{'fraction_non_alphanumeric': 0.033493029479045276, 'fraction_numerical': 0.008632224092537442, 'mean_word_length': 4.451764705882353, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 11, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Noncommutative associative superproduct for general supersymplectic forms

We define a noncommutative and nonanticommutative associative product for general supersymplectic forms, allowing the explicit treatment of non(anti)commutative field theories from general nonconstant string backgrounds like a graviphoton field. We propose a generalization of deformation quantization a la Fedosov to superspace, which considers noncommutativity in the tangent bundle instead of base space, by defining the Weyl super product of elements of Weyl super algebra bundles. Super Poincare symmetry is not broken and chirality seems not to be compromised in our formulation. We show that, for a particular case, the projection of the Weyl super product to the base space gives rise the Moyal product for non(anti)commutative theories.


I. INTRODUCTION
During the last string theory revolutions we have learned that an undeniable relationship between strings and noncommutative field theories exists. As exemplified by the study of D-branes in presence of a finite constant B-field [1, 2], it seems that geometry can be noncommutatively deformed whenever stringy effects are relevant. In particular cases like that of strings in a constant graviphoton background, the deformation produced lies in the Graßmann sector of superspace whose undeformed coordinates are characterised by anticommutators, giving raise to what has been called non(anti)commutative field theories [3].
Most studies introduce a Moyal product to deform the (super)field algebra of known theories, as this is precisely the kind of geometry that results from string constant backgrounds.
The resulting theories usually break Lorentz symmetry and, in the non(anti)commutative case, also either supersymmetry or chirality. The Moyal product construction is very satisfactory when dealing with deformations of Euclidean (super)spaces, but breaks down in curved geometry where it should be replaced by a more general object like the Kontsevich product. As an example, the deformed geometry of D-brane worldvolumes in curved backgrounds is given to third order in the deformation parameter precisely by such product [12], which unfortunately can only be constructed order by order in a series expansion.
On the other hand, ordinary Moyal product-noncommutative field theories suffer from bad causal behaviour when time is treated as a noncommuting coordinate.
More recently, in view of the limitations inherent to such methods, an alternative noncommutative deformation of gauge theories based on the notion of the Weyl bundle has been proposed [9,13]. In these works, authors use the powerful machinery developed by Fedosov [14,15] which consists in a Weyl deformation of a symplectic tangent space, where the symplectic structure plays the role of the deformation parameter. The advantage of using this method is twofold: it allows the description of physical situations where the background field is not constant, and in addition spacetime global symmetries remain unbroken by the deformation itself since it affects the tangent and not the base space. This approach to deformation quantization has been used extensively to study string theory D-branes in non-constant B-fields and more general curved backgrounds [13], supermembrane models [9], the fuzzy sphere [16], the canonical formulation of noncommuative gauge theories [17], and BRST symmetry in quantum field theories [18].
One may naturally ask if its possible to generalize Fedosov's ideas to describe more general physical setups like non(anti)commutative field theories associated to non-constant graviphoton backgrounds. At first sight one could consider an atlas where the problem may be stated in terms of constant fields on each open set, as the Darboux theorem is only valid in the neighbourhood of a point and not globally, but we must then consider highly nontrivial matching conditions on the overlapping of the sets. It is therfore more convenient to directly develop a non(anti)commutative product for general fields extending the known one for constant fields. Also, from a quantum field theoretical point of view, constructing a mechanism that deforms superspace but preserves super Poincaré symmetry (or its Euclidean counterpart) is relevant in its own right. Our work has precisely this goal.
The deformation quantization of certain Poisson brackets in the context of Batchelor supermanifolds using Fedosov's approach was first done by M. Bordemann, [19,20]. In the present paper we are mostly interested in field theories involving fermionic fields, as we explicitly discuss in section V. Although the Batchelor approach is an interesting one from a mathematical and physical point of view it does not allow the introduction of fermionic fields, it is for this reason that the Batchelor definition of supermanifolds is not followed in supersymmetric field theory.
