A Characterization of the Heat Kernel Coefficients

Gregor Weingart^{∗}

May 16, 2001

Abstract We consider the asymptotic expansion of the heat kernel of a generalized Laplacian for t → 0^{+} and characterize the coefficients a_{k}, k ≥ 0, of this expansion by a natural intertwining property. In particular we will give a closed formula for the infinite order jet of these coefficients on the diagonal in terms of the local expressions of the powers of the given generalized Laplacian in normal coordinates.

MSC 2000: 58J50

1 Introduction

A generalized Laplacian Δ Δ acting on sections of a vector bundle E over a smooth manifold M is a second order elliptic differential operator with scalar symbol ([BGV]). Its spectrum is interesting for various geometrical questions and the spectra of the more prominent generalized Laplacians are of fundamental importance in global analysis. The short distance or high frequency behaviour of the spectrum of a generalized Laplacian can be studied using analytic properties of its heat operator K_{t}^{Δ Δ}, which solves the heat equation

\frac{∂}{∂t} ( K_{t}^{Δ Δ}ψ) = − Δ Δ( K_{t}^{Δ Δ}ψ)

associated to Δ Δ for all sections ψ of E. The general existence result of Minakshisundaram– Pleijel ([G],[BGV]) shows that K_{t}^{Δ Δ} is an integral operator for all t > 0 with integral kernel k_{t}^{Δ Δ}(x, y), which is a smooth section of the bundle with fiber Hom(E_{x}, E_{y}) over the point (x, y) of the product M × M. Except for a very limited set of examples the heat kernel is not known explicitly, perhaps the most important exception is the heat kernel of the flat Laplacian Δ acting on sections of a trivial vector bundle V × E over a euclidian vector space V of dimension n:

k_{t}^{Δ}(x, y) = \frac{1}{\sqrt{4πt}^{n}} e−\frac{|x−y|^{2}}{4t} id_{E}

As can be seen from this example the heat kernel k_{t}^{Δ Δ}(x, y) peaks on the diagonal {x = y} of M × M as t → 0^{+} and it turns out that the rate of its divergence on the diagonal carries a tremendous amount of information about the geometry of the underlying manifold M. More precisely the heat kernel has an asymptotic expansion along the diagonal as t → 0^{+} of the form

∗partially supported by the SFB 256 “Nichtlineare partielle Differentialgleichungen”

k_{t}^{Δ Δ}(x, y) ∼_{t→0^{+}}  \frac{1}{\sqrt{4πt}^{n}} e−\frac{dist^{2}(x,y)}{4t} ∑_{k≥0} t^{k} â_{k}(x, y)  (1)

where dist^{2}(x, y) is the square of the geodesic distance from x to y and the â_{k}(x, y) are again sections of the bundle Hom(E_{x}, E_{y}). In a sense these coefficients â_{k}, k ≥ 0, describe how much the heat kernel of the operator Δ Δ differs in divergence from the flat Laplacian Δ in various directions and this difference does not only depend on the coefficients of Δ Δ, but involves the geometry of M in a crucial way.

As a matter of convenience we will not use the coefficients â_{k}(x, y) directly but will multiply them a_{k}(x, y) := j(x, y)â_{k}(x, y) by the jacobian determinant j(x, y) of the differential of the exponential map in y at x, which is a well defined smooth function in a neighborhood of the diagonal. Off this neighborhood we may extend j(x, y) arbitrarily to a smooth function on M × M, because only the germs of the coefficients â_{k}(x, y) along the diagonal are characterized by the asymptotic expansion (1) of k_{t}^{Δ Δ} above. The main theorem of this article exhibits a remarkable intertwining property of these modified coefficients a_{k}, k ≥ 0:

Theorem 2.1 (The Intertwining Property) Let Δ Δ be a generalized Laplacian acting on sections ψ of a vector bundle E over M. The coefficients a_{k} : E_{x} ⟶ E_{y} of the asymptotic expansion of the heat kernel of Δ Δ satisfy

\frac{(−1)^{μ}}{μ!} [ Δ Δ^{μ} ψ] (y) = ∑_{ν=0}^{μ} \frac{(−1)^{ν}}{ν!} [ Δ^{ν} (a_{μ−ν}(exp_{y}·, y) ψ(exp_{y}·)) ] (0)

where a_{μ−ν}(exp_{y}·, y) ψ(exp_{y}·) is considered as a section of the trivial vector bunde T_{y}M × E_{y} over T_{y}M and Δ is the flat Laplacian acting on sections of this trivial bundle.

