aa r X i v : . [ h e p - t h ] S e p On Chiral Quantum Superspaces
D. Cervantes.
Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exicoCircuito Exterior M´exico D.F. 04510, M´exico email: [email protected]
R. Fioresi
Dipartimento di Matematica, Universit`a di BolognaPiazza di Porta S. Donato, 5. 40126 Bologna. Italy. e-mail: fi[email protected]
M. A. Lled´o
Departamento de F´ısica Te`orica, Universitat de Val`encia and IFIC (CSIC-UVEG)Fundaci´o General Universitat de Val`encia.
C/Dr. Moliner, 50, E-46100 Burjassot (Val`encia), Spain. e-mail: maria.lledo@ific.uv.es
Abstract
We give a quantum deformation of the chiral Minkowski superspacein 4 dimensions embedded as the big cell into the chiral conformalsuperspace. Both deformations are realized as quantum homogeneoussuperspaces: we deform the ring of regular functions together with acoaction of the corresponding quantum supergroup.
In his foundational work on supergeometry [22] Manin realized the Minkowskisuperspace as the big cell inside the flag supermanifold of 2 | | | | C | (the chiral conformal superspace). This isnot precisely the same supervariety that Manin considers in his work; theGrassmannian is a simpler one, but it also has a physical meaning. Ourchoice is motivated because in some supersymmetric theories chiral super-fields appear naturally. Chiral superfields, in our approach, are identifiedwith elements of the coordinate superalgebra of the above mentioned Grass-mannian. If one wants to formulate certain supersymmetric field theoriesin a noncommutative superspace one needs to have the notion of quantumchiral superfields. It is not obvious in other approaches how to construct aquantum chiral superalgebra without loosing other properties, as the actionof the group, for example. In our construction the quantum chiral superfieldsappear naturally together with the supergroup action.We plan to explore in a forthcoming paper Manin’s construction in thisnew framework.We shall not go into the details of the proofs of all of our statements, sincean enlarged version of part of this work is available in Ref. [3]; neverthlesswe shall make a constant effort to convey the key ideas and steps of ourconstructions.This is the content of the present paper.In section 2 we briefly outline few key facts of supergeometry, favouringintuition over rigorous definitions. Our main reference will be Ref. [2].In section 3 we discuss the chiral conformal superspace as an homoge-neous superspace identified with the super Grassmannian variety of 2 | |
1. We also providean explicit projective embedding of the super Grassmannian into a suitableprojective superspace. 2n Section 4 we give an equivalent approach via invariant theory to thetheory discussed in Section 3.In Section 5 we introduce the complex super Minkowski space as the bigcell in the chiral conformal superspace. We also provide an explicit descrip-tion of the action of the super Poincar´e group.In Sections 6 and 7 we build a quantum deformation of the Minkowskisuperspace and its compactification together with a coaction of the quantumPoincar´e and conformal supergroups.Finally in Section 8 we discuss some relevant physical applications of thetheory developed so far.
Acknoledgements . The authors wish to thank the UCLA Departmentof Mathematics for the wonderful hospitality during the workshop, that madethe present work possible. The authors wish also to thank prof. V. S.Varadarajan for the many helpful discussions on supergeometry and super-groups.
Supergeometry is essentially Z -graded geometry: any geometrical object isgiven a Z -grading in some natural way and the morphisms are the mapsrespecting the geometric structure and the Z -grading.For instance, a super vector space V is a vector space where we establisha Z -grading by giving a splitting V ⊕ V . The elements in V are called even and the elements in V are called odd . Hence we have a function p called the parity defined only on homogeneous elements. A superalgebra A is a supervector space with multiplication preserving parity. The reduced superalgebra associated with A is A r := A/I odd , where I odd is the ideal generated by theodd nilpotents. Notice that the reduced superalgebra A r may have evennilpotents, thus making the terminology a bit awkward.A superalgebra A is commutative if xy = ( − p ( x ) p ( y ) yx for all x , y homogeneous elements in A . ¿From now on we assume all superal-gebras are to be commutative unless otherwise specified and their category is3enoted with (salg). We also need to introduce the notion of affine superalge-bra . This is a finitely generated superalgebra such that A r has no nilpotents.In ordinary algebraic geometry such A r ’s are associated bijectively to affinealgebraic varieties, as we are going to see.The most interesting objects in supergeometry are the algebraic super-varieties and the differentiable supermanifolds . Both these concepts are en-compassed by the idea of superspace . Definition 2.1.
We define superspace the pair S = ( | S | , O S ) where | S | is atopological space and O S is a sheaf of superalgebras such that the stalk at apoint x ∈ | S | denoted by O S,x is a local superalgebra for all x ∈ | S | .A morphism φ : S −→ T of superspaces is given by φ = ( | φ | , φ ), where φ : | S | −→ | T | is a map of topological spaces and φ : O T −→ φ ∗ O S is asheaf morphism such that φ x ( m | φ | ( x ) ) = m x where m | φ | ( x ) and m x are themaximal ideals in the stalks O T, | φ | ( x ) and O S,x respectively.Let us see an important example.
Example 2.2.
The superspace R p | q is the topological space R p endowed withthe following sheaf of superalgebras. For any U ⊂ open R p O R p | q ( U ) = C ∞ ( R p )( U ) ⊗ R [ ξ , . . . , ξ q ] , where R [ ξ , . . . , ξ q ] is the exterior algebra (or Grassmann algebra ) generatedby the q variables ξ , . . . , ξ q . Definition 2.3. A supermanifold of dimension p | q is a superspace M =( | M | , O M ) which is locally isomorphic to the superspace R p | q , i. e. for all x ∈ | M | there exist an open set V x ⊂ | M | and U ⊂ R p | q such that: O M | V x ∼ = O R p | q | U . We shall now concentrate on the study of algebraic supervarieties, sinceour purpose is to obtain quantum deformations and for this reason the alge-braic approach is to be preferred.There are two equivalent and quite different approaches to both, algebraicsupervarieties and differentiable supermanifolds: the sheaf theoretic and thefunctor of points categorical approach. In the first of these approaches an4lgebraic supervariety (resp. a supermanifold) is to be understood as a su-perspace, that is a pair consisting of a topological space and a sheaf of su-peralgebras. In the special cases of an affine algebraic supervariety (resp.a differentiable supermanifold), the superalgebra of global sections of thesheaf allows us to reconstruct the whole sheaf and the underlying topologicalspace (see [2] ch. 4 and 10). Consequently an affine supervariety (resp. adifferentiable supermanifold) can be effectively identified with a commutativesuperalgebra.This is the super counterpart to the well known result of ordinary com-plex algebraic geometry: affine varieties are in one-to-one correspondencewith their coordinate rings, in other words, we associate the zeros of a set ofpolynomials into some affine space to the ideal generated by such polynomi-als. For example we associate to the complex sphere in C , the coordinatering C [ x, y, z ] / ( x + y + z − Definition 2.4.
