T-duality and noncommutativity in type IIB superstring theory

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1 T-duality and noncommutativity in type IIB superstring theory B. Nikolić and B. Sazdović Institute of Physics, 111 Belgrade, P.O.Box 57, SERBIA Abstract In this article we investigate noncommutativity of D9-brane worldvolume embedded in space-time of type IIB superstring theory. On the solution of boundary conditions, the initial coordinates become noncommutative. We show that noncommutativity relations are connected by N = 1 supersymmetry transformations and noncommutativity parameters are components of N = 1 supermultiplet. In addition, we show that background fields of the effective theory (initial theory on the solution of boundary conditions) and noncommutativity parameters represent the background fields of the T-dual theory. 1. Introduction Dp-branes with odd value of p are stable in type IIB superstring theory. As a particular choice, besides D9-brane, we can embed D5-brane in ten dimensional space-time of the type IIB theory in pure spinor formulation (up to the quadratic terms)[1, 2]. In the present paper we investigate the noncommutativity of D9-brane using canonical approach. The case of D5-brane has been considered in Ref.[3]. Also we investigate the supersymmetry of noncommutativity relations [4]. We choose Neumann boundary conditions for all bosonic coordinates x µ. The boundary condition for fermionic coordinates, (θ α θ α ) π = produces additional one for their canonically conjugated momenta, (π α π α ) π =. We treat boundary conditions as canonical constraints [5, 6, 7, 3]. Using their consistency conditions, we rewrite them in compact σ-dependent form. We find that they are of the second class and on these solutions, we obtain initial coordinates in terms of effective coordinates and momenta. Presence of the momenta is source of noncommutativity. Noncommutativity relations are consistent with N = 1 supersymmetry transformations. Work supported in part by the Serbian Ministry of Science and Technological Development, under contract No address: bnikolic@ipb.ac.rs, sazdovic@ipb.ac.rs 259

2 26 B. Nikolić, B. Sazdović We obtain that noncommutativity parameters contain only odd powers of background fields antisymmetric under world-sheet parity transformation Ω : σ σ and form one N = 1 supermultiplet. This result represents a supersymmetric generalization of the result obtained by Seiberg and Witten [8]. We also show that effective background fields and noncommutative parameters are the background fields of T-dual theory. 2. Type IIB superstring theory and canonical analysis We will investigate pure spinor formulation [2, 9, 7, 3] of type IIB theory, neglecting ghost terms and keeping quadratic ones S = κ d 2 ξ + x µ Π +µν x ν (1) Σ [ + d 2 ξ π α (θ α + Ψ α µx µ ) + + ( θ α + Ψ α µx µ ) π α + 1 ] 2κ π αf αβ π β, Σ where Π ±µν = B µν ± 1 2 G µν and ± = τ ± σ. All background fields are constant. The space-time coordinates are labeled by x µ (µ =, 1, 2,..., 9) and the fermionic ones by same chirality Majorana-Weyl spinors θ α and θ α. The variables π α and π α are canonically conjugated to the coordinates θ α and θ α, respectively. Canonical Hamiltonian is of the form H c = dσh c, H c = T T +, T ± = t ± τ ±, (2) where t ± = 1 4κ Gµν I ±µ I ±ν, I ±µ = π µ + 2κΠ ±µν x ν + π α Ψ α µ Ψ α µ π α, (3) τ + = (θ α + Ψ α µx µ )π α 1 4κ π αf αβ π β, τ = ( θ α + Ψ α µx µ ) π α + 1 4κ π αf αβ π β. Varying Hamiltonian H c, we obtain δh c = δh (R) c [γ () µ δx µ + π α δθ α + δ θ α π α ] π, (4) where δh c (R) is regular term without τ and σ derivatives of supercoordinates and supermomenta variations and γ () µ = Π +µ ν I ν + Π µ ν I +ν + π α Ψ α µ + Ψ α µ π α. (5) As a time translation generator Hamiltonian has to be differentiable with respect to coordinates and their canonically conjugated momenta which produces [ γ µ () δx µ + π α δθ α + δ θ α π ] π α =. (6)

