Noncommutativity relations in type IIB theory and their supersymmetry
aa r X i v : . [ h e p - t h ] M a y Noncommutativity relations in type IIB theory and theirsupersymmetry ∗ B. Nikoli´c † and B. Sazdovi´c ‡ Institute of PhysicsUniversity of BelgradeP.O.Box 57, 11001 Belgrade, Serbia
November 28, 2018
Abstract
In the present paper we investigate noncommutativity of D D N = 1 su-persymmetry transformations and noncommutativity parameters are components of N = 1 supermultiplet. PACS number(s) : 02.40.Gh, 11.30.Pb, 11.25.Uv, 11.25.-w.
In the present paper we investigate the noncommutativity of type IIB superstring theory[1] in pure spinor formulation (up to the quadratic terms) [2] using canonical approach. Weconsider two cases: when D D Dp -branes [1] in the certain super-string theory. The R-R sector of type IIB theory contains gauge fields A (0) , A (2) and ∗ Work supported in part by the Serbian Ministry of Science and Technological Development, undercontract No. 141036. † e-mail address: [email protected] ‡ e-mail address: [email protected] (4) , and consequently, Dp -branes with odd value of p are stable. As a particular choice,besides D D θ α and ¯ θ α ( α = 1 , , . . . , S α in terms of two independent D S α and S α ( α , α = 1 , , . . . ,
8) [1, 3, 4].In the case of D x µ . The boundary condition for fermionic coor-dinates, ( θ α − ¯ θ α ) | π = 0 produces additional one for their canonically conjugated mo-menta, ( π α − ¯ π α ) | π = 0. Choosing Neumann boundary conditions for x i coordinates( i = 0 , , . . . , x a ( a = 6 , . . . , D θ α + ( ⋆ Γ¯ θ ) α ] | π = 0, where ⋆ Γ = Γ Γ Γ Γ Γ Γ is introduced to pre-serve supersymmetry [1]. Corresponding boundary condition for momenta is of the form[ π α + ( ⋆ Γ¯ π ) α ] | π = 0. In terms of D θ α − ¯ θ α ) | π = 0, ( θ α + ¯ θ α ) | π = 0, ( π α − ¯ π α ) | π = 0 and ( π α + ¯ π α ) | π = 0.We treat boundary conditions as canonical constraints [5, 6, 7, 8]. Using their consis-tency conditions, we rewrite them in compact σ -dependent form and find their Poissonbrackets. It turns out that all constraints are of the second class for nonsingular openstring metric G eff = G − BG − B . Solving the second class constraints, we obtain ini-tial coordinates in terms of effective coordinates and momenta. Presence of the momentain the solutions for initial coordinates is source of noncommutativity. Noncommutativ-ity relations are consistent with N = 1 supersymmetry transformations. We obtainedthat noncommutativity parameters contain only odd powers of background fields anti-symmetric under world-sheet parity transformation Ω : σ → − σ . They are componentsof N = 1 supermultiplet. This result represents a supersymmetric generalization of theresult obtained by Seiberg and Witten [9].At the end we give some concluding remarks. D -brane We will investigate pure spinor formulation [2, 10, 7, 8] of type IIB theory, neglecting ghostterms and keeping quadratic ones as in the action of Ref.