Twist-Deformed Supersymmetries in Non-Anticommutative Superspaces

We consider a quantum group interpretation of the non-anticommutative deformations in Euclidean supersymmetric theories. Twist deformations in the corresponding superspaces and Lie superalgebras are constructed in terms of the left supersymmetry generators. Non-anticommutative $\star$-products of superfields are covariant objects in the twist-deformed supersymmetries, and this covariance guarantees the manifest invariance of superfield actions using $\star$-products.


Introduction
The most popular classical noncommutative field theory (see, e.g. review [1]) can be realized on ordinary smooth field functions f (x), g(x) on R 4 using the following pseudolocal representation of the ⋆-product : f ⋆ g = f e P g = f g + i 2 ϑ mn ∂ m f ∂ n g − 1 8 ϑ mn ϑ rs ∂ m ∂ r f ∂ n ∂ s g + . . . , f P g = i 2 ϑ mn ∂ m f ∂ n g. (1.1) where x m are the coordinates of R 4 , ∂ m = ∂/∂x m , and ϑ mn are some constants ( m, n = 1, 2, 3, 4). All products of the functions and their derivatives in the right-hand side are commutative. It is evident that nonlinear interactions in these noncommutative (nonlocal) field theories are not invariant with respect to the standard Lorentz transformations of local fields.
The quantum group structures in this noncommutative algebra of functions were found and analyzed in [2]- [5]. The basic point of this interpretation is connected with the twist operator acting on tensor products of functions F = exp(P), P = i 2 ϑ mn P m ⊗ P n (1.2) where P m f = ∂ m f . The strict definition of the noncommutative product is where µ is the multiplication map in the commutative algebra. Thus, this twist operator is the quantum-group analog of the pseudolocal operator exp(P ) (1.1).
Let us consider generators of the Poincaré group P m and M mn . By definition, the twist-deformed Poincaré group U t (P m , M mn ) has the undeformed Lie algebra of generators; however, its coproduct is deformed ∆ t (P m ) = P m ⊗ 1 + 1 ⊗ P m , ∆ t (M mn ) = exp(−P)(M mn ⊗ 1 + 1 ⊗ M mn ) exp(P). (1.4) The exact constructions of maps between differential operators on commutative and noncommutative algebras of functions were formulated in recent papers of the Munchen group [4]. It was shown that the ⋆-product (1.3) transforms covariantly in U t (P m , M mn ), but the Leibniz rule for deformed transformations is changed according to Eq.(1.4). 4D-space integrals of the covariant ⋆-products of fields are invariant with respect to U t (P m , M mn ). The quadratic free interactions possess also the standard Poincaré invariance.
We shall consider the quantum group interpretation of the non-anticommutative deformations in the Euclidean supersymmetric theories [6]- [8] 1 . The basic ⋆-product of these models is realized on the standard local superfields, and supersymmetry generators can be presented as the 1-st order differential operators on the undeformed superspace. The left-handed Grassmann coordinates of these superspaces do not anticommute with respect to the ⋆-product, but the basic chiral bosonic coordinates commute with all superspace coordinates. The non-anticommutative superspace is defined exactly as the ⋆-product algebra on ordinary functions of the superspace coordinates. The twist elements for the nilpotent deformations can be constructed in terms of the left supersymmetry generators. Section 2 is devoted to the analysis of the twist deformation of the N=( 1 2 , 1 2 ) supersymmetry [10]. We derive the unusual Leibniz rules for the deformed transformations on the products of superfields or the products of component fields. Twist deformation of the Euclidean N=(1, 1) supersymmetry in the chiral and harmonic superspaces is considered in Sect. 3. The corresponding nilpotent operator P is analogous to the basic bi-differential operator of the N=(1, 1) deformations in Refs. [7,8].
The twist interpretation allow us to understand correctly transformation properties of ⋆-products of superfields by analogy with the deformed transformations of the ⋆-products of fields (1.1) in the noncommutative field theory [4]. At the level of the pseudolocal superfield formalism, the t-supersymmetry is equivalent to the ⋆-covariance principle for the noncommutative algebra of superfields which means the similarity of transformations of local superfields and their ⋆-products. The covariance principle allow us to obtain a simplified field-theoretical derivation of the unusual Leibniz rules for the deformed supersymmetry transformations on ⋆-products of superfields. Known deformed supersymmetric actions in the non-anticommutative superspaces are manifestly invariant with respect to the corresponding twist-deformed supersymmetry, and this invariance explains naturally all selection rules of these theories which could seem formal earlier.
