Three Natural Generalizations of Fedosov Quantization

Fedosov's simple geometrical construction for deformation quantization of symplectic manifolds is generalized in three ways without introducing new variables: (1) The base manifold is allowed to be a supermanifold. (2) The star product does not have to be of Weyl/symmetric or Wick/normal type. (3) The initial geometric structures are allowed to depend on Planck's constant.


Introduction
The aim of this paper is to give natural generalizations of Fedosov's simple geometrical construction for deformation quantization on a symplectic manifold M [21,22,23,24]. In short, the term "deformation quantization" refers to the construction of an associative * product that is an expansion in some formal parameterh, and whose leading behavior is controlled by a geometric structure m ij , which usually (but not always) is a symplectic structure [8]. Fedosov quantization, in its most basic form, is a deformation quantization recipe that relies on yet another geometric input in form of a compatible torsion-free tangent bundle connection ∇. It is an important feature of Fedosov's * product that it is a global construction, which applies to a manifold as a whole, and not just a local neighborhood or vector space. Another characteristic feature that sets Fedosov's approach apart from most other methods is the assemblage of a globally welldefined, flat connection D. We discuss in this paper the following three natural generalizations of Fedosov's original construction: 2. We do not assume that m ij is antisymmetric. This is sometimes referred to as the * product does not have to be of Weyl/symmetric type. It is also not necessarily of the Wick/normal type [9,10,11,36,14,18,42,19]. 3. We let the two geometric input data m ij and ∇ depend on Planck's constanth. A priori, such a generalization is a non-trivial matter, since Planck's constanth is charged with respect to the pertinent resolution degree, cf. equation (6.1). Moreover, forh-dependent structures, there is no analogue of Darboux' theorem at our disposal, cf. discussion below equation (3.7).
Certain aspects of each of the three generalizations have appeared in the literature, but never simultaneously. Our main point is that the three generalizations taken together constitute a natural habitat for Fedosov's original variables x i , c i ≡ dx i , y i andh, without destroying the simplicity of his construction.
The paper starts with listing the basic setup and assumptions in Sections 2-3. Then follows an introduction of the relevant tools: An algebra A of covariant tensors in Sections 4-5, the Fedosov resolution degree in Section 6, the • product in Sections 7-9, the Koszul-Tate differential δ and its cohomology in Sections 10-11. Thereafter is given a discussion of Riemann curvature tensors on supermanifolds in Sections 12-13. The flat D connection, whose existence is guaranteed by Fedosov's 1st theorem (Theorem 1), is discussed in Sections 14-16. Fedosov's 2nd theorem (Theorem 2), which establishes an algebra isomorphism between symbols and horizontal zeroforms, is discussed in Section 17. Finally, the * product is constructed in the last Section 18.
We shall focus on the existence of the construction and skip the important question of uniqueness/ambiguity/equivalence/classification of * products for brevity. In detail, this paper is an elaboration of the material covered on p. 138-147 in Fedosov's book [24] subjected to the above generalizations 1-3. In particular, what we refer to as Fedosov's 1st and 2nd theorems (Theorems 1 and 2) are generalizations of Theorems 5.2.2 and 5.2.4 in [24], respectively. Some other references that deal with Fedosov's original construction are [43,31,40,26,45,30]. Historically, De Wilde and Lecomte were the first to prove the existence of an associative * product on every symplectic manifold [46]. The same question for Poisson manifolds (which may be degenerate) was proved by Kontsevich using ideas from string theory [39]. Cattaneo et al. gave an explicit construction in the Poisson case that merges Kontsevich's local formula with Fedosov's flat D connection approach [15,16].
General remarks about notation. We shall work with smooth finite-dimensional supermanifolds in terms of charts and atlases (as opposed to, e.g., ringed spaces or functors of points). We shall sometimes make use of local formulas, because these often provide the most transparent definitions of sign conventions on supermanifolds. We stress that all formulas in this paper, if not written in manifestly invariant form, hold with respect to any coordinate system, and they transform covariantly under general coordinate transformations. Adjectives from supermathematics such as "graded", "super", etc., are implicitly implied. We will also follow commonly accepted superconventions, such as, Grassmann-parity ε is only defined modulo 2, and "nilpotent" means "nilpotent of order 2". The sign conventions are such that two exterior forms ξ and η, of Grassmann-parity ε ξ , ε η and exterior form degree p ξ , p η , respectively, commute in the following graded sense η ∧ ξ = (−1) ε ξ ε η +p ξ pη ξ ∧ η inside the exterior algebra. We will often not write exterior wedges "∧" nor tensor multiplications "⊗" explicitly. Covariant and exterior derivatives will always be from the left, while partial derivatives can be from either left or right. We shall sometimes use round parenthesis "( )" to indicate how far derivatives act, see e.g., equations (2.1) and (11.1).