The rest of the article is devoted to corollaries of Theorem 2.1. In particular we prove the striking fact that the intertwining property alone characterizes the infinite order jet of all coefficients a_{k}(·, y) in y and on the way we will give a different proof of Polterovich’s inversion formula ([P1]) for the endomorphisms a_{k}(y, y), k ≥ 0, of the fiber E_{y} in a somewhat stronger formulation:

Theorem 2.4 (Polterovich’s Inversion Formula) Let Δ Δ be a generalized Laplacian acting on sections of a vector bundle E over M and let dist^{2}(·, y) be the square of the geodesic distance to y ∈ M. For all r ≥ k ≥ 0 and all sections ψ of E the endomorphism a_{k}(y, y) = â_{k}(y, y) of the fiber E_{y} is given by the formula:

[ a_{k} ψ] (y) = ∑_{l=0}^{r} (−\frac{1}{4})^{l} \binom{r + \frac{n}{2}}{r − l} [ \frac{(−1)^{k+l}}{(k + l)!} Δ Δ^{k+l} ( \frac{1}{l!} dist^{2l}(·, y) ψ) ] (y) .

The final result is Theorem 2.7, which provides an explicit formula for the infinite order jets of the coefficients a_{k}(·, y) in y in terms of the local expressions of the powers of the operator Δ Δ in normal coordinates on M and some smooth trivialization of E.

The present version of this article was written down while I was staying at the University of California in Berkeley, whose hospitality is gratefully acknowledged. It is only the first part of a projected article of three and hence will not appear separately in printed form.

2 The Intertwining Property

It is well known that the coefficients â_{k}, k ≥ 0, of the asymptotic expansion of the heat kernel of Δ Δ are local expressions in the curvature of M and the coefficients of Δ Δ ([G]). Consequently we will assume throughout this section that Δ Δ is defined on sections of a trivializable vector bundle E over a starshaped neighborhood of the origin in some euclidean vector space V of dimension n. Moreover we will assume that the identity map V ⟶ V is the exponential map in the origin with respect to the Riemannian metric defined by the symbol of Δ Δ, a fortiori the symbol metric must agree with the scalar product of V in 0. The general case is easily reduced to this local model by taking V = T_{y}M to be the tangent space of the underlying manifold at the point y under consideration and choosing the neighborhood of the origin and the trivializable vector bundle E accordingly.

The preceeding assumptions are crucial to the entire argument but single out the origin as a special point. Consequently all our statements below refer to the fixed target point y = 0. In particular the coefficients â_{k}(x) := â_{k}(x, 0) will be homomorphisms â_{k}(x) : E_{x} ⟶ E from the fiber of E over x to the fiber E of E over the origin. Similarly we will write j(x) := j(x, 0) for the Jacobian determinant of the exponential map in the origin at the point x ∈ V, which is the identity map V ⟶ V by assumption. In other words the Jacobian determinant is characterized by dvol_{g} = j(x)dx and is the square root j(x) := \sqrt{det g(x)} of the determinant of the symbol metric with respect to the flat scalar product. Now by the remarks following ([BGV, Theorem 2.30]) we have an asymptotic expansion for the heat operator

[ K_{t}^{Δ Δ}ψ ] (0) ∼_{t→0^{+}} ∑_{μ≥0} \frac{(−t)^{μ}}{μ!} [ Δ Δ^{μ}ψ] (0)

which complements the asymptotic expansion (1) of its integral kernel k_{t}^{Δ Δ}. Using both asymptotic expansions we find for any smooth section ψ of E with compact support in a sufficiently small neighborhood of the origin

