On the Trace Density in Deformation Quantization

Boris Fedosov^{∗}

Abstract

We derive a variation formula for the trace functional resulting in a differential equation for the trace density. It enables one in principle to calculate completely the trace density. We illustrate this process computing several few terms.

Mathematics Subject Classification: 53D55

1 Introduction

In this paper we consider a standard deformation quantization on a symplectic manifold M, ω and a canonical trace on the algebra of quantum observables, see e.g. [1, 2]. Our result consists in a variation formula for a trace density c which enables one to compute it in principle. We illustrate this process by computing several few terms.

In the simplest scalar-valued case we obtain

c(x, h) = 1 + \frac{h^{2}}{48}∂_{i_{1}}∂_{i_{2}}R_{j_{1}j_{2}i_{3}j_{3}}ω^{i_{1}j_{1}}ω^{i_{2}j_{2}}ω^{i_{3}j_{3}}

−\frac{h^{2}}{64}R_{i_{1}i_{2}i_{3}j_{3}}R_{j_{1}j_{2}i_{4}j_{4}}ω^{i_{1}j_{1}}ω^{i_{2}j_{2}}ω^{[i_{3}j_{3}}ω^{i_{4}j_{4}]} + O(h^{3}).  (1.1)

Here R_{ijkl} is a curvature tensor of a symplectic connection ∂ = ∂^{M} on M, ω^{ij} is inverse to ω_{ij}, square brackets mean alternation (alternative averaging over the symmetric group S_{4}). Of course, we use the Einstein summation convention, moreover, in the sequel we will omit the tensor factors ω^{i_{k}j_{k}} meaning a contraction by ω^{i_{k}j_{k}} with respect to a pair of indices i_{k}, j_{k} with the same subscript k. Thus, (1.1) may be rewritten in a simplified form

∗Mathématique Physique, Université de Bourgogne, BP 47870, F-21078, Dijon Cedex, France. e-mail: fedosov@u-bourgogne.fr

c(x, h) = 1 + \frac{h^{2}}{48}∂_{i_{1}}∂_{i_{2}}R_{j_{1}j_{2}i_{3}j_{3}} − \frac{h^{2}}{64}R_{i_{1}i_{2}[i_{3}j_{3}}R_{|j_{1}j_{2}|i_{4}j_{4}]} + …,

the vertical dashes mean that the indices j_{1}j_{2} between them are excluded from alternation.

Our quantization gives a star-product for which the identity function 1 is a unit with respect to ∗, that is

1 ∗ a = a ∗ 1 = a.  (1.2)

It is always possible to pass to an equivalent star-product ∗′ (the so-called closed star-product introduced in [3] such that the integral

Tr′a = \frac{(2πh)^{−n}}{n!} ∫_{M} a(x, h)ω^{n}

possesses a trace property

Tr′a ∗′ b = Tr′b ∗′ a.

Of course, in general the normalization (1.2) is violated for the closed product ∗′, but may be easily reconstructed. Indeed, let

c(x, h) = 1 + hc_{1} + h^{2}c_{2} + …  (1.3)

be a unit with respect to ∗′, then

a ∗ b = ((ca) ∗′ (cb))\frac{1}{c}

defines a star-product ∗ satisfying (1.2), but the trace functional for ∗ looks more complicated

Tr a = Tr′ca = \frac{(2πh)^{−n}}{n!} ∫_{M} ca\frac{ω^{n}}{n!} .  (1.4)

Thus, the unit in the ∗′-algebra gives a trace density in the ∗- algebra and vice versa. We will use the canonical deformation quantization for which (1.2) is fulfilled automatically and the trace functional is given by (1.4).

If M is compact, we obtain by the index theorem for deformation quantization

Tr 1 = \frac{(2πh)^{−n}}{n!} ∫_{M} cω^{n} = ∫_{M} exp \frac{ω}{2πh}det^{−1/2} (\frac{sinh κ/2}{κ/2})

where

κ = κ_{M} = \frac{1}{4πi}ω^{im}R_{mjkl}dx^{k} ∧ dx^{l}

is the curvature 2-form of the connection ∂^{M}. In the last term of (1.1) one can easily recognize the Pontrjagin polynomial, being multiplied by ω ∧ ω it is proportional to the first Pontrjagin form. The second term in (1.1) is a complete covariant derivative, so it vanishes under integration.

The computation of the higher-oder terms in the trace density is a challenging problem. We mean, of course, precise computation not only up to complete covariant derivatives leading to the index formula. Our direct computation of the lower-order terms may be of interest since it illustrates in detail the process of “miraculous cancelation” resulting in the elegant formula (1.1).

Acknowledgment

I would like to thank the Max-Planck-Institute in Bonn where this work was completed for hospitality and support.

2 Preliminaries

To fix notation, let us recall necessary notions and constructions from [1, 2]. We will consider a canonical quantization with nontrivial coefficients, so we need a vector bundle E with a connection ∂ = ∂^{E}. It defines a connection on the homomorphism bundle K = Hom (E, E) called a coefficient bundle. Further, W means a bundle over M whose fiber W_{x} is the Weyl algebra defined on the tangent space T_{x}M with respect to the constant symplectic form 1/2ω_{ij}(x)dy^{i}∧dy^{j} defined on T_{x}M by the form ω on M. The symplectic connection ∂ = ∂^{M} on M gives a connection on the bundle W which in local Darboux coordinates has the form

∂a = d_{x}a(x, y, h) + [\frac{i}{2h}Γ_{ijk}y^{i}y^{j}dx^{k}, a] .