[10].The action in a flat background S = Z Σ d ξ (cid:16) κ η mn η µν ∂ m x µ ∂ n x ν − π α ∂ − θ α + ∂ + ¯ θ α ¯ π α (cid:17) , (2.1)deformed by integrated form of the massless IIB supergravity vertex operator V SG = Z Σ d ξX TM A MN ¯ X N , (2.2)2roduces the full action S = S + V SG . (2.3)The world sheet (Σ) parameters are ξ m = ( τ , σ ), while D = 10-dimensional space-timecoordinates are labelled by x µ ( µ = 0 , , , . . . , θ α and ¯ θ α . The variables π α and¯ π α are canonically conjugated to the coordinates θ α and ¯ θ α , respectively. The fermioniccoordinates and momenta are Majorana-Weyl spinors.Using equations of motion which are consequences of BRST invariance, requiring forall background fields to be constant and restricted the action to the quadratic terms, thevertex operator gets the form V SG = Z Σ d ξ (cid:20) κ ( 12 g µν + B µν ) ∂ + x µ ∂ − x ν − π α Ψ αµ ∂ − x µ + ∂ + x µ ¯Ψ αµ ¯ π α + 12 κ π α F αβ ¯ π β (cid:21) , (2.4)where g µν is symmetric, B µν is antisymmetric Neveu-Schwarz field, Ψ αµ and ¯Ψ αµ are NS-Rgravitino fields and F αβ is R-R field strength. Adding V SG to flat background action, wehave S = κ Z Σ d ξ (cid:20) η mn G µν + ε mn B µν (cid:21) ∂ m x µ ∂ n x ν (2.5)+ Z Σ d ξ (cid:20) − π α ∂ − ( θ α + Ψ αµ x µ ) + ∂ + (¯ θ α + ¯Ψ αµ x µ )¯ π α + 12 κ π α F αβ ¯ π β (cid:21) , where G µν = η µν + g µν is constant gravitational field.Embedding D x µ so that D D x i ( i =0 , , . . . ,
5) and Dirichlet boundary conditions for orthogonal directions x a ( a = 6 , , , S = κ Z Σ d ξ (cid:20) η mn G ij + ε mn B ij (cid:21) ∂ m x i ∂ n x j + 2 ℜ (cid:26)Z Σ d ξ (cid:2) − π α ( ∂ τ − ∂ σ ) (cid:0) θ α + Ψ α i x i (cid:1) + ( ∂ τ + ∂ σ ) (cid:0) ¯ θ α + ¯Ψ α i x i (cid:1) ¯ π α (cid:3)(cid:27) + 2 ℜ (cid:26)Z Σ d ξ (cid:2) − π α ( ∂ τ − ∂ σ ) (cid:0) θ α + Ψ α i x i (cid:1) + ( ∂ τ + ∂ σ ) (cid:0) ¯ θ α + ¯Ψ α i x i (cid:1) ¯ π α (cid:3)(cid:27) + 1 κ ℜ (cid:26)Z Σ d ξ h π α f α β ¯ π β + π α f α β ¯ π ∗ β − π α f α β ¯ π β − π α f α β ¯ π ∗ β i(cid:27) + 1 κ ℜ (cid:26)Z Σ d ξ h π α f α β ¯ π β − π α f α β ¯ π β + π α f α β ¯ π ∗ β − π α f α β ¯ π ∗ β i(cid:27) , (2.6)where ℜ means real part of some complex number, ∗ means complex conjugation andwith f rs we denoted 8 independent D F αβ (for more details seeAppendix B of Ref.[8]). 3 Canonical analysis
Here we will perform canonical analysis of type IIB superstring theory. Boundary condi-tions will be treated as canonical constraints. Consistency procedure for boundary con-ditions enable us to rewrite them in compact σ -dependent form. It turns out that allconstraints are of the second class. Using standard canonical procedure we find canonical Hamiltonian of type IIB superstringtheory in the form H c = Z dσ H c , H c = T − − T + , T ± = t ± − τ ± , (3.1)where t ± = ∓ κ G µν I ± µ I ± ν , I ± µ = π µ + 2 κ Π ± µν x ′ ν + π α Ψ αµ − ¯Ψ αµ ¯ π α ,τ + = ( θ ′ α + Ψ αµ x ′ µ ) π α − κ π α F αβ ¯ π β , τ − = (¯ θ ′ α + ¯Ψ αµ x ′ µ )¯ π α + 14 κ π α F αβ ¯ π β . (3.2)For the case of embedded D t ± = ∓ κ G ij I ± i I ± j ,I ± i = π i + 2 κ Π ± ij x ′ j + 2 ℜ (cid:0) π α Ψ α i + π α Ψ α i − ¯Ψ α i ¯ π α − ¯Ψ α i ¯ π α (cid:1) ,τ + = 2 ℜ (cid:2)(cid:0) θ ′ α + Ψ α i x ′ i (cid:1) π α + (cid:0) θ ′ α + Ψ α i x ′ i (cid:1) π α (cid:3) − κ ℜ (cid:16) π α f α β ¯ π β + π α f α β ¯ π ∗ β − π α f α β ¯ π β − π α f α β ¯ π ∗ β (cid:17) − κ ℜ (cid:16) π α f α β ¯ π β − π α f α β ¯ π β + π α f α β ¯ π ∗ β − π α f α β ¯ π ∗ β (cid:17) ,τ − = 2 ℜ (cid:2)(cid:0) ¯ θ ′ α + ¯Ψ α i x ′ i (cid:1) ¯ π α + (cid:0) ¯ θ ′ α + ¯Ψ α i x ′ i (cid:1) ¯ π α (cid:3) + 12 κ ℜ (cid:16) π α f α β ¯ π β + π α f α β ¯ π ∗ β − π α f α β ¯ π β − π α f α β ¯ π ∗ β (cid:17) + 12 κ ℜ (cid:16) π α f α β ¯ π β − π α f α β ¯ π β + π α f α β ¯ π ∗ β − π α f α β ¯ π ∗ β (cid:17) , (3.3)and π i , π α , π α , ¯ π α and ¯ π α are canonically conjugated to x i , θ α , θ α , ¯ θ α and ¯ θ α ,respectively. Note, that in both cases energy-momentum tensor components T ± satisfyVirasoro algebra as a consequence of two dimensional diffeomorphisms. As a time translation generator Hamiltonian has to be differentiable with respect to co-ordinates and their canonically conjugated momenta. From this fact, following method4f Ref.