In this paper we extend Fedosov's deformation quantization to N = 1 superspace. We take a symplectic supermanifold and ordinary superspace as the tangent and base space respectively, and propose a new associative Weyl product which is non(anti)commutative, meaning it is noncommutative for even objects that naturally fulfil commutation relations, and nonanticommutative for odd objects naturally satisfying anticommutation relations.
Such product endows the space of superfields with a superalgebra structure. A key feature of our formulation is the preservation of super Poincaré symmetry, a fact derived from a deformation operator acting exclusively on tangent superspace variables. The super Poincaré algebra acting on base superspace is therefore totally transparent to such deformation. We prove that in certain cases, the projection of the Weyl (super)product on the base supermanifold, results in the nonanticommutative product associated to nilpotent Q-and D-deformations such as those studied in [4,6,7,8]. And further explore more general situ-ations where the nonanticommuting parameter is actually a function of the base supermanifold variables. This last subject is not even formulated in the literature as a supersymmetric extension using Kontsevich methods. [21].

II. WEYL ALGEBRA BUNDLE
In this section we review Weyl algebra bundle theory basics [15]. We start with a symplectic manifold (M, ω) of dimension 2n, where the two-form ω defines a symplectic structure The formal Weyl algebra W x is an associative algebra of formal series over which stems from the symplectic space T x (M). Its elements are defined by with being the formal parameter, y ∈ T x (M) with coordinates (y 1 , . . . , y 2n ), and α = α 1 , . . . , α 2n a multi-index such that y α = (y 1 ) α 1 · · · (y 2n ) α 2n . It is possible to introduce an ordering prescription for the terms if we define the degree of the variables to be deg y i = 1, deg = 2. The terms in the series can then be arranged by increasing degrees 2k + |α|.
A more friendly notation is a(y, ) = k,p≥0 k a k,µ 1 ...µp y µ 1 . . . y µp , where it is understood that p = 0 terms correspond to the sum k k a k .
The product of elements in this algebra is given by the Weyl product rule which is associative and independent of the particular choice of basis on the tangent space T x (M). The algebras W x constitute the fibers of the Weyl bundle W of formal Weyl algebras, which is defined through the union of all fibers Differential q-forms on M with values in W are sections of the bundle W ⊗ Ω q defined by it formal series where the coefficients a k,i 1 ...1p,j 1 ...jq (x) are covariant tensors fields symmetric in i 1 . . . i p and antisymmetric in j 1 . . . j q depending on a point in M. They constitute an algebra C ∞ (W ⊗ Ω * ) under the usual Weyl product and the exterior product of forms, with unit identically equal to 1. The total degree deg a = 2k+p corresponding to the variables and y i establishes a filtration within this algebra W ⊗ Ω * ⊃ W 1 ⊗ Ω * ⊃ W 2 ⊗ Ω * ⊃ · · · . For q = 0 we have sections simply as formal series with a k,α (x) covariant symmetric tensor fields on M. The bracket of two forms a ∈ W ⊗ Ω q 1 A form a will be called central if for any b ∈ C ∞ (W ⊗ Ω * ) the commutator (7) vanishes.