3 T-duality and noncommutativity in type IIB superstring theory 261 Embedding D9-brane means that we impose Neumann boundary conditions for x µ coordinates, γ µ () π =. Fermionic boundary condition (θα θ α ) π = preserves half of the initial N = 2 supersymmetry and it produces additional boundary condition for fermionic momenta (π α π α ) π =. Using standard Poisson algebra, the consistency procedure for γ () µ produces an infinite set of constraints γ (n) µ {H c, γ (n 1) µ } (n = 1, 2, 3,... ), (7) which can be rewritten at σ = in the compact σ-dependent form Γ µ n= σ n n! γ(n) µ =Π +µ ν I ν (σ) + Π µ ν I +ν ( σ) + π α ( σ)ψ α µ + Ψ α µ π α (σ). Similarly, from conditions (θ α θ α ) = and (π α π α ) =, we get Γ α (σ) = Θ α (σ) Θ α (σ), Γ π α(σ) π α ( σ) π α (σ), (9) (8) where Θ α (σ) = θ α ( σ) Ψ α µ q µ (σ) 1 2κ F αβ + 1 2κ Gµν Ψ α µ dσ 1 P s (I +ν + I ν ), dσ 1 P s π β (1) Θ α (σ) = θ α (σ) + Ψ α µ q µ (σ) + 1 2κ F βα dσ 1 P s π β (11) + 1 2κ Gµν Ψα µ dσ 1 P s (I +ν + I ν ). We introduced new variables, symmetric and antisymmetric under worldsheet parity transformation Ω : σ σ. For bosonic variables we use the standard notation q µ (σ) = P s x µ (σ), q µ (σ) = P a x µ (σ), p µ (σ) = P s π µ (σ), p µ (σ) = P a π µ (σ), while for fermionic ones we use the projectors on σ symmetric and antisymmetric parts P s = 1 2 (1 + Ω), P a = 1 (1 Ω). (12) 2 From {H c, Γ A } = Γ A, Γ A = (Γ µ, Γ α, Γ π α), it follows that there are no more constraints in the theory. For practical reasons we will find the

4 262 B. Nikolić, B. Sazdović algebra of the constraints Γ A = (Γ µ, Γ α, Γ π α) instead of Γ A. It has the form { Γ A, Γ B } = M AB δ, (13) where the supermatrix M AB is given by the expression M AB = κgeff µν 2(Ψ eff ) γ µ 2(Ψ eff ) α 1 ν κ F αγ eff 2δ α δ. (14) 2δ γ β Here we obtain effective background fields G eff µν = G µν 4B µρ G ρλ B λν, (Ψ eff ) α µ = 1 2 Ψα +µ + B µρ G ρν Ψ α ν, F αβ eff = F αβ a Ψ α µg µν Ψ β ν, (15) where we introduced useful notation Ψ α ±µ = Ψ α µ ± Ψ α µ, F αβ s = 1 2 (F αβ + F βα ), F αβ a = 1 2 (F αβ F βα ). (16) This is supersymmetric generalization of Seiberg and Witten open string metric, G eff µν, because all effective fields contain bilinear combinations of Ω odd fields. The superdeterminant s det M AB is proportional to det G eff. Because we assume that effective metric G eff is nonsingular, we conclude that all constraints Γ A are of the second class. 3. Solution of the constraints and noncommutativity From Γ µ =, Γ α = and Γ π α =, we obtain x µ (σ) = q µ 2Θ µν dσ 1 p ν + 2Θ µα dσ 1 p α, π µ = p µ, (17) θ α (σ) = Φ α (σ) ξ α, π α = p α + p α, (18) θ α (σ) = Φ α (σ) 1 2 ξ α, π α = p α p α, (19) where Φ α (σ) = 1 2 ξα Θ µα dσ 1 p µ Θ αβ dσ 1 p β, ξα P a (θ α θ α ),