Here a(x, y, h) ∈ C^{∞}(M, W) is a section of W, Γ_{ijk} the connection coefficients of ∂^{M}. Next we consider the bundle W ⊗K of Weyl algebras with coefficients in K with a connection ∂ = ∂^{M} ⊗ 1 + 1 ⊗ ∂^{E} on it. Finally, construct a connection

D = δ + ∂ + [s, ·]

where

δa = dx^{i} \frac{∂a}{∂y^{i}} = \frac{i}{h}[ω_{ij}y^{i}dx^{j}, a],

and s subject to the following conditions:

(1) s = ir/h where r ∈ C^{∞}(M, W ⊗ K ⊗ Λ^{1}) with deg r ≥ 3 (recall that deg y^{i} = 1, deg h = 2),

(2) δ^{−1}s = 0, where δ^{−1} is defined as follows

δ^{−1}a_{pq} = \frac{1}{p + q} y^{k}i (\frac{∂}{∂x^{k}} ) a_{pq}

for a homogeneous form a_{pq} of degree p in y^{i} and q in dx^{j} if p + q > 0 and δ^{−1}a_{00} = 0,

(3) D^{2} ≡ 0 (the so-called Abelian property). More precisely, we impose a sufficient condition

δs = R + ∂s + s^{2}  (2.1)

where

R = \frac{1}{2}R_{i_{1}i_{2}}^{E}dx^{i_{1}} ∧ dx^{i_{2}} + \frac{i}{4h}R_{i_{1}i_{2}i_{3}i_{4}}y^{i_{1}}y^{i_{2}}dx^{i_{3}} ∧ dx^{i_{4}}

is the curvature form of the connection ∂ on the bundle W ⊗ K.

These conditions define D uniquely. It may be shown that they are equivalent to the following equation

s = s_{1} + δ^{−1} (∂s + s^{2})  (2.2)

where

s_{1} = \frac{1}{2}R_{i_{1}i_{2}}^{E}y^{i_{1}}dx^{i_{2}} + \frac{i}{8h}R_{i_{1}i_{2}i_{3}i_{4}}y^{i_{1}}y^{i_{2}}y^{i_{3}}dx^{i_{4}}  (2.3)

which may be solved by iterations.

We define W_{D} to be the algebra of flat sections of W ⊗ K with respect to D, that is

W_{D} = {a ∈ C^{∞}(M, W ⊗ K) : Da ≡ 0}.

We also consider a complex of differential forms with values in W ⊗ K

0 ⟶ W ⊗ K ⟶^{D} W ⊗ K ⊗ Λ^{1} ⟶^{D} ….

This complex is exact except the zero-order term where Ker D = W_{D} (see [2]). In particular, Da = 0 has a unique solution such that a|_{y=0} = a_{0} ∈ C^{∞}(M, K). It may be found by iterations of the equation

a = a_{0} + δ^{−1} (∂a + [s, a]) .  (2.4)

The map

Q : a_{0} ↦ a ∈ W_{D}

will be called a canonical quantization map or simply quantization. From (2.4) we can obtain an “explicit” form of the operator Q

Qa = ∑_{k=0}^{∞}(δ^{−1}(∂ + ad s))^{k}a_{0}

= (1 − δ^{−1}(∂ + ad s))^{−1}a_{0}.  (2.5)

In such a form Q may be applied to any section of the bundle W ⊗ K, the result will be not necessarily flat in general.

Further, the equation Da = b with a one-form b satisfying Db = 0 has a unique solution a such that a|_{y=0} = 0. We denote it D^{−1}b. It can be found from the equation similar to (2.4)

a = −δ^{−1}b + δ^{−1} (∂a + [s, a])  (2.6)

which is equivalent to to a system

Da = b, a|_{y=0} = 0

with a solvability condition Db = 0. Using the operator Q the result may be written in the form

D^{−1}b = −Q(δ^{−1}b).  (2.7)

Next we recall the definition of the canonical trace on W_{D}. Locally, using the Darboux charts we can trivialize the manifold M and the bundle E and construct a connection D_{0} on the restriction of the bundle W ⊗ K to the coordinate neighborhood O ⊂ M starting with the de Rham differential d_{x} as an initial connection ∂.

It turns out that the two algebras W_{D}|_{O} and (W |_{O})|_{D_{0}} defined over O are isomorphic, moreover, this isomorphism may be taken in the form

A : W_{D}|_{O} ⟶ (W |_{O})_{D_{0}}
Aa = U ∘ a ∘ U^{−1}.  (2.8)

Here a ∈ W_{D}, U is a section of W |_{O} extended by negative powers of h so that the total degree of each term with respect to y^{i} and h is positive (see [1, 2]), the product ∘ is a fiberwise Weyl product in the bundle W. Applying D_{0} to (2.8), we come to the following equation for U

[U^{−1} ∘ D_{0}U − Δγ, a] = 0

with

Δγ = \frac{i}{2h}Γ_{ijk}y^{i}y^{j}dx^{k} + Γ_{i}^{E}dx^{i} + s.

We set

U^{−1} ∘ D_{0}U = Δγ.  (2.9)

This equation has a unique solution satisfying the normalization condition U|_{y=0} = 1. In fact the equation (2.9) together with U|_{y=0} = 1 is equivalent to the following equation

U = 1 + δ^{−1} (dU + U ∘ Δγ)  (2.10)

which may be solved by iterations.

Now, for a section a ∈ W_{D} with support in a coordinate chart O we define

Tr a = \frac{(2πh)^{−n}}{n!} ∫_{ℝ^{2n}} tr (Aa)|_{y=0} ω^{n}  (2.11)

where tr means coefficient trace in K = Hom (E, E). In general, we define Tr a by linearity using coordinate Darboux covering {O_{i}}, a partition of unity {ρ_{i}} and local trivializations A_{i} defined by (2.8), (2.9). It turns out that this is a correct definition, in particular, (2.11) does not depend on the choice of local coordinates and trivialization isomorphisms. The trace (2.11) is unique up to a normalization factor, the choice of this factor as in (2.11) gives us the so-called canonical trace.