∫_{V} k_{t}^{Δ Δ}(x)ψ(x) d vol_{g} ∼_{t→0^{+}} ∑_{μ≥0} \frac{(−t)^{μ}}{μ!} [ Δ Δ^{μ}ψ] (0)

∼_{t→0^{+}} ∫_{V} \frac{1}{\sqrt{4πt}^{n}} e−\frac{|x|^{2}}{4t} ∑_{μ≥0} t^{μ} â_{μ}(x) ψ(x) dvol_{g}

∼_{t→0^{+}} ∑_{μ≥0} t^{μ} ∫_{V} \frac{1}{\sqrt{4πt}^{n}} e−\frac{|x|^{2}}{4t} ( j(x) â_{μ}(x) ψ(x) ) dx

∼_{t→0^{+}} ∑_{μ≥0} t^{μ} ∑_{ν≥0} \frac{(−t)^{ν}}{ν!} [ Δ^{ν}( a(x) ψ(x) ) ] (0) ,

where a(x) := j(x)â_{k}(x) and Δ is the flat Laplacian acting on sections of the trivial E– bundle V × E over V. Note that the asymptotic expansion of the heat kernel k_{t}^{Δ Δ} is locally uniform in x according to ([BGV, Theorem 2.30]) and hence we may integrate it over the compact support of ψ to obtain an asymptotic expansion of the integral. Existence of asymptotic expansions implies uniqueness and so we need only sort out the different powers of t to prove the intertwining property of the heat kernel coefficients in the form:

Theorem 2.1 (Intertwining Property of the Heat Kernel Coefficients) Let Δ Δ be a generalized Laplacian acting on sections of a trivializable vector bundle E over V. The coefficients a_{k}, k ≥ 0, of the asymptotic expansion of the heat kernel of Δ Δ are sections of the bundle Hom(E, V × E) and so a_{k}ψ is a well defined section of V × E for any section ψ of E. The coefficients a_{k} intertwine the powers of Δ Δ and Δ:

\frac{(−1)^{μ}}{μ!} [ Δ Δ^{μ} ψ] (0) = ∑_{ν=0}^{μ} \frac{(−1)^{ν}}{ν!} [ Δ^{ν} (a_{μ−ν} ψ) ] (0) .  (2)

A direct consequence of Theorem 2.1 is that a generalized Laplacian Δ Δ acting on sections of a vector bundle E remembers to vanish on sections ψ of E, which are harmonic in some smooth trivialization of E, in the following sense:

Remark 2.2 (Excess Vanishing for Harmonic Sections) Consider a section ψ of E, which is a homogeneous, harmonic polynomial Φψ ∈ Sym^{d} V^{∗}⊗E of degree d with Δ(Φψ) = 0 in some smooth trivialization Φ of E. If Δ Δ is a generalized Laplacian acting on sections of E, then [ Δ Δ^{μ}ψ] (0) = 0 vanishes for all μ < d.

Philosophically the intertwining property (2) provides a precise geometric interpretation for the coefficients a_{k} of the heat kernel expansion in terms of Δ Δ and the flat model operator Δ for generalized Laplacians. Similar considerations should apply to other model operators arising e. g. in Heisenberg calculus or in other parabolic calculi. A convenient reformulation of this property (2) can be given with the help of the formal power series e^{−zΔ Δ} and e^{−zΔ} of differential operators on E and V × E respectively and the generating series

a(z) := ∑_{μ≥0} a_{μ} z^{μ}

for the heat kernel coefficients a_{k}, k ≥ 0. Namely the intertwining property (2) is just another way to write down the equality [e^{−zΔ Δ}ψ](0) = [e^{−zΔ}(a(z)ψ)](0) of formal power series in z for all sections ψ, which in turn is equivalent to the commutativity of the pentagram where ev is the evalutation at the origin. Only the infinite order jet of the sections a_{k}, k ≥ 0, of Γ(V, Hom(E, V × E)) in the origin can ever be sensed by jet theory of course, and so the commutativity of this square does only depend on the infinite order jet of the generating series a(z). Strikingly however the commutativity of the pentagon characterizes the infinite order jet of a(z) completely. In fact the unique jet solution of the intertwining property thought of as a set of equations for the unknowns a_{k}, k ≥ 0, is given in Theorem 2.7 below.