Let O ( X ) be an affine superalgebra. We define affine su-pervariety X associated with O ( X ) the superspace ( | X | , O X ), where | X | isthe topological space of an ordinary affine variety, while O X is the (unique)sheaf of superalgebras, whose global sections coincide with O ( X ), and thereexists an open cover U i of | X | such that O X ( U i ) = O ( X ) f i = (cid:26) gf i (cid:12)(cid:12) g ∈ O ( X ) (cid:27) for suitable f i ∈ O ( X ) . (for more details see [8] ch. II and [2] ch. 10).A morphism of affine supervarieties is a morphism of the underlying su-perspaces, though one readily see it corresponds (contravariantly) to a mor-phism of the corresponding coordinate superalgebras: { morphisms X −→ Y } ←→ { morphisms O ( Y ) −→ O ( X ) } We define algebraic supervariety a superspace which is locally isomorphicto an affine supervariety. (cid:3) xample 2.5.
1. The affine superspace.
We define the polynomial superalgebra as: C [ x , . . . , x p , θ , . . . , θ q ] := C [ x , . . . , x p ] ⊗ Λ( θ , . . . , θ q ) . We want to interpret this superalgebra as the coordinate superring of a su-pervariety that we call the affine superspace of superdimension p | q , and weshall denote with the symbol C p | q or A m | n . The underlying topological spaceis A m , that is C m with the Zariski topology, while the sheaf is: O A m | n ( U ) := O A m ( U ) ⊗ Λ( θ . . . θ n ) .
2. The supersphere.
The superalgebra C [ x , x , x ] / ( x + x + x + η x + η x + η x −
1) is the superalgebra of the global sections of an affine super-variety whose underlying topological space is the unitary sphere in A .The first important example of a supervariety which is not affine is givenby the projective superspace . Example 2.6.
1. Projective superspace.
Consider the Z -graded superalgebra S = C [ x . . . x m , ξ . . . ξ n ]. For each r , 0 ≤ r ≤ m , we consider the graded su-peralgebra S [ r ] = C [ x , . . . , x m , ξ , . . . , ξ n ][ x − r ] , deg( x − r ) = − . The subalgebra S [ r ] ⊂ S [ r ] of Z -degree 0 is S [ r ] ≈ C [ u , . . . , ˆ u r , · · · , u m , η , . . . η n ] , u s = x s x r , η α = ξ α x r , (1)(the ‘ ˆ ’ means that this generator is omitted). This is an affine superalgebraand it corresponds to an affine superspace, (see 2.5) whose topological spacewe denote with | U r | and the corresponding sheaf with O U r . Notice thatthe topological spaces | U r | form an affine open cover of | P m | , the ordinaryprojective space of dimension m .A direct calculations shows that: O U r | | U r |∩| U s | = O U s | | U r |∩| U s | , so we conclude that there exists a unique sheaf on the topological space | P m | , that we denote as O P m | n , whose restriction to | U i | is O U i . Hence we6ave defined a supervariety that we denote with P m | n and call the projectivesuperspace of dimension m | n .
2. Projective supervarieties.
Let I ⊂ S = C [ x . . . x m , ξ . . . ξ n ] be a homogeneous ideal; then S/I is also agraded superalgebra and we can repeat the same construction as above. Firstof all, we notice that the reduced algebra (
S/I ) r corresponds to an ordinaryprojective variety, whose topological space we denote with | X | , embeddedinto a projective superspace | X | ⊂ | P m | . Consider the superalgebra of Z -degree zero elements in ( S/I )[ x − i ] (this is called projective localization ): (cid:18) C [ x , . . . x m , ξ . . . ξ n ] I [ x − i ] (cid:19) ∼ = C [ u , . . . , ˆ u i , . . . u m , η . . . η n ] I loc , where I loc are the even elements of Z -degree zero in I [ x − i ].Again this affine superalgebra defines an affine supervariety with topo-logical space | V i | ⊂ | U i | ⊂ | P m | and sheaf O V i . One can check that thesupersheaves O V i are such that O V i | | V i |∩| V j | = O V j | | V i |∩| V j | , so they glue to givea sheaf on | X | . Hence as before there exists a supervariety correspondingto the homogeneous superring S/I . This supervariety comes equipped witha projective embedding, encoded by the morphism of graded superalgebra S −→ S/I , hence ( | X | , O X ) is called a projective supervariety . (cid:3) It is very important to remark that, contrary to the affine case, thereis no coordinate superring associated instrinsecally to a projective super-variety, but there is a coordinate superring associated with the projectivesupervariety and its projective embedding. In other words we can have thesame projective variety admitting non isomorphic coordinate superrings withrespect to two different projective embeddings.We now want to introduce the functor of points approach to the theoryof supervarieties.Classically we can examine the points of a variety over different fields andrings. For example we can look at the rational points of the complex spheredescribed above. They are in one to one correspondence with the morphisms: C [ x, y, z ] / ( x + y + z − −→ Q . In fact each such morphism is specified bythe knowledge of the images of the generators. The idea behind the functorof points is to extend this and consider all morphisms from the coordinatering of the affine supervariety to all superalgebras at once.7 efinition 2.7. Let
A ∈ (salg), the category of commutative superalgebras.We define the A -points of an affine supervariety X as the (superalgebra)morphisms Hom( O ( X ) , A ). We define the functor of points of X as: h X : (salg) −→ (sets) , h X ( A ) = H om ( O ( X ) , A ) . In other words h X ( A ) are the A -points of X , for all commutative superalge-bras A . Example 2.8. If A is a generic (commutative) superalgebra, an A -point of C p | q (see Example 2.5) is given by a morphism C [ x , . . . , x p , θ , . . . , θ q ] −→ A ,which is determined once we know the image of the generators( x , . . . , x p , θ , . . . , θ q ) −→ ( a , . . . , a p , α , . . . , α q ) , with a i ∈ A and α j ∈ A . Notice that the C -points of C p | q are given by( k . . . k p , . . .
0) and coincide with the points of the affine space C p . In thisexample it is clear that the knowledge of the points over a field is by nomeans sufficient to describe the supergeometric object. Remark 2.9.
It is important at this point to notice that just giving a functorfrom (salg) to (sets), does not guarantee that it is the functor of points of asupervariety. A set of conditions to establish this is given in [2] ch. 10.The functor of points for projective supervarieties is more complicatedand we are unable to give a complete discussion here. it would be too longto give a general discussion here. We shall neverthless discuss the functor ofpoints of the projective space and superspace.
Example 2.10.