The center of C ∞ (W ⊗ Ω * ) consists exclusively of forms not containing y, which suggests a projection into the center by setting y to zero. For a ∈ C ∞ (W ) we will denote such projection by It is customary to define operators that rise and lower the rank of differential forms: , and respectively lower and rise the degree of each term in the formal series. These fulfil the following properties where a pq denotes the homogeneous part of Weyl degree p and cohomological degree q of a ∈ W ⊗ Ω q . Note that the last relation is similar to the Hodge-Rahm decomposition, yet δ is a pure algebraic operator on M as we can see from its realization in local coordinates To define a connection on the Weyl bundle one recalls there exists a symplectic connection on any symplectic manifold that is torsion free and preserves the symplectic structure Using Darboux coordinates one can check that the connection symbols Γ ijk are completely symmetric Γ [ijk] = 0. The symplectic connection can be used to define a connection on the bundle W ⊗ Ω * by its action on elements of in Darboux coordinates it reduces to where the connection symbols are Θ = 1 2 Θ ijk y i y j dx k . The properties of the symplectic Weyl bundle connection are More general covariant derivatives D on the bundle may be considered by adding a one form γ globally defined on M with values in W , With this at hand the Weyl curvature may be defined as where R is the curvature associated to the connection ∇. The Weyl curvature satisfies the Bianchi identity In general, transitions on the bundle T (M) will induce transitions on W ⊗ Ω * . The infinitesimal gauge transformations on elements of C ∞ (W ⊗ Ω * ) are expressed as automorphisms given by with infinitesimal λ ∈ C ∞ (W ⊗ Ω * ). The corresponding gauge transformations for the connections D are consequently, There exists a relation between the center and an Abelian subalgebra of the Weyl bundle that maps the Weyl product to the Moyal product through the projection (8). To establish such relation one first defines Abelian connections D A as connections whose Weyl curvature An example of such connections may be expressed as with deg r 3. The Abelian subalgebra is the set There is a one-to-one correspondence between the projections defined above (8) and elements of W A . It is clear that for every a ∈ W A there is a projection projection to the center σ(a) = a 00 .
Now, given a 00 there is a unique element a ∈ W A with such projection. By means of this construction, from the Weyl product of two elements a, b of W A we obtain a globally defined star product on M which coincides with the Moyal product when ω ij is constant and the symplectic connection is zero The main purpose of this work is the generalization of the Weyl bundle construction to superspace in order to study deformations of superfield algebras, which can then be used to extend supersymmetric field theories expressed in the superfield formalism. Following closely the known construction of Weyl bundle, we will need a symplectic structure ω defined on a supermanifold M. In the following section we give the first steps toward such extension of deformation quantization, by defining this concepts and introducing a new Weyl product of superfields.
Like usual manifolds, that are locally isomorphic to Ê p , a supermanifold M of dimension p|q consists of a topological space M together with a sheaf of Ê-algebras called the structure sheaf O M , such that M is locally isomorphic to Ê p|q .
The differential geometry involving supermanifolds is quite similar to the classical one, having similar definitions and properties for super vector bundles, connections and actions of super Lie groups. The most significant difference arises from the existence of differential forms with arbitrary high degree, which hinders a straightforward generalization of integration. Nevertheless, after introducing the Berezinian, integration over supermanifolds can be easily defined.
A super vector bundle of rank p|q over M is a sheaf of locally free O M -supermodules V of dimension p|q. It can be also defined as a fibre bundle over M with typical fiber Ê p|q , having as structure group the super general linear group GL(p|q). The first example of super vector bundle is the tangent bundle of a supermanifold, which can be constructed from the derivations of the superalgebra O M satisfying where p defines the 2 -grading of objects (0 for even, 1 for odd).
The set of vector fields Vect(M) is a super Lie algebra.
The cotangent bundle Ω 1 M of M is the dual of T (M), whose basis we denote by dz A = (dx m , dθâ). One can interpret the duality pairing as an inner product ·, · : In the local basis, such product can be obtained by linearity from The wedge product of 1-forms is defined through where p(A) indicates the parity of an object with index A. One can linearly extend this definition to the exterior product ∧ : Ω k M × Ω l M −→ Ω k+l M of k-forms with l-forms, providing the set of all forms with the structure of a superalgebra which is called the exterior superalgebra Ω * M ≡ ∧ * Ω 1 M . Note that in contrast with the wedge product of forms, supercoordinates and superfield components conform supercommutative superalgebras like as the concepts of cohomological degree of a form, and that of Graßmann parity are independent.
extends uniquely to the exterior derivative d = dz A ∂/∂z A on the exterior superalgebra and satisfies From here onwards we will take p = 2n even and q = 4N representing the physically relevant set of odd variables, thus studying only the supermanifold Ê 2n|4N . In 4|4 dimensions, for example we have A = (m, α,α)., i.e. z A = (x m , θ α ,θα), with α = 1, 2,α =1,2. The notation we use is consistent with that of Wess and Bagger [25] for differential forms in superspace.