One can observe, carefully following the formulas (2.2), (2.4), (2.10), that the integrand in (2.11) is a linear combination of covariant derivatives of the original section a_{0} ∈ C^{∞}(W, K), that is

tr (Aa)|_{y=0} = ∑_{I} tr c^{I}∂_{I}a_{0}

where I = (i_{1}, i_{2}, …, i_{k}) means a set of indices. Integration by parts in (2.11) gives

Tr a = \frac{(2πh)^{−n}}{n!} ∫_{ℝ^{2n}} tr ∑_{I}(−1)^{|I|}∂_{I′}c^{I}a_{0}  (2.12)

where I′ = (i_{k}i_{k−1}, …, i_{1}). The function

c(x, h) = ∑_{I}(−1)^{|I|}∂_{I′}c^{I} ∈ C^{∞}(M, K)  (2.13)

is a correctly defined section of C^{∞}(M, K), in particular, it does not depend upon the choice of local coordinates and trivializations of the bundle E. Thus, we have come to the notion of the trace density which may be expressed by the formula

Tr a = \frac{(2πh)^{−n}}{n!} ∫_{M} tr ca_{0}ω^{n}.

In principle, we are even able to compute it using (2.2)–(2.13), but very soon it becomes evident that the direct approach is rather hopeless. Nevertheless, these formulas lead to the following important conclusions:

1. the trace density depends only on connection coefficients Γ_{ijk} and Γ_{i}^{E} in local coordinates and local frames and their derivatives,

2. to compute c(x, h) at a point x ∈ M, we can use arbitrary local coordinates and frames in a small neighborhood of x, the result does not depend on the choices,

3. using normal Darbooux coordinates (see [2]) centered at x ∈ M, we see that c(x, h) depends on jets of connection coefficients at the origin. They, in turn, may be expressed through curvature tensors and their covariant derivatives.

3 The Variation of the Trace

Let us fix a classical observable a_{0}(x) ∈ C^{∞}(M, K), quantize it obtaining a = Q(a_{0}) ∈ W_{D} and take its canonical trace Tr a. We are interested how this quantity depends on the connection ∂.

Proposition 3.1. The following variation formula holds

\frac{d}{dt}Tr a = Tr ([D^{−1}γ̇, a] − D^{−1}[γ̇, a])  (3.1)

where

γ̇ = Δγ̇ = \frac{i}{2h}Γ̇_{ijk}y^{i}y^{j}dx^{k} + Γ̇_{i}^{E}dx^{i} + ṡ.

Proof. Considering variations of the connection ∂ in a coordinate neighborhood O, we can use the local formula (2.11) implying

\frac{d}{dt}Tr a = \frac{(2πh)^{−n}}{n!} ∫_{M} tr \frac{d}{dt}(Aa)|_{y=0} ω^{n}.

Further, (2.8) yields

\frac{d}{dt}(Aa) = U ∘ \{[U^{−1} ∘ U̇, a] + ȧ\} ∘ U^{−1}.  (3.2)

Differentiating (2.9) and using that D_{0} is a constant connection and thus independent of t, we obtain

U^{−1} ∘ D_{0}U̇ − U^{−1} ∘ U̇ ∘ U^{−1} ∘ D_{0}U = Δγ̇.

The first term may be transformed further

U^{−1} ∘ D_{0}U̇ = D_{0}(U^{−1} ∘ U̇) + U^{−1} ∘ D_{0}U ∘ U^{−1} ∘ U̇,

so the left-hand side of the previous equation may be written as

D_{0}(U^{−1} ∘ U̇) + [U^{−1} ∘ D_{0}U, U^{−1} ∘ U̇]
= D_{0}(U^{−1} ∘ U̇) + [Δγ, U^{−1} ∘ U̇] = D(U^{−1} ∘ U̇).

Here we have made use that U^{−1} ∘ D_{0}U is equal to Δγ in virtue of (2.9) and further that

D_{0} + [Δγ, ·] ≡ D.

Finally, we arrive to the equation for U^{−1} ∘ U̇

D(U^{−1} ∘ U̇) = Δγ̇.  (3.3)

The solvability condition D(Δγ̇) = 0 follows from the construction of D. In terms of the connection D_{0} and the form Δγ we can rewrite the equation (2.1) in the form

D_{0}(Δγ) + (Δγ)^{2} = 0.

The differentiation yields

D_{0}(Δγ̇) + [Δγ, Δγ̇] = D(Δ̇γ) = 0.

Thus, from (3.3) we obtain

U^{−1} ∘ U̇ = (D^{−1}(Δγ̇) + b)

where b is an arbitrary flat section, that is b ∈ W_{D}|_{O}.

Now differentiating Da = 0, we come to the following equation for ȧ

Dȧ + [Δγ̇, a] = 0.

The solvability condition is fulfilled since

D[Δγ̇, a] = [D(Δγ̇), a] + [Δγ̇, Da] = 0.

We also have

ȧ|_{y=0} = ȧ_{0} = 0,

implying

ȧ = −D^{−1}[Δγ̇, a].

Thus, the right-hand side of (3.2) becomes

U ∘ {([D^{−1}γ̇ + b, a] − D^{−1} [γ̇, a])} ∘ U^{−1},

where we have replaced Δγ̇ by a shorter notation γ̇.

Now we can return to global expressions. To this end observe that

[D^{−1}γ̇, a] − D^{−1}[γ̇, a]

is a flat section. Indeed, applying D, we get

D[D^{−1}γ̇, a] = [γ̇, a]

since DD^{−1}γ̇ = γ̇ and Da = 0. On the other hand,

D(D^{−1}[γ̇, a]) = [γ̇, a],

implying the assertion.