Let us consider the functor: h : (alg) −→ (sets), where h ( A )are the projective A -modules of rank one in A n .Equivalently h ( A ) consists of the pairs ( L, φ ), where L is a projective A -module of rank one, and φ is a surjective morphisms φ : A n +1 −→ L . Thesepairs are taken modulo the equivalence relation( L, φ ) ≈ ( L ′ , φ ′ ) ⇔ L a ≈ L ′ , φ ′ = a ◦ φ, If A = C , then projective modules are free and a morphism φ : C n +1 → C
8s specified by a n-tuple, ( a , . . . a n +1 ), with a i ∈ C , not all of the a i = 0.The equivalence relation becomes( a , . . . , a n +1 ) ∼ ( b , . . . b n +1 ) ⇔ ( a , . . . , a n +1 ) = λ ( b , . . . , b n +1 ) , with λ ∈ C × understood as an automorphism of C . It is clear then that h ( C ) consists of all the lines through the origin in the vector space C n +1 ,thus recovering the usual definition of complex projective space.If A is local, projective modules are free over local rings. We then havea situation similar to the field setting: equivalence classes are lines in the A -module A n +1 .Using the Representability Theorem (see [2]) one can show that the func-tor h is the functor of points of a variety that we call the projective spaceand whose geometric points coincide with the projective space P n over thefield k as we usually understand it. (cid:3) This example can be easily generalized to the supercontext: we considerthe functor h P m | n : (salg) −→ (sets), where h P m | n ( A ) is defined as the set theprojective A -modules of rank one in A m | n := A ⊗ C m | n . This is the functorof points of the projective superspace described in Example 2.6.The next question that we want to tackle is how we can define an embed-ding of a (super)variety into the projective (super)space using the functor ofpoints notation.Let X be a projective supervariety and Φ : X −→ P m | n be an injectivemorphism. As we discussed in Example 2.6 this embedding is encoded by asurjective morphism: C [ x , . . . , x m , ξ . . . , ξ n ] −→ C [ x , . . . x m , ξ . . . , ξ n ] / ( f , . . . , f r )In the notation of the functor of points, Φ is a natural transformation betweenthe two functors h X and h P m | n , given byΦ A : h X ( A ) −→ h P m | n ( A )with Φ A injective.If A is a local superalgebra, then an A -point ( a . . . , a m , α . . . , α n ) ∈ h P m | n ( A ) is in φ A ( h X ( A )) if and only if it satisfies the homogeneous polyno-mial relations f ( a . . . a m , α . . . , α n ) = 0 , ... f r ( a . . . a m , α . . . , α n ) = 0 . just on local superalgebras . This willbe our starting point when we shall determine the coordinate superalgebraof the Grassmannian supervariety with respect to its Pl¨ucker embedding. We are interested in the super Grassmannian of (2 | C | , that we denote with Gr. This will be our chiral conformalsuperspace once we establish an action of the conformal supergroup on it.Gr is defined via its functor of points. For a generic superalgebra A ,the A -points of Gr consist of the projective modules of rank 2 | A | := A ⊗ C | . It is not immediately clear that this is the functor of points ofa supervariety, however a fully detailed proof of this fact is available in [3],Appendix A. Another important issue is the fact that once a supervariety isgiven, its functor of points is completely determined just by looking at the local superalgebras, and similarly the natural transformations are determinedif we know them for local superalgebras. This a well known fact that can befound for example in Ref. [16], Appendix A.On a local superalgebra A , h Gr ( A ) consists of free submodules of rank2 | A | (on local superalgebras, projective modules are free). One suchmodule can be specified by a couple of independent even vectors, a and b ,which in the canonical basis { e , e , e , e , E } are given by two column vectorsthat span the subspace π = h a, b i = * a a a a α , b b b b β + , (2)10ith a i , b i ∈ A and α , β ∈ A . Let h GL(4 | ( A ) = c c c c ρ c c c c ρ c c c c ρ c c c c ρ δ δ δ δ d , (3)define the functor of points of the supergroup GL(4 | c ij , d ∈ A and ρ i , δ i ∈ A . We can describe the action of the supergroup GL(4 |
1) overGr as a natural transformation of the functors (for A local), h GL(4 | ( A ) × h Gr ( A ) −→ h Gr ( A ) g, h a, b i 7−→ h g · a, g · b i . Let π = h e , e i ∈ h Gr ( A ). The stabilizer of this point in GL(4 |
1) is theupper parabolic super subgroup P u , whose functor of points is h P u ( A ) = c c c c ρ c c c c ρ c c ρ c c ρ δ δ d ⊂ h GL(4 | ( A ) . Then, the Grassmannian is identified with the quotient h Gr ( A ) = h GL(4 | ( A ) /h P u ( A ) . We want now to work out the expression for the
Pl¨ucker embedding , It isimportant to stress that, contrary to what happens in the classical setting,in the super context we have that a generic Grassmannian supervariety doesnot admit a projective embedding. However for this particular Grassmanniansuch embedding exists, as we are going to show presently.We want to give a natural transformation among the functors p : h Gr → h P ( E ) , where E is the super vector space E = ∧ C | ≈ C | . Given the canonicalbasis for C | we construct a basis for Ee ∧ e , e ∧ e , e ∧ e , e ∧ e , e ∧ e , e ∧ e , E ∧ E , (even) e ∧ E , e ∧ E , e ∧ E , e ∧ E , (odd) (4)11s in the super vector space case, if L is a A -module, for A ∈ (salg), we canconstruct ∧ L Λ L = L ⊗ L/ h u ⊗ v + ( − | u || v | v ⊗ u i , u, v ∈ L. If L ∈ h Gr ( A ), then ∧ L ⊂ ∧ A | . It is clear that if L is a projective A -module of rank 2 |
0, then ∧ L is a projective A -module of rank 1 |
0. In otherwords it is an element of h P ( E ) ( A ), for E = ∧ C | . Hence we have defined anatural transformation: h Gr ( A ) p −−−→ h P ( E ) ( A ) L −−−→ ∧ L. Once we have the natural transformation defined, we can again restrict our-selves to work only on local algebras.Let a, b be two even independent vectors in A | . For any superalgebra A , they generate a free submodule of A | of rank 2 |
0. The natural transfor-mation described above is as follows. h Gr ( A ) p A −−−→ h P ( E ) ( A ) h a, b i A −−−→ h a ∧ b i . The map p A is clearly injective. The image p A ( h Gr ( A )) is the subset of evenelements in h P ( E ) ( A ) decomposable in terms of two even vectors of A | . Weare going to find the necessary and sufficient conditions for an even element Q ∈ h P ( E ) ( A ) to be decomposable. Let Q = q + λ ∧ E + a E ∧ E , with q = q e ∧ e + · · · + q e ∧ e , q ij ∈ A ,λ = λ e + · · · + λ e , λ i ∈ A . (5) Q is decomposable if and only if Q = ( r + ξ E ) ∧ ( s + θ E ) with r = r e + · · · r e , s = s e + · · · s e , r i , s i ∈ A ξ, θ ∈ A , which means Q = r ∧ s +( θr − ξs ) ∧E + ξθ E ∧E equivalent to q = r ∧ s, λ = θr − ξs, a = ξθ. q ∧ q = 0 , q ∧ λ = 0 , λ ∧ λ = 2 a q λa = 0 . Plugging (5) we obtain q q − q q + q q = 0 , (classical Pl¨ucker relation) q ij λ k − q ik λ j + q jk λ i = 0 , ≤ i < j < k ≤ λ i λ j = a q ij ≤ i < j ≤ λ i a = 0 . (6)These are the super Pl¨ucker relations . As we shall see in the next section thesuperalgebra O (Gr) = k [ q ij , λ k , a ] / I P , (7)is associated to the supervariety Gr in the Pl¨ucker embedding describedabove, where I P denotes the ideal of the super Pl¨ucker relations (6). Inother words I P contains all the relations involving the coordinates q ij , λ k and a . Remark 3.1.