A symplectic supermanifold is a 2n|4N dimensional supermanifold M endowed with a closed, non-degenerated two-form ω called the symplectic superform. In coordinates, The Darboux theorem [27,28,29] states that there exist local coordinates on the supermanifold where the symplectic superform ω has the following block shape with Graßmann-even components having definite symmetry under index exchange ω mn = −ω nm and ωαβ = ωβα. The symplectic superform can be used to rise and lower indices.
Symplectic supermanifolds always admit a torsionless affine connection preserving the symplectic superform, called (super)symplectic connection. We will denote the connection by ∇, and define its coefficients Γ A BC by its action on the tangent base Such coefficients can be found from the Leibnitz property of the covariant derivative The conditions for the vanishing of the torsion in superspace can be derived from its very definition in terms of vector fields Y 1 and Y 2 resulting in the following components A torsion free affine connection associated to our symplectic superform is defined by the equation In Darboux coordinates, the resulting conditions on the connection coefficients (36) are A Weyl superalgebra W z defined locally on the tangent superspace T z (M) is an associative superalgebra over , its elements being formal series on the parameter Taking advantage of the nilpotent nature of odd coordinates, we can collect the terms in the formal series into a superfield whose coefficients are superfields of tangent space odd coordinates.
A Weyl superalgebra can be then defined by means of the following Weyl product, Since this product should preserve the degree, we propose deg(η) = 1. Similar structures that appeared in the context of anti-Poisson brackets on Graßmann algebras [26] were developed further into a Fedosov-type procedure on the dual Graßmann algebra bundle of a given vector bundle [19,20]. Note that the product (48) differs from such anti-Moyal products as it acts on the tangent superspace T (M), thus protecting super Poincaré symmetry on M from breaking. In Darboux coordinates [27,28] the Poisson structure ω M N becomes block diagonal allowing a separation of the sum inside the exponential into a purely bosonic and a purely fermionic part. Each term in turn, as a bilinear operator, is even and commutes with the others. For instance which is also true for any other combination of indices (note that a term with mixed indices ω ma is absent). This allow us to split the exponential and expand the Weyl superproduct as the nested application of two products: The usual Weyl rule for standard spacetime, and a new one for the fermionic coordinates Though both this products are formal series of the exponential in the fermionic case the series is truncated due to nilpotency of odd derivations. Nesting the products • and can be understood as the replacement of the standard product in the formal series of • with the product rule of , that is a result of nesting the products already known to be associative. As the defined product is covariant, a completely general proof of associativity can be worked out in Darboux coordinates. This can be verified directly by expanding the product f (g h) for the fermionic part exclusively. One can see why this work by looking at the particular case with only two fermionic coordinates Y A = (y m , η 1 , η 2 ), which has a particular importance in the description of the Weyl bundle over chiral superspace having precisely such coordinates.
The Weyl rule in this setup will looks like where we use the standard spinor index notation for the odd variables α = 1, 2 and the usual shorthand ∂ α = ∂ ∂η α to denote derivation respect to Y α = η α (with its straightforward extension ∂ αβ = ∂ α ∂ β ). Note that the series is truncated to second order, which facilitates direct verification of the associativity A product involving 4N odd variables, as required by more general N-supersymmetric theories, is obtained by repeatedly nesting products as (55) above, resulting in an associative product with a truncated expansion of order 4N. If we include the standard • product in Darboux coordinates one also concludes that this product is associative in general, from the covariant character of ⊚.
The Weyl superalgebra is easily extended to include q-superforms on M, that is, sections on W ⊗ Ω * expressed as These differential forms constitute an algebra with multiplication defined by the combined exterior superproduct of forms given in §III A together with the Weyl product acting on tangent superspace variables.
The commutator of 2 superforms f ∈ W ⊗ Ω q 1 and g ∈ W ⊗ Ω q 2 is A central superform f is such that for any g ∈ W ⊗ Ω * , [f, g] = 0. In (58) the wedge exterior product ∧ acting on forms is understood.