Now for a flat section s ∈ W_{D} with support in a neighborhood O we have by definition

\frac{(2πh)^{−n}}{n!} ∫_{ℝ^{2n}} tr U ∘ s ∘ U^{−1}ω^{n} = Tr s.

It means that the integral of the right-hand side of (3.2) is nothing but

Tr ([D^{−1}γ̇, a] − D^{−1}[γ̇, a] + [b, a]).

The last summand may be dropped since the trace of the commutator vanishes.

To calculate the section D^{−1}γ̇ ∈ C^{∞}(M, W ⊗ K) (which is not flat), let us write equation (2.6) for it. Denoting u = D^{−1}γ̇, we have

u = −δ^{−1}γ̇ + δ^{−1} (∂u + [s, u]) .

Taking into account that δ^{−1}ṡ ≡ 0, so that

δ^{−1}γ̇ = δΓ̇ = \frac{i}{6h}Γ̇_{ijk}y^{i}y^{j}y^{k} + Γ̇_{i}^{E}y^{i},

we obtain an equation

u = −\frac{i}{6h}Γ̇_{ijk}y^{i}y^{j}y^{k} + Γ̇_{i}^{E}y^{i} + δ^{−1} (∂u + [s, u])  (3.4)

which gives a computational algorithm by the iteration method.The result may be represented as

D^{−1}γ̇ = −QΓ̇ = −Q(Γ̇_{i}^{E}y^{i} + \frac{i}{6h}Γ̇_{ijk}y^{i}y^{j}y^{k})  (3.5)

with Q given by (2.5) The formula (3.1) can be immediately translated into the language of trace density. Indeed, by the definition of the trace density the left-hand side of (3.1) is equal to (up to a normalizing factor)

∫_{M} tr ċa_{0}ω^{n}.

Similarly, the right-hand side is

∫_{M} tr c ([D^{−1}γ̇, a] − D^{−1}[γ̇, a])|_{y=0} ω^{n}
= ∫_{M} tr c [D^{−1}γ̇, a]|_{y=0} ω^{n},

since D^{−1} [γ̇, a]|_{y=0} = 0. Thus, the variation formula may be rewritten as

∫_{M} tr ċa_{0}ω^{n} = ∫_{M} tr c [D^{−1}γ̇, Q(a_{0})]|_{y=0} ω^{n}.  (3.6)

Finally, we can get rid of a_{0} since it is an arbitrary function and obtain an equation for c. To this end introduce a pairing

⟨a, b⟩ = ∫_{M} tr a ∘ b|_{y=0} ω^{n}

on the space of all sections a, b ∈ C^{∞}(M, W ⊗ K)

Lemma 3.2. The pairing ⟨·, ·⟩ has the following properties:

(1) it is non-degenerate, that is ⟨a, b⟩ = 0 for any b ∈ C^{∞}(M, W ⊗ K) if and only if a = 0,

(2) for homogeneous sections

a = a_{p} = \frac{1}{p!}a_{i_{1}i_{2}...i_{p}}y^{i_{1}}y^{i_{2}} … y^{i_{p}}, b = b_{q} = \frac{1}{q!}b_{j_{1}j_{2}...j_{q}}y^{j_{1}}y^{j_{2}} … y^{j_{q}}

⟨a, b⟩ = 0 if p ≠ q and
⟨a_{p}, b_{p}⟩ = (−1)^{p}⟨b_{p}, a_{p}⟩,  (3.7)

(3)

⟨a, δ^{−1}∂b⟩ = −\frac{ih}{2}⟨i(ω^{−1})∂δea, b⟩  (3.8)

where ea_{p} = a_{p}/p for p > 0, ea_{0} = 0 and i(ω^{−1}) means a contraction with the tensor ω^{ij}.

Proof. The first two properties are obvious, so we prove only (3.8). Let a_{p}, b_{p−1} be two homogeneous sections of degree p and p − 1 respectively. Then

δ^{−1}∂b = \frac{1}{p!}∂_{j}b_{j_{1}...j_{p−1}}y^{j_{1}} … y^{j_{q−1}}y^{j},

so, the left-hand side is equal to

∫_{M} tr (−\frac{ih}{2})^{p} \frac{1}{p!}a_{i_{1}...i_{p}}∂_{j}b_{j_{1}...j_{p−1}}ω^{i_{1}j_{1}} … ω^{i_{p−1}j_{p−1}}ω^{i_{p}j}ω^{n}.

On the right-hand side we have

δea = \frac{1}{p!}a_{i_{1}...i_{p−1}i_{p}}y^{i_{1}} … y^{p−1}dx^{i_{p}},

i(ω^{−1})∂δea = i(ω^{−1})\frac{1}{p!}∂_{j}a_{i_{1}...i_{p−1}i_{p}}y^{i_{1}} … y^{i_{p−1}}dx^{j} ∧ dx^{i_{p}}

= \frac{1}{p!}∂_{j}a_{i_{1}...i_{p−1}i_{p}}y^{i_{1}} … y^{i_{p−1}}ω^{ji_{p}}, so that

−\frac{ih}{2}tr (i(ω^{−1})∂δea) ∘ b|_{y=0}

= (−\frac{ih}{2})^{p} \frac{1}{p!}tr ∂_{j}a_{i_{1}...i_{p}}b_{j_{1}...j_{p−1}}ω^{ji_{p}}ω^{i_{1}j_{1}} … ω^{i_{p−1}j_{p−1}}.

After integration by parts we obtain the same expression as on the left-hand side.

Non-degeneracy allows one to define a transposition for operators acting on sections, so that

⟨A′a, b⟩ = ⟨a, Ab⟩.