The superalgebra O (Gr) is a sub superalgebra (though not aHopf sub superalgebra) of O (GL(4 | O (GL(4 | In this section we propose an alternative and equivalent way to construct thesuper Grassmannian Gr as a complex supervariety and we give the coordinatesuperring associated to the super Grassmannian in the Pl¨ucker embedding,thus completing the discussion initiated in the previous section.As we have seen in Section 2, the super Grassmannian can be equiva-lently understood as a a pair consisting of the underlying topological space G (2 , S be S = { ( v, w ) ∈ C ⊕ C / rank( v, w ) = 2 } , v, w ) ∼ ( v ′ , w ′ ) ⇔ span { v, w } = span { v ′ , w ′ } , or equivalently( v, w ) ∼ ( v ′ , w ′ ) ⇔ ∃ g ∈ GL(2 , C ) such that ( v ′ , w ′ ) = ( v, w ) g. Then we have that G (2 ,
4) = S/ ∼ .We consider now the set of polynomials on S , Pol( S ), and the subset ofsuch polynomials that is semi-invariant under the transformation of GL(2 , C ),that is f ( v ′ , w ′ ) = f ( u, v ) λ ( g ) , λ ( g ) ∈ C , f ∈ Pol( S ) . This defines the homogeneous ring of G (2 , y ij = v i w j − v j w i , with i < j and λ = det g. These are not all independent, they satisfy the Pl¨ucker relation y y + y y + y y = 0 . Let O be the sheaf of polynomials on S , so for each open set in ˜ U ⊂ S , O ( ˜ U ) = Pol( ˜ U ) and O inv the subsheaf of O corresponding to the semi-invariant polynomials.Let π : S → G (2 ,
4) be the natural projection. It is clear that for U ⊂ open G (2 , U = π − ( U ) ⊂ S is also open in S . We can define the followingsheaf over G (2 , O ( U ) = O inv ( π − ( U )) . This is the structural sheaf of the projective variety G (2 ,
4) with respect tothe Pl¨ucker embedding.Now we turn to the super setting and we want to define the sheaf of super-algebras generalizing the non super construction to the super Grassmannian.We define the superalgebra F ( S ) := Pol( S ) ⊗ Λ[ ξ , ξ ] . v, w ) ∈ S and consider the (5 ×
2) matrix (cid:18) v wξ ξ (cid:19) = v w ... ... v w ξ ξ . The group GL(2 , C ) acts on the right on these matrices (cid:18) v ′ w ′ ξ ′ ξ ′ (cid:19) = (cid:18) v wξ ξ (cid:19) · g, g ∈ GL(2 , C ) . We will write an element f ( v, w, ξ ) ∈ F ( S ) as f ( v, w, ξ ) = X i,j =0 , f ij ( v, w ) ξ i ξ j . We will refer to the elements of F ( S ) as ‘functions’, being this customary inthe physics literature. We now consider the set of semi-invariant functions f ( v ′ , w ′ , ξ ′ ) = f ( v, w, ξ ) λ ( g ) , λ ( g ) ∈ C , f ∈ F ( S ) . The following functions are semi-invariant: y ij = v i w j − v j w i , θ i = v i ξ − w i ξ , a = ξ ξ , (8)with λ ( g ) = det g but they are not all independent. They satisfy the superPl¨ucker relations (6) y y − y y + y y = 0 , (standard Pl¨ucker relation) y ij θ k − y ik θ j + y jk θ i = 0 1 ≤ i < j < k ≤ θ i θ j = ay ij ≤ i < j ≤ θ i a = 0 1 ≤ i ≤ . We want to show that the elements in (8) generate the ring of semi-invariants and that (6) are all the relations among these generators.
Proposition 4.1.
Let f be a homogeneous semi-invariant function, so f ( v ′ , w ′ , ξ ′ ) = f ( v, w, ξ ) λ ( g )15 ith (cid:18) v ′ w ′ ξ ′ ξ ′ (cid:19) = (cid:18) v wξ ξ (cid:19) · g, g ∈ GL(2 , C ) . Then in the decomposition f ( v, w, ξ ) = f ( v, w ) + X i f i ( v, w ) ξ i + f ( v, w ) ξ ξ , (9) one has that f ( v, w ) and f ( v, w ) are standard (non-super) semi-invariantsand X i f i ( v, w ) ξ i = X i h i ( v, w ) θ i , with h i ( v, w ) also a standard semi-invariant.Proof. Let us take g = (cid:18) a bc d (cid:19) , so (cid:18) v ′ w ′ ξ ′ ξ ′ (cid:19) = (cid:18) va + wc vb + wdξ a + ξ c ξ b + ξ d (cid:19) . Then we can see immediately that each term in (9) has to be a semi-invariant,so f ( v ′ , w ′ ) = λ ( g ) f ( v, w ) , X i f i ( v ′ , w ′ ) ξ ′ i = λ ( g ) X i f i ( v, w ) ξ i ,f ( v ′ , w ′ ) ξ ′ ξ ′ = f ( v, w ) ξ ξ . We have that f is an ordinary semi-invariant transforming with λ ( g ), andsince ξ ′ ξ ′ = ξ ξ det g , f ( v, w ) is a ordinary semi-invariant transformingwith λ ( g ) det g − . The odd terms θ i are of the same form as the ordinaryinvariants y ij , since the fact that ξ i is odd plays no particular role here (recallthat we are considering the action of an ordinary group, namely GL(2 , C )).So by the same argument we have in the ordinary case, there are no otherodd invariants, besides those we have already found, that are linear in theodd variable ξ and ξ . Then X i f i ξ i = X i h ( v, w ) i θ i , where h ( v, w ) i transforms with λ ( g ) det g − .16e now wish to give a result that describes completely the relationsamong the invariants.Consider the polynomial superalgebra C [ a ib ], 1 ≤ i ≤
5, 1 ≤ b ≤
2, withtheir parity defined as p ( a ij ) = p ( i ) + p ( j ) , with p ( k ) = 0 if 0 ≤ k ≤ p (5) = 1 . On C [ a ij ] there exists the following action of GL(2 , C ) : C [ a ib ] × GL(2 , C ) −−−→ C [ a ib ]( a ia , g − ) −−−→ P k a ib g − ba We have just proven that the semi-invariants are generated by the polyno-mials d ij = a i a j − a i a j , ≤ i < j ≤ ,d = a a . We have the following proposition:
Proposition 4.2.