[6], we will derive boundary conditions directly in terms of the canonical variables.Varying Hamiltonian H c , we obtain δH c = δH ( R ) c − [ γ (0) µ δx µ + π α δθ α + δ ¯ θ α ¯ π α ] | π , (3.4)where δH ( R ) c is regular term without τ and σ derivatives of supercoordinates and super-momenta variations and γ (0) µ = Π + µν I − ν + Π − µν I + ν + π α Ψ αµ + ¯Ψ αµ ¯ π α . (3.5)Consequently, differentiability of Hamiltonian for type IIB theory demands h γ (0) µ δx µ + π α δθ α + δ ¯ θ α ¯ π α i (cid:12)(cid:12)(cid:12) π = 0 . (3.6)Embedding D x µ coordinates, whichmeans γ (0) µ | π = 0 . (3.7)Boundary condition for fermionic coordinates chosen to preserve half of the initial N = 2supersymmetry is ( θ α − ¯ θ α ) (cid:12)(cid:12)(cid:12) π = 0 , (3.8)and it produces additional boundary condition for fermionic momenta( π α − ¯ π α ) | π = 0 . (3.9)In order to embed D D x i we choose Neumann bound-ary conditions, implying γ (0) i | π = 0 , (3.10) γ (0) i = Π + ij I − j + Π − ij I + j + 2 ℜ (cid:0) π α Ψ α i + π α Ψ α i + ¯Ψ α i ¯ π α + ¯Ψ α i ¯ π α (cid:1) . For othogonal coordinates we choose Dirichlet ones, δx a | π = 0. As in Ref.[8], dynamics of x a directions decouples from the rest part of action and we will not consider this boundarycondition. Fermionic boundary conditions take the form (cid:2) θ α + ( ⋆ Γ¯ θ ) α (cid:3) | π = 0 , [ π α + ( ⋆ Γ¯ π ) α ] | π = 0 , (3.11)where ⋆ Γ = Γ Γ Γ Γ Γ Γ . By convention we introduce ⋆ Γ because if Q and Q are typeIIB supersymmetry charges then, after Dp -brane is embedded, the conserved supersym-metry is the linear combination [1] Q + Γ Γ . . . Γ p Q . (3.12)5ote that arbitrary Majorana-Weyl spinor can be expressed in terms of two oppositechirality D S α = S α S α ( b S ∗ ) α − ( b S ∗ ) α , (3.13)where b is D D θ α − ¯ θ α ) (cid:12)(cid:12)(cid:12) π = 0 , ( θ α + ¯ θ α ) (cid:12)(cid:12)(cid:12) π = 0 , (3.14)( π α − ¯ π α ) | π = 0 , ( π α + ¯ π α ) | π = 0 . (3.15)According with Ref.[6], we will treat the expressions (3.7)-(3.9) and (3.10), (3.14) and(3.15) as canonical constraints for D D We assume that all background fields are constant which enables us to calculate Poissonbrackets. Using standard Poisson algebra, the consistency procedure for γ (0) µ produces aninfinite set of constraints γ ( n ) µ ≡ { H c , γ ( n − µ } ( n = 1 , , , . . . ) , (3.16)which can be rewritten at σ = 0 in the compact σ -dependent formΓ µ ( σ ) ≡ ∞ X n =0 σ n n ! γ ( n ) µ | = Π + µν I − ν ( σ ) + Π − µν I + ν ( − σ ) + π α ( − σ )Ψ αµ + ¯Ψ αµ ¯ π α ( σ ) . (3.17)From conditions ( θ α − ¯ θ α ) | = 0 and ( π α − ¯ π α ) | = 0, we getΓ α ( σ ) = Θ α ( σ ) − ¯Θ α ( σ ) , Γ πα ( σ ) ≡ Π α ( σ ) − ¯Π α ( σ ) , (3.18)where the right-hand side functions are defined asΘ α ( σ ) = θ α ( − σ ) − Ψ αµ ˜ q µ ( σ ) − κ F αβ Z σ dσ P s ¯ π β + 12 κ G µν Ψ αµ Z σ dσ P s ( I + ν + I − ν ) , (3.19)¯Θ α ( σ ) = ¯ θ α ( σ ) + ¯Ψ αµ ˜ q µ ( σ ) + 12 κ F βα