For example, if A = ad c_{p} where c_{p} is homogeneous in y of degree p then, using (3.7), we obtain

⟨a, (ad c)b⟩ = ⟨a, c ∘ b − b ∘ c⟩ = ⟨a ∘ c, b⟩ − (−1)^{p}⟨c ∘ a, b⟩,

so that

(ad c_{p})′a = a ∘ c − (−1)^{p}c ∘ a.

Now, we can remove the operator Q from a_{0} using this transposition. We also use the fact that a_{0} ∈ C^{∞}(M, K), that is does not depend on y. The latter observation implies

δ^{−1}(ad s)a_{0} = (ad δ^{−1}s)a_{0} = 0

because of the condition δ^{−1}s = 0. Thus,

Qa_{0} = a_{0} + Qδ^{−1}(∂ + ad s)a_{0} = a_{0} + Qδ^{−1}∂a_{0},  (3.9)

and we come to the following theorem.

Theorem 3.3. The trace density satisfies the following differential equation

ċ = \frac{ih}{2}ω^{ij}∂_{i} \frac{∂}{∂y^{j}} Q′(ad Q(δ^{−1}Γ̇))′c|_{y=0}  (3.10)

where

Q′ = (1 + \frac{ih}{2}i(ω^{−1})(∂ + (ad s)′δe))^{−1} .

Proof. The equation easily follows from (3.6), (3.5) and (3.9) by Lemma 3.2.

This theorem gives an algorithm to compute the trace density. In a local chart introduce a family of connections with

Γ(t) = t(Γ_{i}^{E}dx^{i} + \frac{i}{2h}Γ_{ijk}y^{i}y^{j}dx^{k}), 0 ≤ t ≤ 1.  (3.11)

It is clear that the coefficients of the trace density are polynomials in t, so the Taylor series gives a complete answer:

c = c(1) = ∑_{k=0}^{∞} \frac{1}{k!}c^{(k)}(0).

On the other hand, the derivatives c^{(k)}(0) may be computed successively from (3.10). This yields an expression for c in terms of connection coefficients and their derivatives. But we know that this expression is independent of the choice of local Darboux coordinates and local frames. So, using normal Darboux coordinates, we can rewrite this expression in terms of curvature tensors and their covariant derivatives.

4 Calculations

In this section we calculate the coefficients c_{1} and c_{2} of the expansion (1.3) using the differential equation (3.10) for the trace density. So, we introduce a family of connections in local coordinates and frames

Γ(t) = Γ_{i}^{E}(t)dx^{i} + \frac{i}{2h}Γ_{i_{1}i_{2}i_{3}}(t)y^{i_{1}}y^{i_{2}}dx^{i_{3}},

so that Γ(0) = 0 and Γ(1) coincides with the given connection form Γ. For example, we may take a linear family (3.11).

Let

s = s_{1} + s_{2} + …

be the solution of (2.2), (2.3) where s_{k} denotes the term of degree k in y, h. We calculate the right-hand side of (3.10) up to degree 4. So,

Q(δ^{−1}Γ̇(t)) = u_{1} + u_{2} + u_{3} + …

with

u_{1} = δΓ̇(t) = Γ̇_{i}^{E}y^{i} + \frac{i}{6h} Γ̇_{i_{1}i_{2}i_{3}}(t)y^{i_{1}}y^{i_{2}}y^{i_{3}}
u_{2} = δ^{−1}∂u_{1}
u_{3} = δ^{−1}[s_{1}, u_{1}] + δ^{−1}∂δ^{−1}∂u_{1}.  (4.1)

Further, define v = v_{1} + v_{2} + v_{3} + … to be

v = (ad Q(δ^{−1}Γ̇))′c

= (1 + hc_{1} + …)(u_{1} + u_{2} + u_{3} + …) + (u_{1} − u_{2} + u_{3} …)(1 + hc_{1} + …) = 2u_{1} + 2u_{3} + h(u_{1}c_{1} + c_{1}u_{1}) + …,

so that v_{2} ≡ 0. Thus, up to degree 4 the right-hand side of (3.10) may be written in the form

\frac{ih}{2}ω^{ij}∂_{i} \frac{∂}{∂y^{j}} Q′v|_{y=0}
= \frac{ih}{2}ω^{ij}∂_{i} \frac{∂}{∂y^{j}} \{v_{1} + v_{3} − \frac{ih}{2}i(ω^{−1})(∂ + (ad s)′)δev_{1}
+ (\frac{ih}{2}i(ω^{−1})∂δe)^{2} v_{1}\}|_{y=0} .  (4.2)

The first observation is that only linear terms with respect to y in curly brackets do contribute to (4.2). Hence, the second term in (4.1) may be dropped since it is at least cubic in y. Retaining only linear terms in the first summand of (4.1), we obtain the following expression for v_{3} in (4.2)

v_{3} = 2δ^{−1} ([s_{1}, u_{1}]|_{y=0}) + h(c_{1}u_{1} + u_{1}c_{1}).  (4.3)

Next, taking into account that v_{1} (which is equal to 2u_{1}) contains linear and cubic terms in y, we see that ∂δev_{1} contains only even-degree terms in y, so, this term does not contribute to (4.2).

Thus, retaining only essential terms, we rewrite (3.10) in the form

hċ_{1} + h^{2}ċ_{2} + …
= ihω^{ij}∂_{i} \frac{∂}{∂y^{j}} u_{1}|_{y=0}
+ \frac{ih^{2}}{2} ω^{ij}∂_{i} \frac{∂}{∂y^{j}} (c_{1}u_{1} + u_{1}c_{1})|_{y=0}
+ihω^{ij}∂_{i} \frac{∂}{∂y_{j}} δ^{−1} ([s_{1}, u_{1}]|_{y=0})
+ \frac{h^{2}}{2} ω^{ij}∂_{i} \frac{∂_{i}}{∂y^{j}} i(ω^{−1})(ad s_{1})′δeu_{1}|_{y=0}
− \frac{ih^{3}}{4} ω^{ij}∂_{i} \frac{∂}{∂y^{j}} (i(ω^{−1})∂δe)^{2}u_{1}|_{y=0} + …  (4.4)

We have replaced s by its leading term s_{1}, this does not affect the terms of degree ≤ 4.