Let O (Gr) be the subring of C [ a ib ] generated by the de-terminants d ij = a i a j − a j a i and d = a a . Then O (Gr) ∼ = C [ a ib ] /I P ,where I P is the ideal of the super Pl¨ucker relations (6). In other words I P contains all the possible relations satisfied by d ij and d .Proof. It is easy to verify that d ij and d satisfy all the above relations, theproblem is to prove that these are the only relations.The proof of this fact is the same as in the classical setting. Let us brieflysketch it. Let I , . . . , I r be multiindices organized in a tableau. We saythat a tableau is superstandard if it is strictly increasing along rows with theexception of the number 5 (that can be repeated) and weakly increasing alongcolumns. A standard monomial in O (Gr) is a monomial d I , · · · , d I r wherethe indices I , . . . , I r form a superstandard tableau. Using the super Pl¨uckerrelation one can verify that any monomial in O (Gr) can be written as a linearcombination of standard ones. This can be done directly or using the sameargument for the classical case (see Ref. [19] pg 110 for more details). Thestandard monomials are also linearly independent, hence they form a basisfor O (Gr) as C -vector space. Again this is done with the same argument asin Ref. [19] pg 110. So given a relation in O (Gr), once we write each termas a standard monomial we obtain that either the relation is identically zero(hence it is a relation in the Pl¨ucker ideal) or it gives a relation among thestandard monomials, which gives a contradiction.17n the end we summarize the main results of Sections 3 and 4 with acorollary. Corollary 4.3.
1. Let Gr be the Grassmannian of | spaces in C | .Then Gr ⊂ P | , that is Gr is a projective supervariety. Such em-bedding is encoded by the superring O (Gr) described above.2. O (Gr) is isomorphic to the ring generated by the determinants d ij , d . In this section we concentrate our attention to determine the big cell insidethe Grassmannian supervariety that we have discussed in the previous sec-tions. We shall identify such big cell with the chiral Minkowski superspace.As in the ordinary setting, the super Grassmannian Gr admits an opencover in terms of affine superspaces: topologically the two covers are thesame.We want to describe the functor of points of the big cell U inside Gr.This is the open affine functor corresponding to the points in which thecoordinate q is invertible.First of all, we write an element of h GL(4 | ( A ) in blocks as (see (3)) C C ρ C C ρ δ δ d . Assuming that det C is invertible, we can bring this matrix, with a trans-formation of h P u ( A ), to the form C C ρ C C ρ δ δ d h P u ( A ) = A α h P u ( A ) ∈ h GL(4 | ( A ) (cid:14) h P u ( A )(10)Consider the subspace π = span { a, b } in h Gr ( A ) for A local. Recall thatin Sec. 3 we made the identification: h Gr ( A ) ∼ = h GL(4 | ( A ) (cid:14) h P u ( A ). Hence: π = span { a, b } ≈ C C ρ C C ρ δ δ d h P u ( A ) ∈ h GL(4 | ( A ) (cid:14) h P u ( A )18ith det C invertible. Then, by a change of coordinate (10) we can bringthis matrix to the standard form detailed above π ≈ A α h P u ( A ) , A = (cid:18) a a a a (cid:19) , α = ( α , α ) , with the entries of A in A and the entries of α in A . Its column vectorsgenerate also the submodule h a, b i .The assumption that det C is invertible is equivalent to assume to be inthe topological open set | U | = | Gr | ∩ | V | , where V is the affine open setcorresponding to the topological open set | V | defined by taking in P ( E ) thecoordinate q to be invertible. Consequently the coordinate superring of theaffine open subvariety U of Gr corresponds to the projective localization ofthe Grassmannian superring in the coordinate q . In other words it consistsof the elements of degree zero in C [ q ij q − , λ j q − , a q − ] ⊂ O (Gr)[ q − ] . As one can readily check, there are no relations among these generators sothat the big cell U of Gr is the affine superspace with coordinate ring O ( U ) = C [ x ij , ξ j ] ≈ C | . (11)where we set x ij = q ij q − , x = a q − , ξ j = λ j q − .We are now interested in the super subgroup of GL(4 |
1) that preservesthe big cell U . This the lower parabolic sub-supergroup P l (see [3]), whosefunctor of points is given in suitable coordinates as type h P l ( A ) = x tx y yηdτ dξ d ⊂ h GL(4 | ( A )where x and y are even, invertible 2 × t is an even, arbitrary 2 × η a 2 × τ, ξ are 1 × d is an invertibleeven element.The action of the supergroup P l on the big cell U is as follows, h P l ( A ) × h U ( A ) −−−→ h U ( A ) x tx y yηdτ dξ d , Aα −−−→ A ′ α ′ , h P u ( A ) to revert the resulting matrix to thestandard form (10), we have A ′ α ′ = y ( A + ηα ) x − + td ( α + τ + ξA ) x − . (12)The subgroup with ξ = 0 is the super Poincar´e group times dilations (com-pare with Eq. (14) in Ref [18]). In that case d = det x det y. In this section we give a quantum deformation of O (Gr), discussed in the pre-vious sections. This will yield a quantum deformation of the chiral conformalsuperspace together with the natural coaction of the conformal supergroupon it. Definition 6.1.
Let us define following Manin [23] the quantum matrixsuperalgebra. M q ( m | n ) = def C q < a ij > /I M where C q < a ij > denotes the free algebra over C q = C [ q, q − ] generated bythe homogeneous variables a ij and the ideal I M is generated by the relations[23]: a ij a il = ( − π ( a ij ) π ( a il ) q ( − p ( i )+1 a il a ij , j < la ij a kj = ( − π ( a ij ) π ( a kj ) q ( − p ( j )+1 a kj a ij , i < ka ij a kl = ( − π ( a ij ) π ( a kl ) a kl a ij , i < k, j > l or i > k, j < la ij a kl − ( − π ( a ij ) π ( a kl ) a kl a ij =( − π ( a ij ) π ( a kj ) ( q − − q ) a kj a il i < k, j < l where p ( i ) = 0 if 1 ≤ i ≤ m , p ( i ) = 1 otherwise and π ( a ij ) = p ( i ) + p ( j )denotes the parity of a ij . 20 q ( m | n ) is a bialgebra with the usual comultiplication and counit:∆( a ij ) = X a ik ⊗ a kj , E ( a ij ) = δ ij . We are ready to define the general linear supergroup which will be mostinteresting for us.
Definition 6.2.
We define quantum general linear supergroup GL q ( m | n ) = def M q ( m | n ) h D − , D − i where D − , D − are even indeterminates such that: D D − = 1 = D − D , D D − = 1 = D − D and D = def P σ ∈ S m ( − q ) − l ( σ ) a σ (1) . . . a mσ ( m ) D = def P σ ∈ S n ( − q ) l ( σ ) a m +1 ,m + σ (1) . . . a m + n,m + σ ( n ) are the quantum determinants of the diagonal blocks.GL q ( m | n ) is H hopf algebra, where the comultiplication and counit arethe same as in M q ( m | n ), while the antipode S is detailed in Ref. [14].We now give the central definition in analogy with the ordinary setting(compare with Prop. 4.3). Definition 6.3.