Z σ dσ P s π β + 12 κ G µν ¯Ψ αµ Z σ dσ P s ( I + ν + I − ν ) , (3.20)6 α ( σ ) = π α ( − σ ) , ¯Π ¯ α ( σ ) = ¯ π ¯ α ( σ ) . (3.21)Similarly, for D i ( σ ) = Π + ij I − j ( σ ) + Π − ij I + j ( − σ )+ 2 ℜ (cid:2) π α ( − σ )Ψ α i + π α ( − σ )Ψ α i + ¯Ψ α i ¯ π α ( σ ) + ¯Ψ α i ¯ π α ( σ ) (cid:3) , (3.22)Γ α ( σ ) = Θ α ( σ ) − ¯Θ α ( σ ) , Γ α ( σ ) = Θ α ( σ ) + ¯Θ α ( σ ) , (3.23)Γ πα ( σ ) = π α ( − σ ) − ¯ π α ( σ ) , Γ πα ( σ ) = π α ( − σ ) + ¯ π α ( σ ) , (3.24)where right-hand side variables are defined asΘ α ( σ ) = θ α ( − σ ) − Ψ α i ˜ q i ( σ ) − κ f α β Z σ dσ P s ¯ π β − κ f α β Z σ dσ P s ¯ π ∗ β + 12 κ f α β Z σ dσ P s ¯ π β + 12 κ f α β Z σ dσ P s ¯ π ∗ β + 12 κ G ij Ψ α i Z σ dσ P s ( I + j + I − j ) , (3.25)¯Θ α ( σ ) = ¯ θ α ( σ ) + ¯Ψ α i ˜ q i ( σ ) + 12 κ f β α Z σ dσ P s π β + 12 κ f ∗ β α Z σ dσ P s π ∗ β + 12 κ f β α Z σ dσ P s π β + 12 κ f ∗ β α Z σ dσ P s π ∗ β + 12 κ G ij ¯Ψ α i Z σ dσ P s ( I + j + I − j ) . (3.26)The expression for Θ α can be obtained from the expression for Θ α using substitution θ α → θ α , π α ↔ π α , Ψ α i → Ψ α i , f → − f , f → − f , f → − f and f ↔− f . We obtain the expression for ¯Θ α from ¯Θ α using similar transition rules (fermionicvariables and background fields have bars).We introduced variables, even and odd under world-sheet parity transformation Ω : σ → − σ . For bosonic variables we use standard notation [6] q µ ( σ ) = P s x µ ( σ ) , ˜ q µ ( σ ) = P a x µ ( σ ) , (3.27) p µ ( σ ) = P s π µ ( σ ) , ˜ p µ ( σ ) = P a π µ ( σ ) , (3.28)while for fermionic ones we explicitly use the projectors on Ω even and odd parts P s = 12 (1 + Ω) , P a = 12 (1 − Ω) . (3.29)For all constraints we apply the consistency procedure at σ = π and obtain similarexpressions, where all variables depending on − σ are replaced by the same variables de-pending on 2 π − σ . That set of constraints is solved by 2 π periodicity of all canonicalvariables as in Ref.[6]. 7 .4 Classification of constraints Let us denote all constraints with Γ A = (Γ µ , Γ α , Γ πα ). From { H c , Γ A } = Γ ′ A ≈ , (3.30)it follows that all constraints weakly commute with canonical Hamiltonian, so there areno more constraints in the theory and the consistency procedure is completed.For practical reasons we will separate the constraints Γ A in two sets: the zero modes( θ α − ¯ θ α ) | and the rest ⋆ Γ A = (Γ µ , Γ ′ α , Γ πα ). The reason for this separation is that Poissonbrackets of constraints ⋆ Γ A close on δ ′ function while those with Γ A , close on δ , δ ′ or stepfunction.First we will classify ⋆ Γ A . The algebra of the constraints ⋆ Γ A has the form { ⋆ Γ A , ⋆ Γ B } = M AB δ ′ , (3.31)where the supermatrix M AB is given by the expression M AB = ( M ) µν ( M ) µγ δ ( M ) αβν ( M ) αβγ δ ! = − κG effµν − eff ) γµ − eff ) αν κ F αγeff − δ αδ − δ β γ . (3.32)Here we introduced effective background fields G effµν = G µν − B µρ G ρλ B λν , (Ψ eff ) αµ = 12 Ψ α + µ + B µρ G ρν Ψ α − ν ,F αβeff = F αβa − Ψ α − µ G µν Ψ β − ν , (3.33)and useful notationΨ α ± µ = Ψ αµ ± ¯Ψ αµ , F αβs = 12 ( F αβ + F βα ) , F αβa = 12 ( F αβ − F βα ) . (3.34)Following [9] we will refer to the fields appearing in matrix M AB as the open string back-ground fields. This is supersymmetric generalization of Seiberg and Witten open stringmetric, G effµν , because all effective fields contain bilinear combinations