5 T-duality and noncommutativity in type IIB superstring theory 263 and 1 2 ξα P s θ α = P s θα, p α P s π α = P s π α, p α P a π α = P a π α, Θ µν = 1 κ (G 1 eff BG 1 ) µν, Θ µα = 2Θ µν (Ψ eff ) α ν 1 2κ Gµν ψ α ν, (2) Θ αβ = 1 2κ F s αβ + 4(Ψ eff ) α µθ µν (Ψ eff ) β ν 1 κ Ψα µ(g 1 BG 1 ) µν Ψ β ν + Gµν κ [ ] Ψ α µ(ψ eff ) β ν + Ψ β µ (Ψ eff ) α ν. (21) Using the solutions of the constraints (17)-(19) and the basic Poisson algebra, after removing the center of mass variables, we get the noncommutativity relations {x µ (σ), x ν ( σ)} = Θ µν (σ + σ), (22) {x µ (σ), θ α ( σ)} = 1 2 Θµα (σ + σ), {θ α (σ), θ β ( σ)} = 1 4 Θαβ (σ + σ). The function (σ + σ) is nonzero only at string endpoints 1 if x = (x) = if < x < 2π, 1 if x = 2π (23) (24) and we conclude that interior of the string is commutative, while string endpoints are noncommutative. 4. Supersymmetry of noncommutativity relations The action of initial theory (1) is invariant under global N = 2 supersymmetry with transformations of supercoordinates and background fields δx µ = ɛ α Γ µ αβ θβ + ɛ α Γ µ αβ θ β, δθ α = ɛ α, δ θ α = ɛ α, (25) δg µν = ɛ α +Γ [µ αβ Ψ β +ν] ɛα Γ [µ αβ Ψ β ν], δb µν = ɛ α +Γ [µ αβ Ψ β ν] ɛα Γ [µ αβ Ψ β +ν], δψ α +µ = 1 16 ɛβ Γ µ βγf γα s 1 16 ɛβ + Γ µ βγf γα a, δψ α µ = 1 16 ɛβ + Γ µ βγf γα s ɛβ Γ µ βγf γα a, (26)

6 264 B. Nikolić, B. Sazdović δa () =, δa (2) µν = ɛ α +Γ [µ αβ Ψ β +ν] ɛα Γ [µ αβ Ψ β ν] + A() δb µν, δa (4) µνρσ = 2ɛ α +Γ [µνρ αβ Ψ β σ] + 2ɛα Γ [µνρ αβ Ψ β +σ] + 6A(2) [µν δb ρσ]. Here we used notation ɛ α ± = ɛ α ± ɛ α = const., Γ µ1 µ 2...µ k Γ [µ1 Γ µ2... Γ µk ], (27) and [ ] in the subscripts of the fields mean antisymmetrization of space-time indices between brackets. The connection between potentials A (), A (2) µν and A (4) µνρσ and RR field strength F αβ is given in [7]. The truncation from N = 2 to N = 1 supersymmetry we can realize omitting transformations for G µν, Ψ +µ and F a [1]. The rest fields make N = 1 supermultiplet with transformation rules δb µν = ɛ α +Γ [µ αβ Ψ β ν], δψα µ = 1 16 ɛβ + Γ µ βγf γα s, δf αβ s =. (28) Now we can find the supersymmetric transformations of the coefficients, Θ µν, Θ µα and Θ αβ, multiplying the momenta in the solutions of boundary conditions. From δx µ (σ) = 1 2 ɛα +Γ µ αβ ξβ 2δΘ µν dσ 1 p ν 2Θ µν dσ 1 δp ν, (29) + 2δΘ µα dσ 1 p α + 2Θ µα dσ 1 δp α = ɛ α +Γ µ αβ θ(σ)β 1 2 ɛα +Γ µ ξ(σ) β αβ, δθ α (σ) = 1 2 ɛα + δθ µα dσ 1 p µ Θ µα dσ 1 δp µ δθ αβ dσ 1 p β Θ αβ dσ 1 δp β = 1 2 ɛα +, (3) we obtain global N = 1 SUSY transformations of the background fields δθ µν = ɛ α +Γ [µ αβ Θν]β, δθ µα = 1 2 ɛβ + Γµ βγ Θγα, δθ αβ =. (31) Consequently, these fields are components of N = 1 supermultiplet. Using N = 1 supersymmetry transformations of SUSY coordinates (25) and background fields (31), we can easily prove that noncommutativity relations are connected by supersymmetry transformations. The SUSY transformation of (22) ɛ α +Γ [µ αβ { x ν], θ β} = 1 2 ɛα +Γ [µ αβ Θν]β (σ + σ), (32)

