1 2 [g 1 1 , 1 2d(r 0 r 1 ) 3 2 r

 1111132 (r 0 , r 1 ) + g 1 (r 1 , r0 )] = 1 d (r 0 r 1 ) -d/2 r d/2 if d = 2m. (A.3) Let g 2 (r 0 , r 1 ) := -d 2 I d/2+1,2 (r 0 , r 1 , r 0 ) + d+2 2 [r 1 I d/2+2,3 (r 0 , r 1 , r 1 , r 0 ) + r 1 I d/2+2,3 (r 1 , r 1 , r 0 , r 1 ) + r 0 I d/2+2,3 (r 0 , r 0 , r 1 , r 0 )]. Then g 2 (r 0 , r 1 ) = -d/2 (d r (r 0 , r 1 ) := (d+2) 2 0 I d/2+2,3 (r 0 , r 0 , r 1 , r 0 ) + (d + 2)(d + 4)[-r 0 2 I d/2+3,4 (r 0 , r 0 , r 0 , r 1 , r 0 ) -1 2 r 0 r 1 I d/2+3,4 (r 0 , r 0 , r 1 , r 1 , r 0 )]. (A.5)

  w or their (at most second order) derivatives. Thus we can apply Lemma 4.1 since, at most, only two A i (relabeled indistinctly B 1 , B 2 ) are different from u. This method generates only four cases modulo B 1 ↔ B 2 : Case 0: only one variable, namely w. Case 1:

  r 1 and we prove below that the map h d (x, y) for (x, y) ∈ (R * + ) 2 is explicit only when d is even: Lemma 5.4 The function h d (x, y) = finite h (1) (x) h (2) (y) with continuous functions h (i) if and only if d is even. Proof It is equivalent to show that function d (xy) d h d (x, y) has or not such decomposition. Assume that d = 2m. Then we have the decomposition

(α-1) (r k-1 -r k ) -1 [(r 0 + s 1 (r 1 -r 0 ) + • • • + s k-1 (r k-1 -r k-2 ) + s k-1 (r k -r k-1 )) -(α-1) -(r 0 + s 1 (r 1 -r 0 ) + • • • + s k-1 (r k-1 -r k-2 )) -(α-1) ]

Acknowledgments

We would like to thank Andrzej Sitarz and Dimitri Vassilevich for discussions during the early stage of this work and Thomas Krajewski for his help with Lemma A.4.

Then, --→ x , let J ν µ := ∂ µ x µ and |J| := det(J ν µ ), so that

- --→ γs (where "g.t." stands for gauge transformation). The proof of the following lemma is a straightforward computation: Lemma A. [START_REF] Gilkey | Heat equation asymptotics of "nonminimal" operators on differential forms[END_REF] The differential operator P is well defined on sections of V if and only if u µν , v µ and w have the following transformations:

--→ γu µν γ -1 , (A.7)

--→ w, w

--→ γwγ -1 + γv µ (∂ µ γ -1 ) + γu µν (∂ µ ∂ ν γ -1 ). (A.9)

As these relations show, neither v µ nor w have simple transformations under changes of coordinates or gauge transformations. It is possible to parametrize P using structures adapted to changes of coordinates and gauge transformations. Let us fix a (gauge) connection A µ on V and denote by ∇ µ := ∂ µ +A µ its associated covariant derivative on sections of V . For any section s of V , one then has:

--→ γ∇ µ s.

(A.10)

Lemma A. [START_REF] Branson | Heat Equation Asymptotics of Elliptic Operators with Non-scalar Leading Symbol[END_REF] The differential operator

is well defined on sections of V if and only if u µν , p µ and q have the following transformations:

--→ q, (A.12)

--→ γu µν γ -1 , p µ g.t.

--→ γp µ γ -1 , q g.t.

--→ γqγ -1 . (A.13)

It is equal to P when (the u µν are the same in P and Q)

Proof Combining (A.10), (A.12) and (A.13), the operator s → -(p µ ∇ µ +q)s is well behaved under changes of coordinates and gauge transformations. Let X := |g| -1/2 ∇ µ |g| 1/2 u µν ∇ ν (a matrix valued "Laplace-Beltrami operator"). Then, using (A.10) and (A.12), one gets --→ γ|g| 1/2 u µν ∇ ν s (it behaves like a section of V ), so that Xs g.t.

--→ γXs. This proves that Q = -X -(p µ ∇ µ + q) is well defined. The expansion of X gives

2 (∂ µ log|g|)u µν A ν + (∂ µ u µν )A ν + u µν (∂ µ A ν ) + A µ u µν A ν which, combined with the contributions of -(p µ ∇ µ + q), gives (A.14) and (A.15). Contrary to the situation in [1, Section 1.2.1], one cannot take directly p = 0 in (A.11) since we cannot always solve A µ in (A.14) to write it in terms of u µν , v µ , w.