of Ω odd fields.When D ⋆ Γ A = (Γ i , Γ ′ α , Γ ′ α , Γ πα , Γ πα ) satisfy the algebra (3.31), where the supermatrix M AB isgiven by the expression M AB = − κG effij − eff ) iγ − eff ) iγ − eff ) α j κ ( f eff ) α γ κ ( f eff ) α γ − δ α δ − eff ) α j − κ ( f eff ) α γ κ ( f eff ) α γ − δ α δ − δ β γ − δ β γ . (3.35)8he open string background fields are defined as G effij = G ij − B ik G kl B lj , (Ψ eff ) α i = 12 Ψ α + i + B ik G kj Ψ α − j , (Ψ eff ) α i = 12 Ψ α − i + B ik G kj Ψ α + j , ( f eff ) α β = ( f a ) α β − Ψ α − i G ij Ψ β − j , ( f eff ) α β = ( f a ) α β − Ψ α + i G ij Ψ β + j , ( f eff ) α β = 12 (cid:16) f α β − f β α (cid:17) − Ψ α − i G ij Ψ β + j . (3.36)From the definition of superdeterminant s det M AB = det( M − M M − M )det M , (3.37)and using the fact that M M − M = 0 , det M = const , (3.38)we obtain from (3.32) s det M AB ∼ det G eff . (3.39)Because we assume that effective metric G eff is nonsingular, we conclude that all con-straints ⋆ Γ A are of the second class. It is easy to check that zero modes, ( θ α − ¯ θ α ) | ,are also of the second class, and consequently, all constraints originating from boundaryconditions, Γ A , are of the second class. Note that the condition s det M AB = 0 is exactlythe same condition as in the bosonic case [6]. Instead to calculate Dirac brackets we prefer to explicitly solve second class constraintsoriginating from boundary conditions. From Γ µ = 0, Γ α = 0 and Γ πα = 0, we obtain x µ ( σ ) = q µ − µν Z σ dσ p ν + 2Θ µα Z σ dσ p α , π µ = p µ , (4.1) θ α ( σ ) = Φ α ( σ ) + 12 ˜ ξ α , π α = p α + ˜ p α , (4.2)¯ θ α ( σ ) = Φ α ( σ ) −
12 ˜ ξ α , ¯ π α = p α − ˜ p α , (4.3)where Φ α ( σ ) = 12 ξ α − Θ µα Z σ dσ p µ − Θ αβ Z σ dσ p β , (4.4)92 ξ α ≡ P s θ α = P s ¯ θ α , ˜ ξ α ≡ P a ( θ α − ¯ θ α ) ,p α ≡ P s π α = P s ¯ π α , ˜ p α ≡ P a π α = − P a ¯ π α , (4.5)and Θ µν = − κ ( G − eff BG − ) µν , Θ µα = 2Θ µν (Ψ eff ) αν − κ G µν ψ α − ν , (4.6)Θ αβ = 12 κ F αβs + 4(Ψ eff ) αµ Θ µν (Ψ eff ) βν − κ Ψ α − µ ( G − BG − ) µν Ψ β − ν + G µν κ h Ψ α − µ (Ψ eff ) βν + Ψ β − µ (Ψ eff ) αν i . (4.7)Using σ -dependent form of boundary conditions (3.22)-(3.24), we get D x i ( σ ) = q i − ij Z σ dσ p j + 4 ℜ (cid:18) Θ iα Z σ dσ p α + Θ iα Z σ dσ p α (cid:19) , π i = p i , (4.8) θ α ( σ ) = Φ α ( σ ) + 12 ˜ ξ α ( σ ) , π α = p α + ˜ p α , (4.9) θ α ( σ ) = Φ α ( σ ) + 12 ˜ ξ α ( σ ) , π α = p α + ˜ p α , (4.10)¯ θ α ( σ ) = Φ α ( σ ) −
12 ˜ ξ α ( σ ) , ¯ π α = p α − ˜ p α , (4.11)¯ θ α ( σ ) = − Φ α ( σ ) + 12 ˜ ξ α ( σ ) , ¯ π α = − p α + ˜ p α , (4.12)where Φ α ( σ ) = 12 ξ α − Θ iα Z σ dσ p i − Θ α β Z σ dσ p β − Θ α β Z σ dσ p β − ⋆ Θ α β Z σ dσ p ∗ β − ⋆ Θ α β Z σ dσ p ∗ β , (4.13)Φ α ( σ ) = 12 ξ α − Θ iα Z σ dσ p i − Θ α β Z σ dσ p β − Θ α β Z σ dσ p β − ⋆ Θ α β Z σ dσ p ∗ β − ⋆ Θ α β Z σ dσ p ∗ β , (4.14)and the coefficients multiplying momenta are of the formΘ ij = − κ ( G − eff BG − ) ij , (4.15)Θ iα = 2Θ ij (Ψ eff ) α j − κ G ij Ψ α − j , Θ iα = 2Θ ij (Ψ eff ) α j − κ G ij Ψ α + j , (4.16)10 α β = 12 κ ( f s ) α β + 4(Ψ eff ) α i Θ ij (Ψ eff ) β j − κ Ψ α − i ( G − BG − ) ij Ψ β − j + G ij κ h Ψ α − i (Ψ eff ) β j + Ψ β − i (Ψ eff ) α j i , (4.17)Θ α β = Θ β α = 14 κ ( f α β + f β α ) + 4(Ψ eff ) α i Θ ij (Ψ eff ) β j − κ Ψ α − i ( G − BG − ) ij Ψ β + j + G ij κ h Ψ α − i (Ψ eff ) β j + Ψ β + i (Ψ eff ) α j i , (4.18) ⋆ Θ α β = 14 κ ( f α β + f ∗ β α ) + 4(Ψ eff ) α i Θ ij (Ψ eff ) ∗ β j − κ Ψ α − i ( G − BG − ) ij Ψ ∗ β − j + G ij κ h Ψ α − i (Ψ eff ) ∗ β j + Ψ ∗ β − i (Ψ eff ) α j i , (4.19) ⋆ Θ α β = ⋆ Θ β α = 14 κ ( f α β + f ∗ β α ) + 4(Ψ eff ) α i Θ ij (Ψ eff ) ∗ β j − κ Ψ α − i ( G − BG − ) ij Ψ ∗ β + j + G ij κ h Ψ α − i (Ψ eff ) ∗ β j + Ψ ∗ β + i (Ψ eff ) α j i . (4.20)The coefficient Θ α β can be obtained from Θ α β after substitution f s → f s , (Ψ eff ) α i → (Ψ eff ) α i and Ψ α − i → Ψ α + i , while ⋆ Θ α β follows from ⋆ Θ α β after substitution f → f ,(Ψ eff ) α i → (Ψ eff ) α i and Ψ α − i → Ψ α + i . Dp -brane world-volume Using the solutions of boundary conditions we will show that Poisson brackets of initial Dp -brane variables are nonzero. D -brane From basic Poisson algebra { x µ ( σ ) , π ν (¯ σ ) } = δ µν δ ( σ − ¯ σ ) , (5.1)and definitions (3.27)-(3.28) we obtain { q µ ( σ ) , p ν (¯ σ ) } = δ µν δ s ( σ , ¯ σ ) , { ˜ q µ ( σ ) , ˜ p ν (¯ σ ) } = δ µν δ a ( σ , ¯ σ ) , (5.2)where δ s ( σ, ¯ σ ) = 12 [ δ ( σ − ¯ σ ) + δ ( σ + ¯ σ )] , δ a ( σ, ¯ σ ) = 12 [ δ ( σ − ¯ σ ) − δ ( σ + ¯ σ )] , (5.3)11re symmetric and antisymmetric delta functions, respectively. Using basic Poisson algebraof fermionic variables { θ α ( σ ) , π β (¯ σ ) } = { ¯ θ α ( σ ) , ¯ π β (¯ σ ) } = − δ αβ δ ( σ − ¯ σ ) , (5.4)we have (cid:8) P s θ α ( σ ) + P s ¯ θ α ( σ ) , P s π β (¯ σ ) + P s ¯ π β (¯ σ ) (cid:9) = − δ αβ δ s ( σ , ¯ σ ) , (5.5)which gives { ξ α ( σ ) , p β (¯ σ ) } = − δ αβ δ s ( σ , ¯ σ ) . (5.6)Similarly we obtain n ˜ ξ α ( σ ) , ˜ p β (¯ σ ) o = − δ αβ δ a ( σ , ¯ σ ) . (5.7)Therefore, the momenta p µ , ˜ p µ , p α and ˜ p α are canonically conjugated to the coordinates q µ , ˜ q µ , ξ α and ˜ ξ α , respectively.Using the solutions of constraints (4.1)-(4.3), we get the noncommutativity relations { x µ ( σ ) , x ν (¯ σ ) } = 2Θ µν θ ( σ + ¯ σ ) , (5.8) { x µ ( σ ) , θ α (¯ σ ) } = − Θ µα θ ( σ + ¯ σ ) , { θ α ( σ ) , ¯ θ β (¯ σ ) } = 12 Θ αβ θ ( σ + ¯ σ ) , (5.9)where θ ( x ) = x = 01 / < x < π . x = 2 π (5.10)After introducing center of mass variables A ( σ ) = A cm + A ( σ ) , A cm = 1 π Z π dσA ( σ ) , (5.11)where A ( σ ) is arbitrary variable, we obtain { x µ ( σ ) , x ν (¯ σ ) } = Θ µν ∆( σ + ¯ σ ) , (5.12) { x µ ( σ ) , θ α (¯ σ ) } = −
12 Θ µα ∆( σ + ¯ σ ) , { θ α ( σ ) , ¯ θ β (¯ σ ) } = 14 Θ αβ ∆( σ + ¯ σ ) . (5.13)The function ∆( σ + ¯ σ ) is nonzero only at string endpoints∆( x ) = 2 θ ( x ) − − x = 00 if 0 < x < π , x = 2 π (5.14)and we conclude that interior of the string is commutative, while string endpoints arenoncommutative. 12 .2 D -brane Applying the same procedure as in the case of D D { x i ( σ ) , x j (¯ σ ) } = Θ ij ∆( σ + ¯ σ ) , (5.15) { x i ( σ ) , θ α (¯ σ ) } = −
12 Θ iα ∆( σ + ¯ σ ) , { x i ( σ ) , θ α (¯ σ ) } = −
12 Θ iα ∆( σ + ¯ σ ) , (5.16) { θ α ( σ ) , ¯ θ β (¯ σ ) } = 14 Θ α β ∆( σ + ¯ σ ) , { θ α ( σ ) , ¯ θ β (¯ σ ) } = −
14 Θ α β ∆( σ + ¯ σ ) , (5.17) { θ α ( σ ) , ¯ θ β (¯ σ ) } = −