[figure: Jet_{0}^{∞} E; Jet_{0}^{∞} E[[ z]]; Jet_{0}^{∞} E[[ z]]; Jet_{0}^{∞} E[[ z]]; E[[ z]]; a(z); e^{−zΔ Δ}; e^{−zΔ}; ev; ev]

At this point we digress a little bit on a very interesting property of the flat Laplacian Δ acting on functions C^{∞}V on a euclidian vector space V. More general we can consider the flat Laplacian Δ acting on sections C^{∞}(V, E) of the trivial E–bundle V × E over V, but the auxiliary vector space E never enters into the argument directly and so we will stick to the case E = ℝ. Along with the flat Laplacian Δ comes the operator |x|^{2} of multiplication by the square of the distance to the origin, in orthogonal coordinates {x_{μ}} on V these two operators can be written:

Δ := − ∑_{μ} \frac{∂^{2}}{∂x_{μ}^{2}} |x|^{2} := ∑_{μ} x_{μ}^{2} .

By restriction they act on the subspace Sym V^{∗} ⊂ C^{∞}V of polynomials on V and their commutator [Δ, |x|^{2}] = (−4) (N + \frac{n}{2}) as operators on Sym V^{∗} involves the number operator N : Sym V^{∗} ⟶ Sym V^{∗}, which multiplies Sym^{r} V^{∗} by r, shifted by half the dimension n of V. In other words X := \frac{1}{2}|x|^{2} and Y := \frac{1}{2}Δ close with H := N + \frac{n}{2} to an 𝔰𝔩_{2}–algebra of operators on Sym V^{∗}. Iterated commutators of X and Y in 𝔰𝔩_{2}–representations can in general be written down using factorial polynomials [x]_{r} := x(x − 1) … (x − r + 1) or binomial coefficients \binom{x}{r} := \frac{1}{r!}[x]_{r} for integral r ≥ 0. In particular the standard relation Y^{r} X^{r} ψ = r! [−λ]_{r} ψ for all ψ with Y ψ = 0 and H ψ = λ ψ implies for the constant function ψ := 1 ∈ Sym^{0} V^{∗} the classical formula:

[ Δ^{r} |x|^{2r} ] (0) = 4^{r} r! [ −\frac{n}{2} ]_{r} .  (3)

Slightly more useful for our purposes is the following derived identity:

Lemma 2.3 For all smooth functions ψ ∈ C^{∞}V and all k, l ≥ 0:

[ \frac{(−1)^{k+l}}{(k + l)!} Δ^{k+l} ( \frac{1}{k!} |x|^{2k} ψ) ] (0) = (−4)^{k} \binom{−\frac{n}{2} − l}{k} [ \frac{(−1)^{l}}{l!} Δ^{l} ψ ] (0) .

Proof: Only a finite number of partial derivates of ψ in the origin 0 are actually involved in this identity and hence we may assume that ψ is a polynomial without loss of generality. Moreover only the homogeneous component of ψ of degree 2l contributes to left and right hand side, which are both evidently SO V –invariant linear functionals in ψ ∈ Sym^{2l} V^{∗}. However there is but one SO V –invariant linear functional on Sym^{2l} V^{∗} up to scale, so that it is sufficient to check the identity in question, which can be rewritten as

[ Δ^{k+l} ( |x|^{2k} ψ) ] (0) = \frac{4^{k+l} (k + l)! [−\frac{n}{2}]_{k+l}}{4^{l} l! [−\frac{n}{2}]_{l}} [ Δ^{l} ψ ] (0) ,

for the single polynomial ψ := |x|^{2l}, for which it is true by the classical formula (3). □

Returning to the general case of a Laplacian Δ Δ acting on sections of a vector bundle E over V we recall that the generating series a(z) for the coefficients in the asymptotic expansion of the heat kernel intertwines the formal power series e^{−zΔ Δ} and e^{−zΔ} of differential operators. Using this intertwining property (2) together with Lemma 2.3 we calculate:

[ \frac{(−1)^{k+l}}{(k + l)!} Δ Δ^{k+l} ( \frac{1}{l!} |x|^{2l} ψ) ] (0) = ∑_{μ=0}^{k+l} [ \frac{(−1)^{μ}}{μ!} Δ^{μ} ( \frac{1}{l!} |x|^{2l} a_{k+l−μ} ψ) ] (0)

= ∑_{μ=0}^{k} [ \frac{(−1)^{l+μ}}{(l + μ)!} Δ^{l+μ} ( \frac{1}{l!} |x|^{2l} a_{k−μ} ψ) ] (0)

= (−4)^{l} ∑_{μ=0}^{k} \binom{−\frac{n}{2} − μ}{l} [ \frac{(−1)^{μ}}{μ!} Δ^{μ} ( a_{k−μ} ψ) ] (0) .

Combining this equation with the binomial inversion formula

∑_{l=0}^{r} \binom{r + \frac{n}{2}}{r − l} \binom{−\frac{n}{2} − μ}{l} = \binom{r − μ}{r} = δ_{μ, 0}

valid as soon as r ≥ μ ≥ 0 we eventually arrive at the following inversion formula for the heat kernel coefficients a_{k}, k ≥ 0, of a generalized Laplacian Δ Δ:

Theorem 2.4 (Polterovich’s Inversion Formula ([P1],[P2])) Let Δ Δ be a generalized Laplacian acting on sections of a vector bundle E over V and let |x| be the radial distance from the origin. For any section ψ ∈ Γ(V, E) and all r ≥ k ≥ 0 we can compute the action of the endomorphism a_{k}(0) of the fiber E of E over the origin on ψ by means of an explicit inversion formula:

[ a_{k} ψ] (0) = ∑_{l=0}^{r} (−\frac{1}{4})^{l} \binom{r + \frac{n}{2}}{r − l} [ \frac{(−1)^{k+l}}{(k + l)!} Δ Δ^{k+l} ( \frac{1}{l!} |x|^{2l} ψ) ] (0) .

Our normalization of the coefficients a_{k}, k ≥ 0, drops the factor (4π)^{− \frac{n}{2}} arising from the value of the euclidian heat kernel for the flat Laplacian Δ on V at the origin to have the intertwining property (2) in as simple a form as possible. In the original formulation of Polterovich this factor is part of a_{k} and of course in every conceivable application this factor has to be reinserted by hand.

Example 2.5 (The Local Index Theorem for the Gauß–Bonnet Operator) Choose a smooth Riemannian metric g on V satisfying our standing assumptions and a vector bundle E over V with a connection ∇. For a generalized Laplacian on E of the form Δ Δ_{ℏ} = ∇^{∗}∇ + ℏF for some smooth endomorphism F of E we evidently have

\frac{(−1)^{k+l}}{(k + l)!} ( ∇^{∗}∇ + ℏ F )^{k+l} = \frac{(−ℏ)^{k}}{k!} F^{k} \frac{(−1)^{l}}{l!} Δ^{l} + terms of lower order in ℏ

+ differential operators of lower order for all k, l ≥ 0. Plugging this into Polterovich’s Inversion formula we verify immediately

a_{k}(0) = \frac{(−ℏ)^{k}}{k!} F^{k} + terms of lower order in ℏ

for all k ≥ 0, which may have been anticipated from Theorem 2.1. Remarkably this simple calculation together with a few general considerations concerning the order of the coefficients a_{k} is already sufficient to prove the local index theorem for the Gauß–Bonnet operator!

In conclusion the intertwining property (2) alone is sufficient to determine the value of the coefficients a_{k}, k ≥ 0, of the heat kernel expansion at 0. Before we proceed to show that in fact the jets of infinite order jet_{0}^{∞}a_{k}, k ≥ 0, are determined by (2) we want to make a few general remarks concerning our guiding philosphy in the calculations to come. We want to avoid formulas involving compositions of differential operators, because such formulas are hardly if ever of any use in explicit calculations. In favourable situations it may still be possible to calculate the values ev D^{r}, r ≥ 0, of powers of a differential operator D in the origin without knowing the partial derivatives of their coefficients. Note that the usual arguments of symbol calculus become meaningless if we have no control over the partial derivatives of the differential operators in a singular point like the origin.