We shall use the notation S(4|2, 2) or C(4|2, 0) for the supercommutative algebras of general or chiral superfields. The bilinear multiplication map µ connects the tensor product of superfields with the local supercommutative product in S(4|2, 2) (2.5) The standard coproduct map is defined on the generators of SUSY( 1 2 , 1 2 ) (2.4) It determines the action of these generators on the tensor product of superfields and yields the standard Leibniz rule for supersymmetry transformations on the local product of superfields δ(AB) = (δA)B + AδB.
The non-anticommutative deformationẑ = (y m ,θ α ,θα) of the coordinates of the Euclidean N=( 1 2 , 1 2 ) superspace was considered in [6]. The basic operator relation of the non-anticommutative superspace is where C αβ are some constants. The operator superfieldsÂ(y,θ,θ) andB(y,θ,θ) with the antisymmetric ordering of theθ α decomposition contain the highest terms ∼ ε αβθ α ⋆θ β . In the pseudolocal representation, we consider the usual superfields A(z) and B(z) as the supercommutative images of these operator superfields. The corresponding ⋆-product of superfields A(z) and B(z) is defined via the generators of the left N=( 1 2 , 0) supersymmetry where p(A) is the Z 2 grading, and P is the nilpotent bi-differential operator. The deformed algebras S ⋆ (4|2, 2) and C ⋆ (4|2, 0) use this noncommutative product for general or chiral superfields, respectively.
The twist operator in this supersymmetry was introduced in [10] (see also discussions in [11,14]) The bilinear map µ ⋆ in S ⋆ (4|2, 2) can be defined via this twist operator so F is the quantum group analog of the pseudolocal operator e P (2.7).
By analogy with the map between differential operators on commutative and noncommutative algebras of functions [4], one can easily define the corresponding differential operatorX D on S ⋆ (4|2, 2) for any differential operator D on the supercommutative algebra. In the case of the 1-st order operator D 1 = ξ M (z)∂ M , the imageX D 1 contains, in general, terms with higher derivatives on S ⋆ (4|2, 2) where p(D 1 ) is the Z 2 grading of D 1 . For instance, the deformed images of generatorsQα and L β a (2.3) are the following second-order differential operators on S ⋆ (4|2, 2): The Lie superalgebra of the deformed generators with hats is isomorphic to the Lie superalgebra of the undeformed supersymmetry generators (2.3).
Acting by the composition of µ ⋆ and coproduct ∆ t (Qǭ) on the tensor product of superfields, one can obtain the following relation: (2.14) The last formula can be derived from the pseudolocal definition of A ⋆ B (2. Note that Eq.(2.14) can be treated as the deformed Leibniz rule forδǭ.
The deformed transformations act noncovariantly on the supercommutative product of superfields AB. For instance, it is not difficult to consider the following transformations of the ordinary product of the even chiral superfields: The first terms coincide with the transformations of the undeformed supersymmetry. Using the θ-decomposition of these superfield formulas one can obtain the deformed transformations of the products of component fields, for instance, Let us consider two even chiral superfields φ 1 and φ 2 in the chiral basis The θ-decomposition of the ⋆-product of two chiral superfields depends on these components and constants C αβ These relations can be treated as a deformed tensor calculus for chiral component multiplets. The t-supersymmetry transformations of the composite components (2.20) are completely analogous to the transformations of the basic components a i , ψ αi , f î The non-anticommutative deformation of the Euclidean model for an arbitrary number of the chiral and antichiral superfields φ a andφ a is based on the superfield action S ⋆ (φ a ,φ a ) [6]. Each term of the ⋆-polynomial decomposition of this action is separately invariant with respect to SUSY t ( 1 2 , 1 2 ), while the quadratic terms like d 8 z φ a ⋆φ a = d 8 z φ aφa possess also the undeformed supersymmetry.