14 Θ α β ∆( σ + ¯ σ ) . (5.18)The parameters multiplying complex conjugated momenta denoted by star are absentin noncommutativity relations, because the solutions for initial fermionic coordinates donot depend on complex conjugated effective coordinates.On the solutions of the boundary conditions original string variables depend both oneffective coordinates and effective momenta, and that is a source of noncommutativity. Inthe supersymmetric case the presence of Ω odd fields B µν , Ψ α − µ and F αβs leads to noncom-mutativity of the supercoordinates. Nontrivial B µν leads to nonzero of all noncommutativeparameters, Θ µν , Θ µα and Θ αβ . If only Ψ α − µ is nontrivial, we have Θ µν = 0, but Θ µα andΘ αβ are nonzero. Finally, if only F αβs is nontrivial then Θ µν = 0 and Θ µα = 0, and onlyΘ αβ is nonzero. The last case corresponds to the noncommutativity relations used in [11],where bosonic variables are commutative. This discussion is the same for D α ± µ = Ψ αµ ± ¯Ψ αµ → Ψ α ± µ = Ψ αµ ∓ ( ⋆ Γ ¯Ψ µ ) α ,F αβs = 12 ( F αβ + F βα ) → F αβs = − h ( F ⋆ Γ) αβ + ( F ⋆ Γ) βα i . (5.19) Here we will explicitly show that noncommutativity relations of D N = 1 supersymmetry transformations. Because of therelation between D D D N = 2 supersymmetry withparameters ǫ and ¯ ǫ . The supersymmetry transformations of the variables x µ , θ α and ¯ θ α [12] are δx µ = ¯ ǫ α Γ µαβ θ β + ǫ α Γ µαβ ¯ θ β , δθ α = ǫ α , δ ¯ θ α = ¯ ǫ α , (6.1)13hile the transformation rules of constant background fields are δG µν = ǫ α + Γ [ µ αβ Ψ β + ν ] − ǫ α − Γ [ µ αβ Ψ β − ν ] , δB µν = ǫ α + Γ [ µ αβ Ψ β − ν ] − ǫ α − Γ [ µ αβ Ψ β + ν ] , (6.2) δ Ψ α + µ = − ǫ β − Γ µ βγ F γαs − ǫ β + Γ µ βγ F γαa , δ Ψ α − µ = 116 ǫ β + Γ µ βγ F γαs + 116 ǫ β − Γ µ βγ F γαa , (6.3) δA (0) = 0 , δA (2) µν = − ǫ α + Γ [ µ αβ Ψ β + ν ] − ǫ α − Γ [ µ αβ Ψ β − ν ] + A (0) δB µν , (6.4) δA (4) µνρσ = 2 ǫ α + Γ [ µνρ αβ Ψ β − σ ] + 2 ǫ α − Γ [ µνρ αβ Ψ β + σ ] + 6 A (2)[ µν δB ρσ ] . (6.5)Here we used notation ǫ α ± = ǫ α ± ¯ ǫ α = const. , Γ µ µ ...µ k ≡ Γ [ µ Γ µ . . . Γ µ k ] , (6.6)and [ ] in the subscripts of the fields mean antisymmetrization of space-time indices be-tween brackets. The potentials A (0) and A (4) µνρσ correspond to the symmetric part of F αβ F αβs = 12 ( F αβ + F βα ) , (6.7)and A (2) µν to antisymmetric one F αβa = 12 ( F αβ − F βα ) . (6.8)More about connection between two descriptions of R-R sector is given in Ref.[4] andAppendix B of Ref.[8].From the solution of boundary conditions (4.2)-(4.3) and supersymmetry transforma-tions (6.1), we have δθ α ( σ ) = δ Φ α ( σ ) + 12 δ ˜ ξ α ( σ ) = ǫ α , (6.9) δ ¯ θ α ( σ ) = δ Φ α ( σ ) − δ ˜ ξ α ( σ ) = ¯ ǫ α , (6.10)which gives δ Φ α ( σ ) = 12 ǫ α + , δ ˜ ξ α ( σ ) = ǫ α − . (6.11)From the boundary conditions (3.8), with the help of supersymmetry transformations(6.1), it holds ǫ α − = 0 . (6.12)The starting N = 2 supersymmetry transformations (6.1), on the solution of boundaryconditions, reduces to N = 1 supersymmetry transformations δx µ ( σ ) = ǫ α + Γ µαβ Φ β ( σ ) = ǫ α + Γ µαβ θ β ( σ ) − ǫ α + Γ µαβ ˜ ξ β ( σ ) , δθ α = δ ¯ θ α = 12 ǫ α + , (6.13)14hich gives δq µ = 12 ǫ α + Γ µαβ ξ β , δξ α = ǫ α + , δ ˜ ξ α = 0 . (6.14)From Ref.[13] we read the supersymmetry transformations for the momenta δp α = 12 ǫ β + Γ µβα p µ , δp µ = 0 . (6.15)The truncation from N = 2 to N = 1 supersymmetry we can realize omitting trans-formations for G µν , Ψ + µ and F a [12]. The rest fields make N = 1 supermultiplet withtransformation rules δB µν = ǫ α + Γ [ µ αβ Ψ β − ν ] , δ Ψ α − µ = 116 ǫ β + Γ µ βγ F γαs , δF αβs = 0 . (6.16)Using N = 1 SUSY transformations (6.13)-(6.15), we can find the supersymmetrictransformations of the coefficients Θ µν , Θ µα and Θ αβ multiplying the momenta in thesolutions of boundary