Basic settings and assumptions
Consider a manifold M with local coordinates x i of Grassmann-parity ε i ≡ ε(x i ). The classical limit of the sought-for associative * multiplication is prescribed by a Grassmann-even contravariant (2, 0) tensor (The letter "m" is a mnemonic for the word "multiplication".) The tensor can in principle be a formal power series in Planck's constanth. (The quantum corrections to m ij do not enter actively into the classical boundary condition (2.1), but they will nevertheless affect the Fedosov implementation of the * multiplication at higher orders inh, as we shall see in equation (7.1) below.) Usually one demands [8] that the classical unit function 1 ∈ C ∞ (M ) serves as a unit for the full quantum algebra (A 00 , +, * ): (2.2) (The notation A 00 will be explained in Section 4.) Let m T denote the transposed tensor, It will be necessary to assume that the antisymmetric part of the tensor m ij is non-degenerate, i.e., that there exists an inverse matrix ω ij such that (Note that this does not necessarily imply that m ij itself has to be non-degenerate.) Next, let there be given a torsion-free connection ∇ : ] that preserves the m-tensor In local coordinates, the covariant derivative ∇ X along a vector field X = X i ∂ ℓ i is given as [12] Here ∂ r i ≡ (−1) ε i ∂ ℓ i are not usual partial derivatives. In particular, they do not act on the Christoffel symbols Γ k ij in equations (2.5) and (2.6). Rather they are a dual basis to the one-forms Phrased differently, the ∂ r i are merely bookkeeping devices, that transform as right partial derivatives under general coordinate transformations. (To be able to distinguish them from true partial derivatives, the differentiation variable x i on a true partial derivative ∂/∂x i is written explicitly.) The assumption (2.4) reads in local coordinates where we have introduced a reordered Christoffel symbol Note that the connection ∇ will also preserve the transposed tensor m T , and therefore, by linearity, the antisymmetric part ω ij . We shall later explain why it is crucial that the connection ∇ is torsion-free, see comment after equation (16.1). The Christoffel symbols Γ k ij = Γ k ij (x;h) for the connection ∇ is allowed to be a formal power series in Planck's constanth. Finally, we mention that one traditionally imposes a reality/hermiticity condition on the connection ∇, the multiplicative structure m ij and the * product. However, we shall for simplicity skip the reality/hermiticity condition in this paper.