In general the value of a differential operator D acting on sections of E at the origin will be an element of Hom( Jet_{0}^{∞} E, E). For the trivial E–bundle V × E however we may identify Jet_{0}^{∞}(V × E) with the formal power series completion of Sym V^{∗} ⊗ E and hence Hom( Jet_{0}^{∞}(V × E), E) with Sym V ⊗ End E in the usual way. The scalar product of the euclidian vector space V extends to a scalar product on Sym V^{∗} defined via Gram’s permanent and characterized by ⟨ e^{α}, e^{β} ⟩_{Sym V^{∗}} = e^{⟨ α, β ⟩} for all α, β ∈ V^{∗} with a slight abuse of notation. Alternatively we may choose orthonormal coordinates {x_{μ}} on V and write down the scalar product directly

⟨ ψ, ψ̃ ⟩_{Sym V^{∗}} := ∑_{r≥0} \frac{1}{r!} ∑_{μ_{1},...,μ_{r}} [ \frac{∂^{r}}{∂x_{μ_{1}} … ∂x_{μ_{r}}} ψ ] (0) [ \frac{∂^{r}}{∂x_{μ_{1}} … ∂x_{μ_{r}}} ψ̃ ] (0)
= ( (ev ⊗ ev ) ∘ e^{⟨ ∇, ∇ ⟩} ) ( ψ ⊗ ψ̃)

where ⟨ ∇, ∇ ⟩ is the bidifferential operator ψ ⊗ ψ̃ ⟼ ∑_{μ} \frac{∂}{∂x_{μ}} ψ ⊗ \frac{∂}{∂x_{μ}} ψ̃ and ev is the evaluation at 0 as before. Written in this form it is clear that the scalar product extends to the formal power series completion of Sym V^{∗} or even to smooth functions provided that the defining sum converges. Moreover the musical isomorphism ♯ : Sym V ⟶ Sym V^{∗} of this scalar product is the natural extension of the musical isomorphism of V and extends to the formal power series completion of Sym V, too. The value of a differential operator D acting on sections of V × E has an image (ev D)^{♯} under this musical isomorphism, which is the unique polynomial on V satisfying [Dψ] (0) = ⟨ (ev D^{♯}), ψ ⟩_{Sym V^{∗}} for all smooth sections ψ of V × E.

The scalar product on Sym V^{∗} will enter the formulas through the operator ⟨∇, ∇⟩ introduced above, which in turn makes its appearance via Green’s identity for the flat Laplacian Δ and the multiplication map m

( Δ ∘ m) ( ψ ⊗ ψ̃) = ( m ∘ ( Δ ⊗ id − 2⟨ ∇, ∇ ⟩ + id ⊗ Δ ) ) ( ψ ⊗ ψ̃)

which features three commuting operators Δ ⊗ id, id ⊗ Δ and ⟨ ∇, ∇ ⟩. Hence we are free to put these operators in arbitrary order upon exponentiation. For the ensuing calculations we need to choose a trivialization Φ of E, which we think of as a family of homomorphisms Φ(x) : E_{x} ⟶ E with Φ(0) = id_{E}, so that Φψ is a smooth section of V × E for every section ψ of E. With this in mind we find

[ e^{−zΔ} ( a(z) ψ) ] (0) = ( ev ∘ e^{−zΔ} ∘ m) ( a(z)Φ^{−1} ⊗ Φψ)
= ( (ev ⊗ ev ) ∘ e^{2z⟨ ∇, ∇ ⟩} ) ( e^{−zΔ}(a(z)Φ^{−1}) ⊗ e^{−zΔ}(Φψ) )
= ⟨ (2z)^{N} (e^{−zΔ} (a(z)Φ^{−1}), (e^{−zΔ} (Φψ)) ⟩_{Sym V^{∗}}
= ⟨ e^{z|x|^{2}} (2z)^{N} (e^{−zΔ} (a(z)Φ^{−1}), Φψ ⟩_{Sym V^{∗}}

while [e^{−zΔ Δ}ψ] (0) = ⟨ (ev e^{−zΦΔ ΔΦ^{−1}})^{♯}, Φψ ⟩_{Sym V^{∗}} by definition. As the scalar product ⟨ , ⟩ is non–degenerate on Sym V^{∗} and ψ can be chosen arbitrarily we conclude:

( ev e^{−zΦΔ ΔΦ^{−1}} )^{♯} = e^{z|x|^{2}} (2z)^{N} (e^{−zΔ} (a(z)Φ^{−1})) .  (4)

Evidently the operators e^{z|x|^{2}} and e^{−zΔ} are invertible with inverses e^{−z|x|^{2}} and e^{zΔ} respectively, thus we can solve equation (4) uniquely for the infinite order jet of a(z) leading to:

jet_{0}^{∞}( a(z)Φ^{−1} ) = e^{zΔ} (2z)^{−N} e^{−z|x|^{2}} ( ev e^{−zΦΔ ΔΦ^{−1}} )^{♯}
= e^{zΔ} (2z)^{−N} (ev e^{−zΦΔ ΔΦ^{−1}} e^{zΔ} )^{♯} .  (5)

Concerning the last step we remark that the polynomial e^{−z|x|^{2}} corresponds to the differential operator e^{zΔ} with constant coefficients. Hence it is possible to replace e^{−z|x|^{2}} ( ev e^{−zΦΔ ΔΦ^{−1}} )^{♯} by ( ev e^{−zΦΔ ΔΦ^{−1}}e^{zΔ} )^{♯}, but not by ( ev e^{zΔ}e^{−zΦΔ ΔΦ^{−1}} )^{♯}, because the latter would involve partial derivatives of the coefficients of e^{−zΦΔ ΔΦ^{−1}} in the origin. Nevertheless we cheated a little bit, because while certainly injective the operator (2z)^{N} is definitely not surjective. It is quite remarkable in itself that the value of the differential operator e^{−zΦΔ ΔΦ^{−1}} e^{zΔ} in the origin lies in the image of (2z)^{N} by equation (4), because its coefficients for the different powers z^{r}, r ≥ 0, of z must have order less than or equal to r to have this true! Hence the inverse (2z)^{−N} is well defined and refers to the unique preimage of (ev e^{−zΦΔ ΔΦ^{−1}}e^{zΔ})^{♯} under (2z)^{N}.

Definition 2.6 (The Difference Operator of a Generalized Laplacian) Consider a generalized Laplacian Δ Δ and the flat Laplacian Δ acting on sections of E and the trivial vector bundle V × E over V respectively. Associated to Δ Δ and some smooth trivialization Φ of E is the formal power series

�(z) := e^{−z(ΦΔ ΔΦ^{−1})} e^{zΔ} = 1 + z �_{1} + z^{2} �_{2} + z^{3} �_{3} + ⋯

of differential operators on V × E called the difference operator. The values ev �_{r} of its coefficients have order less than or equal to r, so that ev �(z) lies in the image of (2z)^{N}.

At least in principle the recursion relation \frac{d}{dz}�(z) = �(z)Δ − (ΦΔ ΔΦ^{−1})�(z) or equivalently r�_{r} = �_{r−1}Δ − (ΦΔ ΔΦ^{−1})�_{r−1} for r ≥ 1 together with �_{0} = 1 allows us to calculate �(z) to arbitrary order in z. However the composition (ΦΔ ΔΦ^{−1})�_{r−1} requires knowledge of the full operator �_{r−1} and not only of its value at the origin. Note that the recursion relation in itself does not imply that the values of the �_{r}, r ≥ 0, have order less than or equal to r, because the origin will be a singular point for the full operator �(z) in general. Reformulating equation (5) according to the definition of �(z) we arrive at the following theorem, which in a sense is the converse of Theorem 2.1:

Theorem 2.7 (Characterization of the Heat Kernel Coefficients) The infinite order jet of the generating series a(z) for the heat kernel coefficients of a generalized Laplacian Δ Δ acting on sections of a vector bundle E over V is uniquely characterized by its intertwining property [e^{−zΔ Δ}ψ] (0) = [e^{−zΔ}(a(z) ψ)] (0). In fact the infinite order jet of any solution a(z) to this equation is given in terms of some smooth trivialization Φ of E and the corresponding difference operator �(z) by:

jet_{0}^{∞}( a(z)Φ^{−1} ) = e^{zΔ} (2z)^{−N} (ev �(z))^{♯} .