It is convenient to use the following differential representation of the SUSY(1,1) generators on S(4, 2|4, 4): where L β α (y) and Rβα(y) are defined above (2.3), and partial harmonic derivatives act as follows ∂ ∓l u ± k = δ l k . For our purposes, it is convenient to consider the following combinations of SUSY(1,1) generators and corresponding parameters: The N=(1, 1) twist operator F = exp (P) contains the nilpotent operator where C αβ kl are some constants. The non-anticommutative product in the corresponding deformed algebra S ⋆ (4, 2|4, 4) can be defined by equivalent formulas where µ and µ ⋆ are product maps for S(4, 2|4, 4) and S ⋆ (4, 2|4, 4) and P is the basic operator from [7,8] The non-anticommutative algebras of the N=(1, 1) chiral or G-analytic superfields can be defined analogously.
The ⋆-products of arbitrary N=(1, 1) superfields preserve covariance with respect to all deformed transformations of SUSY t (1,1) (3.9) The deformed Leibniz rules can be derived directly from these covariant relations.
The twisted supersymmetry acts noncovariantly on the supercommutative product of superfields, for instance, It is easy to define the t-deformed transformations on the products of the N=(1, 1) component fields using the corresponding Grassmann decompositions.
In the special case of the singlet deformation [12,13], the twist operator is defined by the parameter I and the SU(2)×SU(2) L invariant constant tensor The P s -twist deformation vanishes for SU (2) and SU(2) L transformations.
The Leibniz rules for differential operators D = (∂ m , D k α ,Dα k , ∂/∂u ± k ) are standard for the general Q-deformation (3.12) The ⋆-product preserves differential constraints of chirality, antichirality and Grassmann analyticity [7,8]. All superfield actions using ⋆-products in the non-anticommutative N = (1, 1) harmonic superspace [12,13] are invariant with respect to the quantum group SUSY t (1,1), and this invariance is a natural basic principle of these deformed theories.
Free quadratic parts of these actions possess also the undeformed N = (1, 1) supersymmetry. The simple examples of the superfield-density terms for the analytic hypermultiplet q + ,q + and the U(1) gauge potential V ++ in the deformed theory arẽ q + ⋆ (D ++ q + + [V ++ , q + ] ⋆ ) + λq + ⋆ q + ⋆q + ⋆q + . (3.13) The t-supersymmetry transformations of any term L +4 ⋆ in this density arê so the analytic-superspace integrals of these variations vanish. We hope that the manifest SUSY t (1,1) covariance could help to prove the nonrenormalization theorems in t-deformed harmonic-superfield theories by analogy with the corresponding undeformed theories.

Conclusions
We analyzed the twist deformations of the Euclidean N=( 1 2 , 1 2 ) and N=(1, 1) supersymmetries. By analogy with the formalism of the deformed Minkowski space [4], we construct explicitly the map between differential operators on ordinary and deformed superspaces (2.10). This map connects the standard representation for the supersymmetry generators with the corresponding operator representation on the deformed superspace. It is shown that the noncommutative ⋆-products of primary superfields transform covariantly in these t-deformed supersymmetries. This covariance is a basic principle of the superfield formalism of the deformed theories. The Grassmann-coordinate decompositions of the ⋆-product superfields define the deformed tensor calculus for the components of primary superfields. The ordinary supercommutative products of primary superfields or component fields are not covariant with respect to the deformed supersymmetries.
Any polynomial terms of the superfield actions in the non-anticommutative N=( 1 2 , 1 2 ) [6] and N=(1, 1) [7,8] superspaces are manifestly invariant with respect to the corresponding t-deformed supersymmetries. The bilinear free parts of these actions are also invariant under the standard supersymmetry transformations. The deformation constants of the non-anticommutative superfield theories break some undeformed (super)symmetries, however, these parameters can be treated as 'coupling constants' compatible with the deformed supersymmetries. We hope that t-supersymmetries would help to analyze nonrenormalization theorems using the superfield effective actions in these theories.
I am grateful to P.P. Kulish for stimulating discussions of quantum-group symmetries in the noncommutative field theory. This work was partially supported by DFG grant 436 RUS 113/669-2 , by RFBR grants 03-02-17440 and 04-02-04002, by NATO grant PST.GLG.980302 and by grants of the Heisenberg-Landau and Votruba-Blokhintsev programs.