The two-form ω is symplectic
The inverse matrix ω ij with lower indices has the following graded skewsymmetry cf. equation (2.3). That equation (3.1) should be counted as an skewsymmetry (as opposed to a symmetry) is perhaps easiest to see by defining the slightly modified matrix Note that the two matrices ω ij andω ij are identical for bosonic manifolds. Then the equation (3.1) translates into the more familiar type of graded antisymmetry, The skewsymmetry means that the inverse matrix can be viewed as a two-form Here is the usual basis of one-forms, and denotes the de Rham exterior derivative on M . It follows from assumption (2.4) that the connection ∇ preserves the two-form where the lowered Christoffel symbol Γ k,ij is defined as The two-form ω is closed (dω) = 0, (3.5) or equivalently, The closeness relation (3.5) is not an extra assumption. It follows from equation (3.4), because the connection ∇ is torsion-free, 6) or equivalently, in terms of the lowered Christoffel symbol, It is practical to call a non-degenerate closed two-form ω ij a symplectic structure, even if it depends on Planck's constanth. The inverse structure, i.e., the corresponding Poisson structure ω ij satisfies the Jacobi identity Note that we cannot rely on Darboux' theorem, i.e., we will not be guaranteed a cover of Darboux coordinate patches in which the ω ij is constant. The issue is that, on one hand, the symplectic structure is allowed to depend on Planck's constanth, but, on the other hand, we shall not allow coordinate transformations x i → x ′j = x ′j (x) that depend onh. Luckily, as we shall see, Darboux patches play no rôle in the Fedosov construction. In fact, as we have mentioned before, all formulas in this paper, if not written in manifestly invariant form, hold with respect to any coordinate system, and they transform covariantly under general coordinate transformations. The classical Poisson bracket is given by the famous quantum correspondence principle [17] {f, is the * commutator, and It is easy to show that every symplectic manifold (M ; ω) has a torsion-free ω-preserving connection ∇, see Section 2.5 in [24] for the bosonic case. However, it is not true that every manifold (M ; m) with a multiplicative structure m ij supports a torsion-free m-preserving connection ∇, cf. assumption equation (2.4). The symmetric part of the tensor m ij needs to be compatible with the symplectic structure ω in a certain sense. In the special case where g ij = 0, we return to the usual Fedosov quantization m ij = ω ij , which corresponds to a Weyl/symmetric type * product. In the generic case where g ij has full rank, there will exist an inverse matrix g ij , which constitute a (pseudo) Riemannian metric, and there will hence exist a corresponding unique Levi-Civita connection ∇ LC . In this non-degenerate case, the necessary and sufficient conditions are ∇ = ∇ LC and (∇ LC i ω jk ) = 0. This is for instance satisfied for (pseudo) Kähler manifolds (M ; ω; g), cf. [9,10,11,36,14,18,42,19].

Covariant tensors
be the vector space of (0, m + n)-tensors a i 1 ...imj 1 ...jn (x;h) that are antisymmetric with respect to the first m indices i 1 . . . i m and symmetric with respect to the last n indices j 1 . . . j n . Phrased differently, they are m-form valued symmetric (0, n)-tensors. As usual, it is practical to introduce a coordinate-free notation Here the Fedosov variables y i and the "∨" symbol are symmetric counterparts to the one-form basis c i ≡ dx i and the "∧" symbol, i.e., We will be interested in covariant derivatives ∇ i a i 1 ...imj 1 ...jn of the above tensors. The covariant derivative ∇ i from equation (2.5) can be implemented on coordinate-free objects (4.2) by the following linear differential operator If both the numbers of antisymmetric and symmetric indices are non-zero m = 0 ∧ n = 0, i.e., if the tensor a i 1 ...imj 1 ...jn has mixed symmetry, the covariant derivative ∇ i a i 1 ...imj 1 ...jn will not belong to any of the A •• spaces (4.1). We repair this by antisymmetrizing with respect to the indices i, i 1 , . . . , i m . Such antisymmetrization can be implemented on coordinate-free objects (4.2) with the help of the following one-form valued Grassmann-even differential operator where we have followed common practice, and given the differential operator (4.4) the same name as the connection. In the second equality of equation (4.4) we have used that the connection is Grading [38,37] consider a hybrid model where torsion is allowed in the y-sector but not in the c-sector in such a way that equation (4.4) remains valid.) Since the ∇ operator is a first-order differential operator, i.e., where a and b are two coordinate-free objects (4.2), it is customary to refer to ∇ as a linear connection. (The order of the exterior factor m (T * M ) and the symmetric factor n (T * M ) in expression (4.1) is opposite the standard convention to ease the use of covariant derivatives ∇ that acts from the left.)