7 T-duality and noncommutativity in type IIB superstring theory 265 produces the first relation in (23). Similarly, SUSY transformation of the first relation in (23) ɛ β + Γµ βγ {θγ, θ α } = 1 4 ɛβ + Γµ βγ Θγα (σ + σ), (33) produces the second relation in (23). 5. Type IIB theory and T-duality We suppose that action (1) has a global shift symmetry in all x µ directions. So, we have to introduce gauge fields v µ ± and make a change ± x µ D ± x µ ± x µ + v µ ±. (34) For the fields v µ ± we introduce additional term in the action S gauge (y, v ± ) = 1 2 κ d 2 ξy µ ( + v µ v µ + ), (35) Σ which produces vanishing of the field strength + v µ v µ + if we vary with respect to the Lagrange multipliers y µ. The full action has the form S (x, y, v ± ) = S(D ± x) + S gauge (y, v ± ). (36) Let us note that on the equations of motion for y µ we have v µ ± = ±x µ and the original dynamics survives unchanged. Now we can fix x µ to zero and obtain the action quadratic in the fields v µ ± which can be integrated out classically. On the equations of motion for v µ ± we obtain expressions for these gauge fields in terms of y µ and momenta ( ) v µ 2 + = 2 κ π αψ α ν + + y ν (G 1 eff Π G 1 ) νµ, (37) ( ) v µ 2 = 2(G 1 eff Π G 1 ) µν κ Ψ α ν π α + y ν. (38) Substituting them in the action S we obtain the dual action from which we read the dual background fields G µν = (G 1 eff )µν, B µν = κθ µν = (G 1 eff BG 1 ) µν, (39) [ [ Ψ µα = 2κΘ µν (G 1 eff )µν] Ψ α ν, Ψµα = 2κΘ µν + (G 1 eff )µν] Ψα ν, F αβ s = 2κΘ αβ, It is easy to see that F αβ a (4) = F αβ eff + 4(Ψ eff ) α µ(g 1 eff )µν (Ψ eff ) α ν. (41) Ψ µα + = 2κΘµα, Ψ µα = 2(G 1 eff )µν (Ψ eff ) α µ. (42)

8 266 B. Nikolić, B. Sazdović 6. Concluding remarks In this paper we considered noncommutativity properties and related supersymmetry transformations of D9 brane in type IIB superstring theory. We used the pure spinor formulation of the theory introduced in Refs.[2, 9]. We treated all boundary conditions at string endpoints as canonical constraints. Solving the second class constraints we obtain the initial coordinates x µ, θ α and θ α in terms of effective ones, q µ, ξ α and ξ α and momenta p µ and p α. The presence of momenta is a source of noncommutativity (22)-(23). The result of the present paper can be considered as a supersymmetric generalization of the result obtained for bosonic string [8]. Beside B µν, its superpartners Ψ α µ and Fs αβ are also a source of noncommutativity. In special case, when Ψ α = Ψ α µ, the noncommutativity relations (22)-(23) correspond to the relations of Ref.[9]. The N = 2 supermultiplet of T-dual theory is split in two N = 1 supermultiplets: the first, Ω even one, (G eff µν, (Ψ eff ) α µ, F αβ eff ), represents the background fields of the effective theory (initial theory on the solution of the boundary conditions) [7], while the second, Ω odd one, (Θ µν, Θ µα, Θ αβ ), contains the noncommutativity parameters. References [1] J. Polchinski, String theory - Volume II, Cambridge University Press, 1998; K. Becker, M. Becker and J. H. Schwarz, String Theory and M-Theory - A Modern Introduction, Cambridge University Press, 27. [2] N. Berkovits and P. Howe, Nucl. Phys. B635 (22) 75; P. A. Grassi, G. Policastro and P. van Nieuwenhuizen, JHEP 11 (22) 4; P. A. Grassi, G. Policastro and P. van Nieuwenhuizen, Adv. Theor. Math. Phys. 7 (23) 499; P. A. Grassi, G. Policastro and P. van Nieuwenhuizen, Phys. Lett. B553 (23) 96. [3] B. Nikolić and B. Sazdović, Nucl. Phys. B 836 (21) [4] B. Nikolić and B. Sazdović, JHEP 8 (21) 37. [5] F. Ardalan, H. Arfaei, M. M. Sheikh-Jabbari, Nucl. Phys. B576, 578 (2); C. S. Chu, P. M. Ho, Nucl. Phys. B568, 447 (2); T. Lee, Phys. Rev. D62 (2) [6] B. Sazdović, Eur. Phys. J C44 (25) 599; B. Nikolić and B. Sazdović, Phys. Rev. D74 (26) 4524; B. Nikolić and B. Sazdović, Phys. Rev. D75 (27) 8511; B. Nikolić and B. Sazdović, Adv. Theor. Math. Phys. 14 (21) 1. [7] B. Nikolić and B. Sazdović, Phys. Lett. B666 (28) 4. [8] N. Seiberg and E. Witten, JHEP 9 (1999) 32. [9] J. de Boer, P. A. Grassi and P. van Nieuwenhuizen, Phys. Lett. B574 (23) 98. [1] F. Riccioni, Phys. Lett. B56 (23) 223; N. Berkovits, ICTP Lectures on Covariant Quantization of the Superstring, hep-th/2959.