A. Abelian subalgebra and deformations
In the purely bosonic case the Weyl product is linked to deformations on the base space through a correspondence between C ∞ (M) and the Abelian subalgebra that, in Darboux coordinates, results in the well known Moyal product, as was proved in [15]. In this section we will show that this is also the case for the Weyl superproduct, which for the chiral case give us precisely the nonanticonmutative product extensively studied in this decade, see for example [3,6,7].
We start with the concept of Abelian connection. A connection D A on the Bundle W is called Abelian if for any section f ∈ W ⊗ Ω * Inspired by Fedosov construction, we will lift the supersymplectic connection defined in §III A to the Weyl superalgebra bundle D is a covariant derivative acting on C ∞ (W ⊗ Ω * ) and obeying the following properties Another important operators involved in the construction of the Abelian connection are δ : W p ⊗ Ω q → W p−1 ⊗ Ω q+1 that raises the rank of forms and acts as a sort of exterior superderivative, and δ * : W p ⊗ Ω q → W p+1 ⊗ Ω q−1 , that lowers the rank of forms and acts as a contraction operator. In coordinate basis they are realized as Using (31), (30) and and observing their action on each term in the formal series one can check their properties Here f pq is the coefficient with p times Y and q times dz, so that (p + q)δ −1 f pq = δ * f pq .
The proof follows the corresponding one in Fedosov [15, §5.2], where mainly the properties of δ, δ −1 and degree counting are used. The arguments are exactly the same. The curvature of the Abelian connection is We consider then r satisfying which ensures that D A is Abelian. Eq (69) has a unique solution belonging to C ∞ (W 3 ⊗ Ω 1 ) and satisfying δ −1 r = 0. The main point in Fedosov's proof, which can be extended to the supersymplectic case, is to show that has a unique solution belonging to C ∞ (W 3 ⊗ Ω 1 ). This follows from the iteration procedure defined by r n+1 = δ −1 R + δ −1 Dr n + i r 2 n , n = 0, 1, 2, . . .
From degree counting one obtains deg(r n − r n+1 ) ≥ n + 3 implying that the terms for any fixed degree of the sequence are the same for large enough n. There exists then a unique solution in C ∞ (W 3 ⊗ Ω 1 ). It is then straightforward to show that this solution satisfies (69) as δ −1 r = 0 holds by construction. Uniqueness arises from acting on (69) with δ −1 and using the above property of r.
With this at hand we propose an Abelian connection of the simplest form Once we have the Abelian connection, we use it to find the elements of the Abelian There is a one-to-one correspondence be- Let us consider a section a ∈ C ∞ (W), a = a(x, θ, y, η), and focus in the Abelian subalgebra D A a = 0. As in Fedosov [15] we take being a a 0-form for which δ −1 a = 0 holds, the Hodge-De Rham decomposition results in a = a| y=η=0 + δ −1 (D A + δ)a.
Considering bosonic coordinates only, it is known that the deformation induced in base space by a Weyl product in Darboux coordinates is precisely the Moyal product [9,15].
More succinctly, let σ : be the projection to the center that maps σ : f → f 00 and σ : g → g 00 , then σ(f • g) = f 00 ⋆ g 00 , where ⋆ stands for the Moyal product. In the next section we will obtain the nonanticommutative ⋆ product, proposed in [4,6,7] as a projection of the ⊚ superproduct to the base space using Darboux chiral coordinates in superspace.
The antisymmetric connection coefficients are defined up to an arbitrary completely antisymmetric tensor Θ αβγ .
An Abelian connection can then be constructed via The associated Weyl subalgebra is obtained following the same procedure as for (81), leading to an analogous correspondence between undeformed sections and Abelian ones, now including a more general covariant derivative. The projection of the Weyl product results in an expression which is formally the same as before (83) due to nilpotency of the Grassmann basis, which truncates the series to second order in the deformation, preventing the appearance of terms involving the curvature. This product does not correspond to the standard Moyal product, as it includes connection coefficients. Explicit expressions for more general non(anti)commutative products following the approach introduced in this work will be presented in a forthcoming paper, this includes the description of D-deformations.