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B. Sazdovic - Noncommutativity and T-duality

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The SEENET-MTP Workshop BW2011
Particle Physics from TeV to Plank Scale
28 August – 1 September 2011, Donji Milanovac, Serbia

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B. Sazdovic - Noncommutativity and T-duality

  1. 1. Noncommutativity and T-duality Lj. Davidovi´, B. Nikoli´ and B. Sazdovi´ c c c Institute of Physics, Belgrade, Serbia• We will discuss relation between – Open string parameters Gef f (G, B) and θ µν (G, B) µν – and T-dual background fields Gµν (G, B) and B µν (G, B) as functions of the initial background fields: metric tensor Gµν and Kalb-Ramond field Bµν• Noncommutativity of Dp-brane world volume The quantization of the open bosonic string whose ends are attached to the Dp-brane leads to noncommutativity of Dp-brane world volume The noncommutativity parameter θ µν (G, B)Noncommutativity and T-duality BSW 2011
  2. 2. • Effective theory On the solution of boundary conditions the initial theory turns to the effective one with effective metric tensor Gef f (G, B) and vanishing effective Kalb-Ramond field µν• T-duality T-duality in presence of background fields leads T-dual background fields Gµν (G, B) and B µν (G, B)• We will extend these investigations considering 1. II B superstring theory instead of bosonic one – Bosonic duality – Fermionic duality 2. ”Weakly curved background” Bµν [x] = bµν + 1 Bµνρxρ 3 instead of the flat one Bµν = bµν = const.Noncommutativity and T-duality BSW 2011
  3. 3. 1 The actiondescribing the open string propagation in curved background 2 g αβ αβ µ νS=κ d ξ −g Gµν (x)+ √ Bµν (x) ∂αx ∂β x , Σ 2 −g• xµ(ξ), µ = 0, 1, ..., D − 1 the coordinates of the D-dimentional space-time• ξ α(ξ 0 = τ, ξ 1 = σ) parametrize 2-dim world-sheet• gαβ (ξ) intrinsic world-sheet metric (g = detgαβ )• background fields – Gµν (x) space-time metric – Bµν (x) Kalb-Ramond antisymmetric fieldNoncommutativity and T-duality BSW 2011
  4. 4. 2 Action principle for stringEvolution from the initial to final configuration is such that theaction is stationary τf σfS= τi dτ σi dσL(xµ, xµ, x µ, gαβ ) ˙ ∂L ∂L ∂L µ δS = dτ dσ − ∂τ µ − ∂σ µ δx ∂xµ ∂x ˙ ∂x ˙ ∂L µ σ=π + dτ δx ∂ xµ ˙ σ=0From the action principle we get1) equation of motionxµ = x¨ µ − 2B µ xν x ρ, νρ ˙Bµνρ = ∂µBνρ + ∂ν Bρµ + ∂ρBµν is field strength2) boundary condition µ µγ0 δxµ = 0, γ0 ≡ ∂L ∂ xµ ˙ = x µ − 2(G−1B)µν xν ˙ σ=0,πNoncommutativity and T-duality BSW 2011
  5. 5. 3 The Boundary conditions• The closed string fulfills the boundary condition because xµ(0) = xµ(π)• For the open string we can impose 1) Neumann boundary condition δxµ , δxµ 0 π are arbitrary i.e. string end-points can move freely 0 0 γµ = 0 , γµ =0 σ=0 σ=π 2) Dirichlet boundary condition δxµ = 0, δxµ =0 σ=0 σ=π The edges of the string are fixedNoncommutativity and T-duality BSW 2011
  6. 6. 4 Noncommutativity and effective theory bosonic open string in flat space-time I• We impose Neumann boundary conditions• We treat boundary conditions as constraints 0• Constraint γµ must be conserved in time 1 ˙0 – Secundary constraint γµ = γµ n ˙ n−1 – Infinite set of constraints γµ = γµ , (n = 1, 2, · · ·) – Two σ -dependent constraints ∞ σn µ Γµ(σ) ≡ n=0 (n)! γn σ=0 ¯ ∞ (σ−π)n µ Γµ(σ) ≡ n=0 (n)! γn σ=π• 2π -periodicity xµ(σ) = xµ(σ + 2π) solve constraint at σ = π ¯ Γµ(σ) = 0 → Γµ(σ) = 0Noncommutativity and T-duality BSW 2011
  7. 7. 5 Noncommutativity and effective theory bosonic open string in flat space-time II• Solving the constraints – In canonical formalism {Γµ(σ), Γν (¯ )} = −κGE δ (σ − σ ) σ µν ¯ For GE = 0 µν Γµ(σ) are SSC – Introduce world-sheet parity Ω Ω : σ → −σ , Ωxµ(σ) → xµ(−σ) and new variables q µ = 1 (1 + Ω)xµ 2 q µ = 1 (1 − Ω)xµ ¯ 2 – Solve Ω odd parts in terms of Ω even ∗ q = f1(q, p), ¯ p = f2(q, p) ¯ ∗ xµ = q µ − 2θ µν dσ1pν , π µ = pµNoncommutativity and T-duality BSW 2011