Comparing linear terms in h, we get

ċ_{1} = iω^{ij}∂_{i} \frac{∂}{∂y^{j}} u_{1}|_{y=0} = iω^{ij}∂_{i}Γ̇_{j}^{E}
= \frac{i}{2}ω^{ij} (∂_{i}Γ̇_{j}^{E} − ∂_{j}Γ̇_{i}^{E}) = \frac{i}{2} Ṙ_{ij}^{E}ω^{ij},

so that

c_{1} = \frac{i}{2}R_{ij}^{E}

(we again drop ω^{ij} in the final expression). Substitute this result into (4.4) and compare the coefficients at h^{2}. The second term in (4.4) yields

−\frac{1}{4}∂_{i} (R_{i_{1}j_{1}}^{E}Γ̇_{j}^{E} + Γ̇_{j}^{E}R_{i_{1}j_{1}}^{E})  (4.5)

(contraction in i, j and i_{1}, j_{1} is understood). Passing to the third term, we first compute

[s_{1}, u_{1}]|_{y=0} = [\frac{1}{2}R_{i_{1}i_{2}}^{E}y^{i_{1}}dx^{i_{2}}, Γ̇_{i}^{E}y^{i}]|_{y=0}
+ [\frac{i}{8h}R_{i_{1}i_{2}i_{3}i_{4}}y^{i_{1}}y^{i_{2}}y^{i_{3}}dx^{i_{4}}, \frac{i}{6h} Γ̇_{i_{5}i_{6}i_{7}}y^{i_{5}}y^{i_{6}}y^{i_{7}}]|_{y=0}
= −\frac{ih}{4} (R_{i_{1}i_{2}}^{E}Γ̇_{j_{1}}^{E} + Γ̇_{j_{1}}^{E}R_{i_{1}i_{2}}^{E}) dx^{i_{2}}
−\frac{ih}{32}R_{i_{1}i_{2}i_{3}i_{4}} Γ̇_{j_{1}j_{2}j_{3}}dx^{i_{4}}.

Thus, the third term yields

\frac{1}{4}∂_{i_{2}} (R_{i_{1}j_{2}}^{E}Γ̇_{j_{1}}^{E} + Γ̇_{j_{1}}^{E}R_{i_{1}j_{2}}^{E})
+\frac{1}{32}∂_{i_{4}} (R_{i_{1}i_{2}i_{3}j_{4}} Γ̇_{j_{1}j_{2}j_{3}}) .  (4.6)

The fourth term requires more work. First of all we have

(ad s_{1})′δeu_{1} = δeu_{1} ∘ s_{1} − s_{1} ∘ δeu_{1},

the sign − before the second summand appears since we have to change the sign twice: once because s_{1} contains the terms of odd degrees in y, and once more because s_{1} and δeu_{1} are one-forms. The terms linear in y are

\frac{1}{2} (Γ̇_{i_{1}}^{E}R_{i_{2}i_{3}}^{E} + R_{i_{2}i_{3}}^{E}Γ̇_{i_{1}}^{E}) y^{i_{2}}dx^{i_{1}} ∧ dx^{i_{3}}
+\frac{1}{16} Γ̇_{i_{1}i_{2}i_{3}}R_{(j_{1}j_{2}i_{4})i_{5}}y^{i_{4}}dx^{i_{3}} ∧ dx^{i_{5}},

where parentheses mean symmetrization. Thus, the fourth summand becomes

\frac{1}{4}∂_{i_{1}} (Γ̇_{i_{2}}^{E}R_{j_{1}j_{2}}^{E} + R_{j_{1}j_{2}}^{E}Γ̇_{i_{2}}^{E})
+\frac{1}{32}∂_{i_{4}} (Γ̇_{i_{1}i_{2}i_{3}}R_{(j_{1}j_{2}j_{4})j_{3}}) .

The completely symmetric component R_{((j_{1}j_{2}j_{4})j_{3})} of the tensor R_{(j_{1}j_{2}j_{4})j_{3}} vanishes, hence

R_{(j_{1}j_{2}j_{4})j_{3}} + R_{(j_{1}j_{2}j_{3})j_{4}} + R_{(j_{1}j_{3}j_{4})j_{2}} + R_{(j_{3}j_{2}j_{4})j_{1}} = 0.

After contraction with the completely symmetric tensor Γ̇_{i_{1}i_{2}i_{3}} this identity yields

3Γ̇_{i_{1}i_{2}i_{3}}R_{(j_{1}j_{2}j_{4})j_{3}} = −Γ̇_{i_{1}i_{2}i_{3}}R_{j_{1}j_{2}j_{3}j_{4}},

so that the fourth term in (4.4) becomes

\frac{1}{4}∂_{i_{1}} (Γ̇_{i_{2}}^{E}R_{j_{1}j_{2}}^{E} + R_{j_{1}j_{2}}^{E}Γ̇_{i_{2}}^{E})
−\frac{1}{96}∂_{i_{4}} (Γ̇_{i_{1}i_{2}i_{3}}R_{j_{1}j_{2}j_{3}j_{4}}) .  (4.7)

Finally, the last term in (4.4) gives

\frac{1}{24}∂_{i_{1}}∂_{i_{2}}∂_{i_{3}} Γ̇_{j_{1}j_{2}j_{3}}.  (4.8)