Let the notation be as above. We define quantum superGrassmannian of 2 | | Gr q generated by the following quantum super minorsin GL q (4 | D ij = a i a j − q − a i a j , ≤ i < j ≤ , D = a a D i = a i a − q − a i a , ≤ i ≤ . D , D , D , D , D , D , D , D , D , D , D Notice that when q = 1 this is the coordinate ring of the super Grass-mannian.We need to work out the commutation relations and the quantum Pl¨uckerrelations in order to be able to give a presentation of the quantum Grass-mannian in terms of generators and relations.Let us start with the commutation relations. With very similar calcula-tions to the ones in Ref. [9] one finds the following relations: • If i, j, k, l are not all distinct we have (1 ≤ i, j, k, l ≤ D ij D kl = q − D kl D ij , ( i, j ) < ( k, l )where < refers to the lexicographic ordering. • If i, j, k, l are instead all distinct we have: D ij D kl = q − D kl D ij , ≤ i < j < k < l ≤ D ij D kl = q − D kl D ij − ( q − − q ) D ik D jl , ≤ i < k < j < l ≤ D ij D kl = D kl D ij , ≤ i < k < l < j ≤ • The only commutation relations that we are left to be shown are thefollowing: D ij D , D i D j , D i D After some computations one gets: D ij D = q − D D ij , ≤ i < j ≤ D i D j = − q − D j D i − ( q − − q ) D ij D ≤ i < j ≤ D i D = D D i = 0 , ≤ i ≤ . D D − q − D D + q − D D = 0 D ij D k − q − D ik D j + q − D i D jk = 0 , ≤ i < j < k ≤ D i D j = qD ij D , ≤ i < j ≤ . The next proposition summarizes all of our calculations and the proofcan be found in Ref. [3].
Proposition 6.4. • The quantum Grassmannian ring is given in terms of generators andrelations as: Gr q = C q h X ij i /I Gr where I Gr is the two-sided ideal generated by the commutations andPl¨ucker relations in the indeterminates X ij . Moreover Gr q / ( q − ∼ = O (Gr) (see Section 3). • The quantum Grassmannian ring is the free ring over C q generated bythe monomials in the quantum determinants: D i j , . . . , D i r j r where ( i , j ) , . . . , ( i r , j r ) form a semistandard tableau (for its definitionrefer to [3]). The quantum Grassmannian that we have constructed admits a coactionof the quantum supergroup GL q (4 | Proposition 6.5. Gr q is a quantum homogeneous superspace for the quan-tum supergroup GL q (4 | , i. e., we have a coaction given via the restrictionof the comultiplication of GL q (4 | : ∆ | Gr q : Gr q −→ GL q (4 | ⊗ Gr q . Quantum Minkowski superspace
We now turn to the quantum deformation of the big cell inside Gr q ; it willbe our model for the quantum Minkowski superspace.In Section 5 we wrote the action of the lower parabolic supergroup P l using the functor of points (12). We want now to translate it into the coactionlanguage in order to make the generalization to the quantum setting.Let O ( P l ) be the superalgebra: O ( P l ) := O (GL(4 | / I where I is the (two-sided) ideal generated by g j , g j , for j = 3 , γ , γ . This is the Hopf superalgebra coordinate superring of the lower parabolicsubgroup P l , with comultiplication naturally inherited by O (GL(4 | A local, we have h P l ( A ) = g g g g g g g g γ g g g g γ γ γ γ γ g ⊂ h GL( m | n ) ( A ) . (13)The superalgebra representing the big cell U can be realized as a subalgebraof O ( P l ). In order to see this better, let us make the following two differentchanges of variables in P l : g g g g g g g g γ g g g g γ γ γ γ γ g = x tx y yη ˜ τ x dξ d = x tx y yηdτ dξ d (14)Notice that the only difference between the two sets of variables is that wereplace τ with ˜ τ and we have: dτ = ˜ τ x, (15)24he next proposition tells us that these are sets of generators for O ( P l )and that having ˜ τ is essential to describe the big cell. Again for the proof werefer the reader to Ref. [3], while the explicit expressions for the generatorscome from a direct calculation. Proposition 7.1.
1. The Hopf superalgebra O ( P l ) is generated by the following sets ofvariables: • x , y , t , ˜ τ , ξ , η and d ; • x , y , t , τ , ξ , η and d defined as x = (cid:18) g g g g (cid:19) , y = (cid:18) g g g g (cid:19) ,t = (cid:18) − d d − d d − − d d − d d − (cid:19) d = g ˜ τ = ( − d d − , d d − ) τ = ( g − γ , g − γ ) η = (cid:18) d − γ d − γ (cid:19) ξ = (cid:0) g − γ g − γ (cid:1) (16) where for ≤ i < j ≤ d ij = g i g j − g j g i , d i = g i γ − γ g i , d = g g − g g .
2. The subalgebra of O ( P l ) generated by ( t, ˜ τ ) coincides with the big cellsuperring O ( U ) as defined in (11). It is given by the projective localizationof O (Gr) with respect to d .3. There is a well defined coaction ˜∆ of O ( P l ) on O ( U ) induced by thecoproduct in O ( P l ) , ˜∆ : O ( U ) ˜∆ −−−→ O ( P l ) ⊗ O ( U )25 hich explicitly takes the form: ˜∆ t ij = t ij ⊗ y ia S ( x ) bj ⊗ t ab + y i η a S ( x ) bj ⊗ ˜ τ jb , ˜∆˜ τ j =( d ⊗ τ a ⊗ ξ b ⊗ t ba + 1 ⊗ ˜ τ a )( S ( x ) aj ⊗ , The reader should notice right away that this is the dual to the expression(12).
We now turn to the quantum setting. In order to keep our notation min-imal, we use the same letters as in the classical case to denote the generatorsof the quantum big cell and the quantum supergroups.Let O ( P l,q ) be the superalgebra: O ( P l,q ) := O (GL q (4 | / I q where I q is the (two-sided) ideal in O (GL q (4 | g j , g j , for j = 3 , γ , γ . (17)This is the Hopf superalgebra of the lower parabolic subgroup, again withcomultiplication the one naturally inherited from O (GL q (4 | O ( P l,q ) explicitly: x = (cid:18) g g g g (cid:19) , t = (cid:18) − q − D D − D D − − q − D D − D D − (cid:19) y = (cid:18) g g g g (cid:19) , d = g , ˜ τ = (cid:0) − q − D D − D D − (cid:1) , ξ = (cid:0) g − γ g − γ (cid:1) η = y − (cid:18) γ γ (cid:19) = ( D ) − (cid:18) g − q − g − qg g (cid:19) = (cid:18) − q − D − D D − D (cid:19) It is not hard to see that O ( P l,q ) is also generated by x, y, d, η, ξ and ˜ τ .26 emark 7.2. The quantum Poincar´e supergroup times dilations is the quo-tient of O ( P l,q ) by the ideal ξ = 0. In fact as one can readily check with asimple calculation, if O ( P o ) denotes the function algebra of the super (un-quantized) Poincar´e groups times dilations, we have that( O ( P l,q ) / ( ξ )) / ( q − ∼ = O ( P o ) . One can also easily check that ( ξ ) is a Hopf ideal, so the comultiplicationgoes to the quotient. The quantum Poincar´e supergroup times dilations isthen generated by the images in the quotient of x, y, d, η and ˜ τ . In matrixform, one has x tx y yη ˜ τ x d . Explicitly in these coordinates its presentation is given as follows: O ( P l,q ) / ( ξ ) = C q < t, x, y, η, τ > /I P o,q where I P o,q is the ideal generated by the following relations. The indetermi-nates x and y behave respectively as quantum (even) matrices, that is, theirentries are subject to the relations 6.1. In other words we have for x (andsimilarly for y ): x x = q − x x , x x = q − x x , x x = q − x x x x = q − x x , x x = x x , x x − x x = ( q − − q ) x x Moreover the entries in x and y commute with each other. x and t , ˜ τ commutein the following way. Let i = 1 , j = 3 , x i t j = q − t j x i , x i t j = t j x i ,x i ˜ τ = q − ˜ τ x i , x i ˜ τ = ˜ τ x i ,x i ˜ τ = ˜ τ x i , x i ˜ τ = q − ˜ τ x i x commutes with η and d . y , t and ˜ τ satisfy similar relations as x , t and τ that we leave to the reader as an exercise (the rows are exchanged with the27olumns). y and η commute following the rules of quantum super matrices,very much the same calculation and relations expressed in 7.4. y and d commute. The commutation among t and ˜ τ are expressed in prop. 7.4. t and η commute. t , τ and d satisfy the following relations. t ij d = dt ij − ( q − − q ) η i ˜ τ i ˜ τ j d = d ˜ τ j ˜ τ and η commute with each other, while finally η j d = q − dη j . In analogy with the classical (non quantum) supersetting, we give thefollowing definition.