The A algebra
The direct sum of the A mn spaces (4.1) is an algebra with multiplication given by the tensor multiplication. It is both associative and commutative. As indicated in equation (5.1), the elements can be viewed as quantum functions a = a(x, c; y;h) on the Whitney sum where c i are identified with the parity-inverted fiber coordinates for the fiber-wise parityinverted tangent bundle ΠT M , and y j are identified with the fiber coordinates for the tangent bundle T M. The word "quantum function" just means that it is a formal power series in Planck's constanth.

The Fedosov resolution degree
The Fedosov degree "deg" is a (non-negative) integer grading of the A algebra defined as and zero for the two other remaining variables x j and c k , cf. Table 1. The Fedosov degree will play the rôle of resolution degree in the sense of homological perturbation theory [29,7,27,1,2,25,35]. We shall therefore often organize the algebra according to this grading: Similarly, one may write the algebra element as a direct sum of elements a (n) of definite Fedosov degree deg(a (n) ) = n.

The • product
One now builds an associative • product A × A → A on the A algebra as a Moyal product [34,41] in the y-variables, The • product is associative, because the m jk -tensor is independent of y-variables. (The yvariables have been interpreted by Grigoriev and Lyakhovich [33,5] as conversion variables for the conversion of second-class constraints into first-class [3,4,6,28].) The • product "preserves" the following gradings The connection ∇ respects the • product, as a consequence of the assumption (2.4).

The Poisson bracket
It is useful to define a Poisson bracket as (It should not be confused with the classical Poisson bracket (3.8).) The Poisson bracket (8.1) "preserves" the following gradings The connection ∇ respects the Poisson bracket: 9 The • commutator The • commutator is defined as Note the following useful observations: The • commutator may be expanded according to the Fedosov degree: 10 The Koszul-Tate differential δ The Koszul-Tate differential is defined as In the second equality in equation (10.1) we have indicated that the Koszul-Tate differential is an inner derivation in the algebra (A, +, 1 ih [· • , ·]) with generator ̟ := c i ω ij y j = y j c iω ij ∈ A 11 , deg(̟) ≥ 1, which we shall refer to as the Hamiltonian for δ. The Koszul-Tate differential δ will serve as the leading term in a resolution expansion of a deformed connection D, see Section 14 [29,7,27,1,2,25,35]. Since the ω ij tensor is covariantly preserved, cf. equation (3.4), it follows immediately that (∇̟) = −y j c i c k (∇ kωij ) = 0 even without using the skewsymmetry (3.1) or the antisymmetry (3.2) (or the torsion-free condition for that matter). As a corollary, The δ-differential is nilpotent and it respects the • product The Poincaré lemma and the homotopy operator δ −1 There exists a version of the Poincaré lemma where the rôle of the de Rham exterior derivative d ≡ c i ∂ ℓ /∂x i has been replaced by the Koszul-Tate differential δ ≡ c i ∂ ℓ /∂y i , or equivalently, where the x-coordinates are replaced by the y-coordinates. As we shall see below, the equation (δb) = a, a ∈ A, (11.1) may be solved locally with respect to an algebra element b ∈ A whenever a ∈ A is a given δclosed algebra element with no 00-part a 00 = 0. (The 00-part a 00 is the part of the algebra element a that is independent of the cand the y-variables, cf. Section 4.) In fact, a local solution b to equation (11.1) may be extended to a global solution, since the total space is contractible in the y-directions. We shall use this crucial fact to guarantee the existence of global solutions to differential equations, whose differential operator is a deformation of the Koszul-Tate differential δ, see Theorems 1 and 2. As usual when dealing with the Poincaré Lemma, it is useful to consider the inner contraction which is dual to δ with respect to a y ↔ c permutation. The commutator is a Euler vector field that counts the number of c's and y's. The homotopy operator δ −1 is defined as ∀ a ∈ A mn : δ −1 a := 1 m+n (δ * a) for (m, n) = (0, 0), 0 for (m, n) = (0, 0), and extended by linearity to the whole algebra A. The homotopy operator δ −1 is not a firstorder differential operator, in contrast to δ * . One easily obtains the following version of the Poincaré lemma.