conditions. From δx µ ( σ ) = 12 ǫ α + Γ µαβ ξ β − δ Θ µν Z σ dσ p ν − µν Z σ dσ δp ν , (6.17)+ 2 δ Θ µα Z σ dσ p α + 2Θ µα Z σ dσ δp α = ǫ α + Γ µαβ θ ( σ ) β − ǫ α + Γ µαβ ˜ ξ ( σ ) β ,δθ α ( σ ) = 12 ǫ α + − δ Θ µα Z σ dσ p µ − Θ µα Z σ dσ δp µ − δ Θ αβ Z σ dσ p β − Θ αβ Z σ dσ δp β = 12 ǫ α + , (6.18)we obtain global N = 1 SUSY transformations of the background fields δ Θ µν = ǫ α + Γ [ µαβ Θ ν ] β , δ Θ µα = − ǫ β + Γ µβγ Θ γα , δ Θ αβ = 0 . (6.19)Consequently, these fields are components of N = 1 supermultiplet. The coefficients,Θ µν , Θ µα and Θ αβ , are the background fields of the T-dual theory. This explains thefact that their SUSY transformations have the same form as the transformations of thecorresponding dual partners B µν , Ψ α − µ and F αβs (6.16).Using N = 1 supersymmetry transformations of SUSY coordinates (6.13) and back-ground fields (6.19), we can easily prove that noncommutativity relations, (5.8) and (5.9),are connected by supersymmetry transformations. The SUSY transformation of (5.8) ǫ α + Γ [ µαβ n x ν ] , θ β o = − ǫ α + Γ [ µαβ Θ ν ] β θ ( σ + ¯ σ ) , (6.20)produces the first relation in (5.9). Similarly, SUSY transformation of the first relation in(5.9) ǫ β + Γ µβγ { θ γ , θ α } = 12 ǫ β + Γ µβγ Θ γα θ ( σ + ¯ σ ) , (6.21)produces the second relation in (5.9). 15 Concluding remarks
In this paper we considered noncommutativity properties and related supersymmetrytransformations of D D G eff all constraints are of thesecond classand. We can solve them and obtain the initial coordinates x µ , θ α and ¯ θ α interms of effective ones, q µ , ξ α and ˜ ξ α (momenta independent parts of the solutions forinitial supercoordinates x µ , θ α and ¯ θ α ) and momenta p µ and p α (canonically conjugatedto q µ and ξ α ).The fact that original string variables depend both on effective coordinates and effectivemomenta is a source of noncommutativity (5.12)-(5.13). The solution for initial variablesdo not depend on momenta ˜ p α (canonically conjugated to ˜ ξ α ). So, Ω odd variables,denoted with tilde, do not contribute to noncommutativity relations. Absence of thefermionic coordinates in the solution for x µ implies that Poisson bracket { x µ , x ν } is thesame as in pure bosonic case. Similar conclusions are valid for D α = ¯Ψ αµ , the noncommutatativity relations(5.12)-(5.13) correspond to the relations of Ref.[10].The result of the present paper can be considered as a supersymmetric generalizationof the result obtained for bosonic string [9]. Beside B µν , its superpartners Ψ α − µ and F αβs are also a source of noncommutativity. For noncommutativity of bosonic coordinates it isnecessary to have nontrivial B µν . Noncommutativity of bosonic and fermionic coordinatescan be caused by both B µν and Ψ α − µ , while noncommutativity of two fermionic coordinatescan be caused by all components of noncommutative supermultiplet, B µν , Ψ α − µ and F αβs .Note that fermionic boundary conditions split N = 2 supermultiplet (consisting ofbackground fields G µν , B µν , Ψ α + µ , Ψ α − µ and F αβ ) into two N = 1 supermultiplets. One, Ωeven ( G µν , Ψ α + µ , F αβa ), represents background fields of type I theory, and the second one,Ω odd ( B µν , Ψ α − µ , F αβs ) is source of noncommutativity of supercoordinates ( x µ , θ α ). References [1] J. Polchinski,
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