Summing up (4.5) - (4.8), we obtain

ċ_{2} = −\frac{1}{4}∂_{i_{1}} (Γ̇_{j_{1}}^{E}R_{i_{2}j_{2}}^{E} + R_{i_{2}j_{2}}^{E}Γ̇_{j_{1}}^{E})
+\frac{1}{2}∂_{i_{1}} (Γ̇_{i_{2}}^{E}R_{j_{1}j_{2}}^{E} + R_{j_{1}j_{2}}^{E}Γ̇_{i_{2}}^{E})
+\frac{1}{24} (∂_{i_{1}}∂_{i_{2}}∂_{i_{3}} Γ̇_{j_{1}j_{2}j_{3}} − ∂_{i_{4}}(Γ̇_{i_{1}i_{2}i_{3}}R_{j_{1}j_{2}j_{3}j_{4}})) .  (4.9)

Now we can watch an amazing interplay of the symmetry properties of connection coefficients and curvature tensors resulting in the fact that the right-hand side is a complete derivative in t. Clearly, the terms corresponding to ∂^{E} and ∂ may be treated separately. First consider the terms corresponding to ∂^{E} dropping temporarily the superscript E:

−\frac{1}{4}(R_{i_{1}j_{1}}∂_{i_{2}} Γ̇_{j_{2}} + ∂_{i_{2}} Γ̇_{j_{2}}R_{i_{1}j_{1}})
−\frac{1}{4}(∂_{i_{2}}R_{i_{1}j_{1}} Γ̇_{j_{2}} + Γ̇_{j_{2}}∂_{i_{2}}R_{i_{1}j_{1}})
+\frac{1}{2}(∂_{i_{1}} Γ̇_{i_{2}}R_{j_{1}j_{2}} + R_{j_{1}j_{2}}∂_{i_{1}} Γ̇_{i_{2}})
+\frac{1}{2}(Γ̇_{i_{2}}∂_{i_{1}}R_{j_{1}j_{2}} + ∂_{i_{1}}R_{j_{1}j_{2}} Γ̇_{i_{2}}).

Note, that the second and the fourth terms cancel out. Indeed, changing summation indices, rewrite them as

−\frac{1}{4}(∂_{i_{2}}R_{i_{1}j_{1}} + 2∂_{i_{1}}R_{j_{1}i_{2}})Γ̇_{j_{2}}
−\frac{1}{4} Γ̇_{j_{2}}(∂_{i_{2}}R_{i_{1}j_{1}} + 2∂_{i_{1}}R_{j_{1}i_{2}}).

Expressions in parentheses vanish by Bianchi’s identity:

∂_{i_{2}}R_{i_{1}j_{1}} + 2∂_{i_{1}}R_{j_{1}j_{2}} = ∂_{i_{2}}R_{i_{1}j_{1}} + ∂_{i_{1}}R_{j_{1}j_{2}} − ∂_{j_{1}}R_{i_{1}j_{2}}
= ∂_{i_{2}}R_{i_{1}j_{1}} + ∂_{i_{1}}R_{j_{1}j_{2}} + ∂_{j_{1}}R_{j_{2}i_{1}} = 0.

In the remaining summands everything may be expressed in terms of the curvature tensor. Since R_{j_{1}j_{2}} is antisymmetric and because j_{1}, j_{2} are contracted with i_{1}, i_{2}, we may antisymmetrize with respect to i_{1}, i_{2}, that is

∂_{i_{1}} Γ̇_{i_{2}}R_{j_{1}j_{2}} = \frac{1}{2}(∂_{i_{1}} Γ̇_{i_{2}} − ∂_{i_{2}} Γ̇_{i_{1}})R_{j_{1}j_{2}}
= \frac{1}{2} Ṙ_{i_{1}i_{2}}R_{j_{1}j_{2}}.

Thus, the remaining terms give

−\frac{1}{8}(R_{i_{1}j_{1}} Ṙ_{i_{2}j_{2}} + R_{i_{2}j_{2}}R_{i_{1}j_{1}})
+\frac{1}{4}(Ṙ_{i_{1}i_{2}}R_{j_{1}j_{2}} + R_{j_{1}j_{2}} Ṙ_{i_{1}i_{2}})
= −\frac{1}{8} \frac{d}{dt}(R_{i_{1}j_{1}}R_{i_{2}j_{2}} − 2R_{i_{1}i_{2}}R_{j_{1}j_{2}})
= −\frac{1}{8} \frac{d}{dt}(R_{i_{1}j_{1}}R_{i_{2}j_{2}} + R_{i_{1}i_{2}}R_{j_{2}j_{1}} + R_{i_{1}j_{2}}R_{j_{1}j_{2}})
= −\frac{3}{8} \frac{d}{dt}R_{[i_{1}j_{1}}R_{i_{2}j_{2}]}.

The terms corresponding to ∂ are proportional to

∂_{i_{1}}∂_{i_{2}}∂_{i_{3}} Γ̇_{j_{1}j_{2}j_{3}} − ∂_{i_{4}}(Γ̇_{i_{1}i_{2}i_{3}}R_{j_{1}j_{2}j_{3}j_{4}}).  (4.10)

We transform the first term using the relation

∂_{i_{3}} Γ̇_{j_{1}j_{2}j_{3}}ω^{i_{3}j_{3}} = \frac{1}{2}(∂_{i_{3}} Γ̇_{j_{1}j_{2}j_{3}} − ∂_{j_{3}}Γ_{j_{1}j_{2}j_{3}})ω^{i_{3}j_{3}}
= \frac{1}{2} Ṙ_{j_{1}j_{2}i_{3}j_{3}}.