Lemma 1 (Poincaré lemma).
There is only non-trivial δ-cohomology in the 00-sector with neither c's nor y's. A more refined statement is the following: For all δ-closed elements a ∈ A with no 00-part, there exists a unique δ * -closed element b ∈ A with no 00-part, such that a = (δb). Or equivalently, in symbols: The unique element b is given by the homotopy operator δ −1 a.

The Riemann curvature
See [12] and [32] for related discussions. The Riemann curvature R is defined as (half) the commutator of the ∇ connection (2.6), where it is implicitly understood in the second equality of equation (12.1) that ∇ i does not act on dx j . If ∇ i is supposed to act on dx j , one should include a torsion-term cf. equation (13.1). Note that the order of indices in the Riemann curvature tensor R n ijk is non-standard. This is to minimize appearances of Grassmann sign factors. Alternatively, the Riemann curvature tensor may be defined as are vector fields of even Grassmann-parity. The Riemann curvature tensor R ij n k reads in local coordinates It is sometimes useful to reorder the indices in the Riemann curvature tensors as For a symplectic connection ∇, we prefer to work with a (0, 4) Riemann tensor (as opposed to a (1, 3) tensor) by lowering the upper index with the symplectic metric (3.3). In terms of Christoffel symbols it is easiest to work with expression (12.2): In the second equality of equation (12.5) we have use the symplectic condition (3.4). If we use the symplectic condition (3.4) one more time on the first term in equation (12.5), we derive the following symmetry This symmetry becomes clearer if we instead start from expression (12.4) and define R ij,kn := R ijk mω mn = −(−1) ε n (ε i +ε j +ε k ) R n,ijk .
Then the symmetry (12.6) simply translates into a symmetry between the third and fourth index: We note that the torsion-free condition has not been used at all in this Section 12.
13 The curvature two-form R Let us now calculate the commutator of two ∇ i operators using the realization (4.3) of the covariant derivative: Using realization (4.3) the curvature two-form reads is a Hamiltonian for the curvature two-form R. In this formalism the connection ∇ i is supposed to act on c k ≡ dx k , so the torsion-term (12.3) is included in the curvature two-form (13.1). In the fourth equality of equation (13.1) we use the torsion-free condition (3.6) and the first Bianchi identity for a torsion-free connection 0 = i,j,k cycl.
to ensure that the xand c-derivative term in equation (13.1) vanishes. As we shall see below, it is vital that there is no xand c-derivative term in the R curvature (13.1). This is the main reason why the connection ∇ is assumed to be torsion-free. (See also the comment after equation (16.1).) The first Bianchi identity (13.2) also implies that the Hamiltonian curvature two-form R is δ-closed: Similarly, the second Bianchi identity for a torsion-free connection implies that the ∇ operator preserves the Hamiltonian curvature two-form R: 14 Higher-order D connection with the help of a so-called deformation one-form r = ⊕ n≥0 r (n) ∈ A 1• , deg(r (n) ) = n.
As we soon shall see, it is better to think of D as a deformation of (minus) the Koszul-Tate differential δ rather than the connection ∇. The word "higher-order" refers to that D is not necessarily a linear derivation of the tensor algebra (A, +, ⊗), cf. equation (4.5). However, it is a linear derivation of the • algebra (A, +, •), similar to equations (7.2) and (10.3). The ∇ and D connections may be expanded in Fedosov degree: Note that the connection D does not depend on r (0) . Also note that it will be necessary to assume that the (1)-sector vanishes to ensure that (minus) the Koszul-Tate differential δ is the sole leading term in the D expansion (14.1).