Definition 7.3.
We define the quantum big cell O q ( U ) as the subring of O ( P lq ) generated by t and ˜ τ .We compute now the quantum commutation relations among the gen-erators of the quantum big cell O q ( U ) , which is our chiral Minkowskisuperspace, and see that the quantum big cell admits a well defined coactionof the quantum supergroup O ( P lq ). Proposition 7.4.
The quantum big cell superring O q ( U ) has the followingpresentation: O q ( U ) := C q h t ij , ˜ τ j i (cid:14) I U , ≤ i ≤ , j = 1 , where I U is the ideal generated by the relations: t i t i = q t i t i , t j t j = q − t j t j , ≤ j ≤ , ≤ i ≤ t t = t t , t t = t t + ( q − − q ) t t , ˜ τ ˜ τ = − q − ˜ τ ˜ τ , t ij ˜ τ j = q − ˜ τ j t ij , ≤ j ≤ t i ˜ τ = ˜ τ t i , t i ˜ τ = ˜ τ t i + ( q − − q ) t i ˜ τ . As in the classical setting we have the following proposition.28 roposition 7.5.
The quantum big cell O q ( U ) admits a coaction of O ( P l,q ) obtained by restricting suitably the comultiplication in O ( P l,q ) . In other wordswe have a well defined morphism: ˜∆ : O q ( U ) −→ O ( P l,q ) ⊗ O q ( U ) satisfying the coaction properties and give explicitly by: (see 7.1), ˜∆ t ij = t ij ⊗ y ia S ( x ) bj ⊗ t ab + y i η a S ( x ) bj ⊗ ˜ τ jb , ˜∆˜ τ j = ( d ⊗ τ a ⊗ ξ b ⊗ t ba + 1 ⊗ ˜ τ a )( S ( x ) aj ⊗ . by choosing as before generators x , y , t , d , τ , η , ξ for O ( P l,q ) and t , ˜ τ for O q ( U ) with dτ = ˜ τ x .Furthermore, this coaction goes down to a well defined coaction for thequantization of the super Poincar´e group (see remark 7.2). To compare with other deformations of the Minkowski space, we writehere the even part of O q ( U in terms of the more familiar generators t = x µ σ µ = (cid:18) x + x x − ix x + ix x + x (cid:19) . The commutation relations of the generators x µ are then [28] x x = 2 q − + q x x + i q − − qq − + q x x ,x x = 2 q − + q x x − i q − − qq − + q x x ,x x = x x ,x x = i ( q − + q )2 (cid:0) − ( x ) + ( x ) + x x − x x (cid:1) ,x x = 2 q − + q x x − i q − − qq − + q x x ,x x = 2 q − + q x x + i q − − qq − + q x x . Chiral superfields in Minkowski superspace
In this section we wish to motivate the importance of the chiral conformalsuperspace and its quantum deformation in physics. We introduce chiralsuperfields in Minkowski superspace as they are used in physics. We startby introducing the complexified Minkowski space: the chiral superfields area sub superalgebra of the coordinate superalgebra of Minkowski space. Theycan also be seen as the coordinate superalgebra of the chiral Minkowskisuperspace, which is complex.
We consider the complexified Minkowski space C . The N = 1 scalar super-fields on the complexified Minkowski space are elements of the commutativesuperalgebra O ( C | ) ≡ C ∞ ( C ) ⊗ Λ[ θ , θ , ¯ θ , ¯ θ ] , (18)where Λ[ θ , θ , ¯ θ , ¯ θ ] is the Grassmann (or exterior) algebra generated by theodd variables θ , θ , ¯ θ , ¯ θ .We will denote the coordinates (or generators) of the superspace as x µ , µ = 0 , , , θ α , ¯ θ ˙ α , α, ˙ α = 1 , field components , asΨ( x, θ, ¯ θ ) = ψ ( x ) + ψ α ( x ) θ α + ψ ′ ˙ α ( x )¯ θ ˙ α + ψ αβ ( x ) θ α θ β + ψ α ˙ β ( x ) θ α ¯ θ ˙ β + ψ ′ ˙ α ˙ β ( x )¯ θ ˙ α ¯ θ ˙ β + ψ αβ ˙ γ ( x ) θ α θ β ¯ θ ˙ γ + ψ ′ α ˙ β ˙ γ ( x ) θ α ¯ θ ˙ β ¯ θ ˙ γ + ψ αβ ˙ γ ˙ δ ( x ) θ α θ β ¯ θ ˙ γ ¯ θ ˙ δ . Action of the Lorentz group SO(1,3).