The essence of Theorem 2.7 is not particularly evident in the abstract formulation above and so we want to close this section expanding this formula for the jets of the heat kernel coefficients into powers of z. This exercise will show in particular that it is just as difficult to calculate the infinite order jets of the heat kernel coefficients a_{k}, k ≥ 0, as it is to calculate the value of the powers Δ Δ^{r}, r ≥ 0, of Δ Δ in 0 using normal coordinates and the chosen trivialization Φ of E. In other words calculating the powers Δ Δ^{r} explicitly is the problem of calculating the coefficients of the heat kernel expansion en nuce! Let us start by expanding the value ev �(z) into powers of z and homogeneous pieces

( ev �(z) )^{♯} = ∑_{r≥s≥0} z^{r} �_{r,s} ,

where each �_{r,s} ∈ Sym^{s} V^{∗} ⊗ End E is homogeneous polynomial of order s, and employ Theorem 2.7 to conclude

jet_{0}^{∞}a(z) = ∑_{r≥s≥2t≥0} \frac{z^{t}}{t!} Δ^{t} ( \frac{z^{r−s}}{2^{s}} �_{r,s}^{♯}) = ∑_{k,l≥0} z^{k} ∑_{t=0}^{k} \frac{1}{2^{l+2t} t!} Δ^{t}�_{k+l+t,l+2t}^{♯}  (6)

by reordering the sum and setting k := r−s+t and l := s−2t. In particular the infinite order jet of the first coefficient a_{0} picks up the principal symbols ev �_{l,l}, l ≥ 0, of the operators �_{l}, l ≥ 0, in the origin. Now by Definition 2.6 of the difference operator �(z) we find for the homogeneous pieces of its value

�_{r,s}^{♯} = ∑_{μ=0}^{r} (−1)^{r} \frac{1}{μ!}|x|^{2μ} \frac{1}{(r − μ)!}(ev (ΦΔ ΔΦ^{−1})^{r−μ})_{s−2μ}^{♯}

where (ev (ΦΔ ΔΦ^{−1})^{r−μ})_{s−2μ}^{♯} ∈ Sym^{s−2μ} V^{∗} ⊗End E is the homogeneous piece of order s−2μ of the polynomial corresponding to ev (ΦΔ ΔΦ^{−1})^{r−μ}. Inserting this expression into equation (6) above we get a closed formula for the homogeneous pieces of the infinite order jets of the heat kernel coefficients a_{k}, k ≥ 0. Though explicit this formula looks rather akward and so we refrain from writing it down. In fact the main point is to understand the fantastic cancellation properties of the Δ Δ_{r}^{♯}, r ≥ 0, which make the ev �_{r}, r ≥ 0, operators of degree less than or equal to r in the first place! Without proper understanding of these cancellations we won’t be able to take advantage of either formula to calculate the heat kernel coefficients explicitly.

References

[BGV] Berline, N. , Getzler, E. & Vergne, M. : Heat Kernels and Dirac Operators, Grundlehren der mathematischen Wissenschaften 298, Springer, 1991

[G] Gilkey, Peter: The Index Theorem and the Heat Equation, Mathematical Lecture Series, Publish or Perish, 1974

[P1] Polterovich, Iossif: Heat invariants of Riemannian manifolds, Israel Journal of Mathematics 119 (2000), 239—252

[P2] Polterovich, Iossif: Heat kernel asymptotics for Laplace type operators and matrix KdV hierarchy, electronic preprint math.DG/0010057