The R D curvature
The curvature two-form R D for the D connection is where the Hamiltonian R D is In the third and fourth equality of equation (15.1) we use that ∇ respects the • product (7.2) and the Jacobi identity for the • product, respectively. In the second equality of equation (

Flat/nilpotent D connection
The next main principle of Fedosov quantization is to choose the D connection to be flat, or equivalently, nilpotent: In other words, the odd D operator is a deformation of the odd Koszul-Tate differential δ, such that the nilpotency is preserved, cf. equations (10.2) and (16.1). (This setup is similar to the construction of an odd nilpotent BRST operator [29,7,27,1,2,25,35].) Since we want to achieve the nilpotency (16.1), it now becomes clear why it was so important that the xand c-derivative terms in the R curvature two-form (13.1) vanish. This is because there are no other xand c-derivatives in the R D curvature two-form (15.1) to cancel them. (All other derivatives in equation (15.1) are y-derivatives.) This crucial point is the main reason that the ∇ connection is assumed to be torsion-free. (References [38,37] consider a hybrid model where torsion is allowed in the y-sector but not in the c-sector to avoid the xand c-derivatives. Note, however, that they restrict the possible torsion by imposing both the independent conditions (∇ iωjk ) = 0 and (dω) = 0 at the same time.) For the curvature two-form R D to be zero, it is enough to let the Hamiltonian curvature two-form R D be Abelian, i.e., to let it belong to the center of the algebra (A, +, •). In other words, there should exist an Abelian two-form C ∈ A 20 , such that Here ω is just the symplectic two-form ( Then there exists a unique one-form valued r-solution to Fedosov's equation (16.3) such that r is δ * -closed, (δ * r) = 0, (16.5) and satisfies the boundary condition for the (1)-sector. As a consequence, it turns out that the first three sectors r (0) , r (1) and r (2) are identically zero.
Proof . Let us split the Abelian condition (16.3) in Fedosov degree: for n ≥ 0. (16.8) In equation (16.8) the Hamiltonian curvature two-form R and the closed Abelian two-form C have also been expanded in Fedosov degree A priori it is known that the (0)-sector r (0) = c i η i (x) is a yandh-independent one-form. The equation (16.7) is therefore automatically satisfied. It follows from equation (16.5) that Therefore the (0)-sector vanishes identically. Equation (16.8) with n = 0 is automatically satisfied because of the two boundary conditions (16.4) and (16.6). Putting n = 1 in equation (16.8) yields δr (2) = 0. Hence the (2)-sector r (2) is a one-form that is both δ-closed and δ * -closed, and therefore it must be identically zero as well: Since r (0) , r (1) and r (2) are zero, the right-hand side expression for δr (n+1) in equation (16.8) will only depend on previous entries r (≤n) . Hence equation (16.8) is a recursive relation. The consistency relation for the Abelian condition (16.3) is that the right-hand side should be δclosed. This is indeed the case: In the second equality of equation (16.9) we have used that δ respects the • product. In the third equality we have used that (δR) = 0, that [δ, ∇] = 0, and that δ is nilpotent. In the fifth (=last) equality we have used that (∇R) = 0, that (dC) = 0, that ∇ 2 = {R, ·}, that ∇ respects the • product, and the Jacobi identity for the • product. We now prove by induction on the Fedosov degree (n) that there exists a unique solution r (n+1) to equation (16.8) if there exists a unique solution for all the previous entries r (≤n) . This is essentially a consequence of the Poincaré lemma 1. Uniqueness: The difference between two solutions r ′ (n+1) and r ′′ (n+1) must satisfy the homogeneous version δ(∆r (n+1) ) = 0 of equation (16.8), i.e., with no right-hand side. Hence the difference ∆r (n+1) is a one-form that is both δ-closed and δ * -closed, and therefore it must be identically zero. Existence: Define where RHS (n) is the two-form valued right-hand side of equation (16.8). This clearly defines a δ * -closed one-form r (n+1) . To check equation (16.8), it is enough to check that the two-form RHS (n) is δ-closed. But this follows by linearity from the consistency relation (16.9), because ∇ and • both carry positive Fedosov degree, and the first three r-sectors vanish to cancel the negative Fedosov degree coming from theh −1 -factor, so that only previous entries r (≤n) can participate to the (n)-sector.
We emphasize that the unique deformation one-form r from Fedosov's 1st theorem is globally well-defined, since it basically appeared from inverting the Koszul-Tate δ differential, cf. Section 11. Normally, one would choose a trivial Abelian two-form C ≡ 0. Also note that the two-form R + C is the lone source term that forces r to be non-trivial. We list here the first few unique r-terms: Similarly, the first few terms in the D expansion read