Thus,

∂_{i_{1}}∂_{i_{2}}∂_{i_{3}} Γ̇_{j_{1}j_{2}j_{3}} = \frac{1}{2} \frac{d}{dt}(∂_{i_{1}}∂_{i_{2}}R_{j_{1}j_{2}i_{3}j_{3}})
−\frac{1}{2}∂_{i_{1}} ˙∂_{i_{2}}R_{j_{1}j_{2}i_{3}j_{3}} − \frac{1}{2} ˙∂_{i_{1}}∂_{i_{2}}R_{j_{1}j_{2}i_{3}j_{3}}.

The last summand is equal to zero. Indeed, the only free index in ∂_{i_{2}}R_{j_{1}j_{2}i_{3}j_{3}} is j_{1}, so

˙∂_{i_{1}}∂_{i_{2}}R_{j_{1}j_{2}i_{3}j_{3}} = Γ̇_{i_{1}j_{1}i_{4}}∂_{i_{2}}R_{j_{4}j_{2}i_{3}j_{3}} = 0

since Γ_{pqr} is symmetric in all the indices, so that Γ̇_{i_{1}j_{1}i_{4}}ω^{i_{1}j_{1}} ≡ 0. Now,

˙∂_{i_{2}}R_{j_{1}j_{2}i_{3}j_{3}} = Γ̇_{i_{2}j_{1}i_{4}}R_{j_{4}j_{2}i_{3}j_{3}} + Γ̇_{i_{2}j_{2}i_{4}}R_{j_{1}j_{4}i_{3}j_{3}},

and again the last summand is equal to 0, so (4.10) becomes

\frac{1}{2} \frac{d}{dt}(∂_{i_{1}}∂_{i_{2}}R_{j_{1}j_{2}i_{3}j_{3}}) − \frac{1}{2}∂_{i_{4}}(Γ̇_{i_{1}i_{2}j_{4}}R_{j_{1}j_{2}i_{3}j_{3}})
−∂_{i_{4}}(Γ̇_{i_{1}i_{2}i_{3}}R_{j_{1}j_{2}j_{3}j_{4}}).

The second and the third terms considered as tensors in i_{3}, j_{3}, i_{4}, j_{4} (the indices i_{1}, j_{1} and i_{2}, j_{2} in both terms are contracted and we consider them as dumb ones) are treated similarly to the ∂^{E}-case. Namely,

−\frac{1}{2}∂_{i_{4}} Γ̇_{i_{1}i_{2}j_{4}}R_{j_{1}j_{2}i_{3}j_{3}} − ∂_{i_{4}} Γ̇_{i_{1}i_{2}i_{3}}R_{j_{1}j_{2}j_{3}j_{4}}
= −\frac{1}{4} Ṙ_{i_{1}i_{2}i_{4}j_{4}}R_{j_{1}j_{2}i_{3}j_{3}} − \frac{1}{2} Ṙ_{i_{1}i_{2}i_{4}i_{3}}R_{j_{1}j_{2}j_{3}j_{4}}
= −\frac{1}{8} \frac{d}{dt}(R_{i_{1}i_{2}i_{4}j_{4}}R_{j_{1}j_{2}i_{3}j_{3}} + R_{i_{1}i_{2}i_{4}i_{3}}R_{j_{1}j_{2}j_{3}j_{4}} + R_{i_{1}i_{2}i_{4}j_{3}}R_{j_{1}j_{2}j_{4}i_{3}})
= −\frac{3}{8} \frac{d}{dt}R_{i_{1}i_{2}[i_{3}j_{3}}R_{|j_{1}j_{2}|i_{4}j_{4}]},

while the remaining terms

−\frac{1}{2}Γ_{i_{1}i_{2}j_{4}}∂_{i_{4}}R_{j_{1}j_{2}i_{3}j_{3}} − Γ̇_{i_{1}i_{2}i_{3}}∂_{i_{4}}R_{j_{1}j_{2}j_{3}j_{4}}

vanish in virtue of the Bianchi identity.

Let us formulate the final result as a proposition.

Proposition 4.1. The trace density up to degree 4 is given by the expression

c = 1 + \frac{ih}{2}R_{ij} − \frac{3}{8}h^{2}R_{[i_{1}j_{1}}R_{i_{2}j_{2}]}
−\frac{h^{2}}{64}R_{i_{1}i_{2}[i_{3}j_{3}}R_{|j_{1}j_{2}|i_{4}j_{4}]} + \frac{h^{2}}{48}∂_{i_{1}}∂_{i_{2}}R_{j_{1}j_{2}i_{3}j_{3}}.

After multiplication by the volume form ω^{n}/(2πh)^{n}n! one can easily recognize characteristic forms of connections ∂^{E} and ∂^{M}. So, the first three terms give

1 + (−\frac{1}{2πi} \frac{1}{2}R_{ij}^{E}dx^{i} ∧ dx^{j}) + \frac{1}{2} (−\frac{1}{2πi} \frac{1}{2}R_{ij}^{E}dx^{i} ∧ dx^{j})^{2}

times exp(ω/2πh), the forth term yields

\frac{1}{12}tr (\frac{1}{2πi} \frac{1}{4}R_{ij}dx^{i} ∧ dx^{j})^{2} \frac{ω^{n−2}}{(2πh)^{n−2}(n − 2)!}.

References

[1] Fedosov, B.: A simple geometrical construction of deformation quantization. J. Diff. Geom., 40 (1994), 213-238.

[2] Fedosov, B.: Deformation Quantization and Index Theory. Math. Top., 9, Akademie Verlag, Berlin (1996).

[3] Connes, A., Flato, M., Sternheimer, D.: Closed star-products and cyclic cohomology. Lett. Math. Phys. 24 (1992), 1-12.