There is an action of the doublecovering of the complexified Lorentz group, Spin(1 , c ≈ SL(2 , C ) × SL(2 , C )over C | . The even coordinates x µ transform according to the fundamentalrepresentation of SO(1 ,
3) ( V ), x µ Λ µν x ν , while θ and ¯ θ are Weyl spinors (or half spinors). More precisely, the coordi-nates θ transform in one of the spinor representations, say S + ≈ (1 / ,
0) and¯ θ transform in the opposite chirality representation, S − ≈ (0 , / θ α S αβ θ β , ¯ θ ˙ α ˜ S ˙ α ˙ β θ ˙ β . x, θ, ¯ θ ) = ( R Ψ)(Λ − x, S − θ, ˜ S − ¯ θ ) , where R Ψ is the superfield obtained by transforming the field components Rψ ( x ) = ψ ( x ) , Rψ α ( x ) = S αβ ψ β ( x ) , . . . The hermitian matrices σ = (cid:18) (cid:19) , σ = (cid:18) (cid:19) , σ = (cid:18) − ii 0 (cid:19) , σ = (cid:18) (cid:19) , define a Spin(1 , S + ⊗ S − −−−→ Vs α ⊗ t ˙ α −−−→ s α σ µα ˙ α t ˙ α . Derivations. A left derivation of degree m = 0 , A isa linear map D L : A 7→ A such that D L (Ψ · Φ) = D L (Ψ) · Φ + ( − mp Ψ Ψ · D L (Φ) . Graded left derivations span a Z -graded vector space (or supervector space ).In general, linear maps over a supervector space are also a Z -gradedvector space. A map has degree 0 if it preserves the parity and degree 1 if itchanges the parity. For the case of derivations of a commutative superalgebra,an even derivation has degree 0 as a linear map and an odd derivation hasdegree 1 as a linear map.In the same way one defines right derivations , D R (Ψ · Φ) = ( − mp Φ D R (Ψ) · Φ + Ψ · D R (Φ) . Notice that derivations of degree zero are both, right and left derivations.Moreover, given a left derivation D L of degree m one can define a rightderivation D R also of degree m in the following way D R Ψ = ( − m ( p Ψ +1) D L Ψ . (19)31et us now focus on the commutative superalgebra O ( C | ). We definethe standard left derivations ∂ Lα Ψ = ψ α + 2 ψ αβ θ β + ψ α ˙ β ¯ θ ˙ β + 2 ψ αβ ˙ γ θ β ¯ θ ˙ γ + ψ ′ α ˙ β ˙ γ ¯ θ ˙ β ¯ θ ˙ γ + 2 ψ αβ ˙ γ ˙ δ θ β ¯ θ ˙ γ ¯ θ ˙ δ ,∂ L ˙ α Ψ = ψ ′ ˙ α − ψ β ˙ α θ β + 2 ψ ′ ˙ α ˙ β ¯ θ ˙ β + ψ γβ ˙ α θ γ θ β − ψ ′ β ˙ α ˙ γ θ β ¯ θ ˙ γ + 2 ψ γβ ˙ α ˙ δ θ γ θ β ¯ θ ˙ δ . Also, using (19) one can define ∂ Rα , ∂ R ˙ α .We consider now the odd left derivations Q Lα = ∂ Lα − i σ µα ˙ α ¯ θ ˙ α ∂ µ , ¯ Q L ˙ α = − ∂ L ˙ α + i θ α σ µα ˙ α ∂ µ . They satisfy the anticommutation rules { Q Lα , ¯ Q L ˙ α } = 2i σ µα ˙ α ∂ µ , { Q Lα , Q Lβ } = { ¯ Q L ˙ α , ¯ Q L ˙ β } = 0 , with ∂ µ = ∂/∂x µ . Q L and ¯ Q L are the supersymmetry charges or super-charges . Together with P µ = − i ∂ µ , they form a Lie superalgebra, the supertranslation algebra , which then actson the superspace C | .Let us define another set of (left) derivations, D Lα = ∂ α + i σ µα ˙ α ¯ θ ˙ α ∂ µ , ¯ D L ˙ α = − ∂ ˙ α − i θ α σ µα ˙ α ∂ µ , with anticommutation rules { D Lα , ¯ D L ˙ α } = − σ µα ˙ α ∂ µ , { D Lα , D Lβ } = { ¯ D L ˙ α , ¯ D L ˙ β } = 0 . They also form a Lie superalgebra, isomorphic to the supertranslation alge-bra. This can be seen by taking Q L → − D L , ¯ Q L −→ ¯ D L . It is easy to see that the supercharges anticommute with the derivations D L and ¯ D L . For this reason, D L and ¯ D L are called supersymmetric covariantderivatives or simply covariant derivatives , although they are not related toany connection form.We go now to the central definition.32 efinition 8.1. A chiral superfield is a superfield Φ such that¯ D L ˙ α Φ = 0 . (20)Because of the anticommuting properties of D ′ s and Q ′ s , we have that¯ D L ˙ α Φ = 0 ⇒ ¯ D L ˙ α ( Q Lβ Φ) = 0 , ¯ D L ˙ α ( ¯ Q L ˙ β )Φ = 0 . This means that the supertranslation algebra acts on the space of chiralsuperfields.On the other hand, due to the derivation property,¯ D L ˙ α (ΦΨ) = ¯ D L ˙ α (Φ)Ψ + ( − p Φ Φ ¯ D L ˙ α (Ψ) , we have that the product of two chiral superfields is again a chiral superfield. One can solve the constraint (20). Notice that the quantities y µ = x µ + i θ α σ µα ˙ α θ ˙ α , θ α (21)satisfy ¯ D L ˙ α y µ = 0 , ¯ D L ˙ α θ α = 0 . Using the derivation property, any superfield of the formΦ( y µ , θ ) , satisfies ¯ D L ˙ α Φ = 0and so it is a chiral superfield. This is the general solution of (20).We can make the change of coordinates x µ , θ α , ¯ θ ˙ α −→ y µ = x µ + iθ α σ µα ˙ α ¯ θ ˙ α , θ α , ¯ θ ˙ α . A superfield may be expressed in both coordinate systemsΦ( x, θ, ¯ θ ) = Φ ′ ( y, θ, ¯ θ ) . The covariant derivatives and supersymmetry charges take the form D Lα Φ ′ = ∂ L Φ ′ ∂θ α + 2 iσ µα ˙ α ¯ θ ˙ α ∂ L Φ ′ ∂y µ ¯ D L ˙ α Φ ′ = − ∂ L Φ ′ ∂ ¯ θ ˙ α , ¯ Q L ˙ α Φ ′ = − ∂ L Φ ′ ∂ ¯ θ ˙ α + 2 iθ α σ µα ˙ α ∂ L Φ ′ ∂y µ Q Lα Φ ′ = ∂ L Φ ′ ∂θ α .
33n the new coordinate system the chirality condition is simply ∂ L Φ ′ ∂ ¯ θ ˙ α = 0 , so it is similar to a holomorphicity condition on the θ ’s.This shows that chiral scalar superfields are elements of the commutativesuperalgebra O ( C | ) = C ∞ ( C ) ⊗ Λ[ θ , θ ]. In the previous sections werealized this superspace as the big cell inside the chiral conformal superspace,which is the Grassmannian of 2 | C | .The complete (non chiral) conformal superspace is in fact the flag spaceof 2 | | C | . On this supervariety one canput a reality condition, and the real Minkowski space is the big cell insidethe flag. It is instructive to compare Eq. (21) with the incidence relation forthe big cell of the flag manifold in Eq. (12) of Ref. [18]. We can then beconvinced that the Grassmannian that we use to describe chiral superfieldsis inside the (complex) flag.There are supersymmetric theories in physics (like Wess-Zumino models,or super Yang-Mills) that include in the formulation chiral superfields. In pre-vious approaches it has been difficult to formulate them on non commutativesuperspaces (with non trivial commutation relations of the odd coordinates).The reason was that the covariant derivatives are not anymore derivationsof the noncommutative superspace, and the chiral superfields do not form asuperalgebra [10, 11]. Some proposals to solve these problems include thepartial (explicit) breaking of supersymmetry [26, 11]. In our approach toquantization of superspace, the quantum chiral ring appears in a naturalway, thus making possible the formulation of supersymmetric theories in noncommutative superspaces. Also, the super variety and the supergroup actingon it become non commutative, the group law is not changed, so the physicalsymmetry principle remains intact. This is a virtue of the deformation basedon quantum matrix groups. References [1] F. A. Berezin,
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