Horizontal sections
Fedosov's 1st theorem establishes the existence of a globally well-defined, unique, flat/nilpotent D connection. Since this higher-order connection D is flat, it is possible to solve the horizontal condition (Da) = 0 locally for a zero-form valued section a ∈ A 0• . (In other words, the flatness relation (16.1) is the local consistency relation for the horizontal condition.) As we shall see below there is no obstruction in patching together local horizontal sections a into global horizontal sections, basically because D is a deformation of (minus) the Koszul-Tate δ-differential. Proof . First note that that a zero-form a ∈ A 0• is automatically δ * -closed: Note that the right-hand side expression for δa (n+1) only depends on previous entries a (≤n) , because r (0) , r (1) and r (2) are zero. Hence equation (17.3) is a recursive relation. The consistency relation for the horizontal condition (17.2) is that the right-hand side should be δ-closed. This is indeed the case: In the second equality of equation (17.4) we have used that δ respects the • product. In the third equality we have used that [δ, ∇] = 0. In the fourth (=last) equality we have used that ∇ 2 = {R, ·}, that ∇ respects the • product, and the Jacobi identity for the • product. We now prove by induction on the Fedosov degree (n) that there exists a unique solution a (n+1) to equation (17.3) if there exists a unique solution for all the previous entries a (≤n) . This is essentially a consequence of the Poincaré lemma 1. Uniqueness: The difference ∆a (n+1) := a ′ (n+1) − a ′′ (n+1) between two solutions a ′ (n+1) and a ′′ (n+1) must satisfy the homogeneous versions δ(∆a (n+1) ) = 0 and ∆a (n+1)00 = 0 of the horizontal condition (17.3) and boundary condition (17.1), i.e., with no right-hand sides. Hence the difference ∆a (n+1) is both δ-closed, δ * -closed and with no 00-sector. Therefore it must be identically zero. Existence: Define for n ≥ 0, where RHS (n) is the one-form valued right-hand side of equation (17.3). This clearly defines a zero-form a (n+1) that satisfies the boundary condition (17.1). To check equation (17.3), it is enough to check that the one-form RHS (n) is δ-closed. But this follows by linearity from the consistency relation (17.4), because ∇ and • both carry positive Fedosov degree, and the first three r-sectors vanish to cancel the negative Fedosov degree coming from theh −1 -factor, so that only previous entries a (≤n) can participate to the (n)-sector.
We list here the solution to the unique first-order correction a (1) : ∇ a (0) ) = y i → ∂ ℓ ∂x i f (0) , which we'll use in the next Section 18.
18 The * product That the vector space (W D , +, •) is an subalgebra, i.e., closed with respect to the • product, follows basically because the connection ∇ respects the • product, cf. equation (7.2). The * product in diagram (18.1) is by definition induced from the • product as f * g := Q −1 (Q(f ) • Q(g)) .
This * product obviously inherits associativity from the • product, and the Q map (18.1) is obviously an algebra isomorphism. Moreover, the inverse map Therefore lim h→0 f * g = f (0) g (0) , It is also easy to check condition (2.2).
Note added. The author has kindly been informed by a referee of the existence of the Ph.D. Thesis [20], which the author has been unable to obtain, and which considers Fedosov quantization on supermanifolds.