  8. 8. 5• Effective action and background fields – S ef f = κ d2ξ 1 η αβ GE ∂αq µ∂β q ν 2 µν – Gµν → Gef f = GE , µν µν Bµν → Bµνf = 0 efGE ≡ [G − 4BG−1B]µν µνeffective metric• Noncommuatativity   −1 σ, σ = 0 ¯ {X µ(σ), X ν (¯ )} = θ µν σ 1 σ, σ = π ¯ .  0 otherwise θ µν ≡ − κ (G−1BG−1)µν 2 Enon-commutativity parameterNoncommutativity and T-duality BSW 2011
  9. 9. 6 T0-duality of closed string– trivial background• Background: – One spatial dimension is curled up into circle – Remaining dimensions are described as Minkowski space-time – All others background fields vanish, Bµν = 0, Φ = 0• – x25(σ + π) = x25(σ) + 2πRm, (m ∈ Z) – m– winding number• Consequences of compactification: n – Momentum along circle is quantized, p = R (n ∈ Z), Lost some states – New states that wrap around circle arise, winding states Gained some statesNoncommutativity and T-duality BSW 2011
  10. 10. 7 Surprising symmetry as stringy property• Mass square of the states n2 m2 R 2 M2 = R2 + + contributions from oscilators α2• – Complementary behavior of momentum and winding states – M 2(R, n, m) = M 2( α , m, n) R – R ←→ α ˜ ≡ R, n ←→ m, ˜ R — Dual radius R• T0 duality for closed string Compactification with radius R is physically indistinguishable from ˜ Compactification with radius R = α R• T0 dual coordinate – Equation of motion ∂+∂−x = 0 =⇒ x = x+(τ + σ) + x−(τ − σ) – T0 dual coordinate x ≡ x+(τ + σ) − x−(τ − σ) ˜Noncommutativity and T-duality BSW 2011
  11. 11. 8 T-duality – nontrivial background I• – Background fields are independent of the circular coordinate – We take all coordinate to be circular → Gµν , Bµν = const Toroidal duality of all cordinates• Lagrange multiplier method S[y, v+, v−] = 2 µ ν 2 µ µ κ d ξv+(B + 1 G)µν v− + 2 d ξyµ(∂+v− − ∂−v+), – yµ – Lagrange multiplier• Integration over y returns to the original action µ µ µ ∂+v− − ∂−v+ = 0 ⇒ v± = ∂±xµ• Integrating out vector field v± µ v±(y) = −2[θ µν 1 −1 µν κ (GE ) ]∂± yνGE ≡ [G − 4BG−1B]µν , µν θ µν ≡ − κ (G−1BG−1)µν 2 Eare the open string background fields:effective metric and non-commutativity parameterNoncommutativity and T-duality BSW 2011
  12. 12. 9 T-duality–nontrivial background II• S[∂+y, ∂−y] = 2 d2ξ∂+yµ[θ µν + κ (G−1)µν ]∂+yµ 1 E =κ d2ξ∂+yµ( B + 1 2 G)µν ∂+yµ• Dual background fields B µν = κ θ µν , 2 Gµν = ( κ )2(G−1)µν 2 E• Turn off background fields 2 – Bµν → 0, Gµν → (ηµν , G25,25 = G), κ→ α 2π √ 2π √ – 2πR = 0 ds = G 0 dθ = 2π G ⇒ G = R2 , ˜ G = R2 2 ˜ – G = α G−1 ⇒ RR = α – ∂±x = ±∂±y ⇒ y=x ˜Noncommutativity and T-duality BSW 2011
  13. 13. 10 Relation between T-duality, effective theory and noncommuatativity• T-duality 2 Gµν = α G−1µν , E B µν = α θ µν• Effective theory Gef f = GE µν µν /• Noncommuatativity / θ µν• The same background fields: effective metric – GE ≡ [G − 4BG−1B]µν µν and non-commutativity parameter θ µν ≡ − κ (G−1BG−1)µν 2 ENoncommutativity and T-duality BSW 2011
  14. 14. 11 Type II B theory• Type IIB theory in pure spinor formulation 2 1 mn mn µ ν S=κ d ξ η Gµν + ε Bµν ∂m x ∂n x Σ 2 2 α α µ ¯α ¯α µ π 1 αβ + d ξ −πα ∂− (θ + Ψµ x ) + ∂+ (θ + Ψµ x )¯ α + πα F πβ¯ Σ 2κ• Variables ¯ xµ, θ α and θ α• Background fields – NS-NS Gµν , Bµν – NS-R ¯ Ψα, Ψα , gravitinos µ µ – R-R F αβ ∼ A0, A2, A4, dA4-self dualNoncommutativity and T-duality BSW 2011
  15. 15. 12 Type II B theory Neumann boundary conditions• Boundary conditions π (0) γi δx i ¯α ¯ α + παδθ + δ θ πα =0 0 (0) j j α ¯ α¯ γi = Π+i I−j + Π−i I+j + παΨi + Ψi πα• For bosonic coordinates Neumann boundary conditions (0) π γi 0 =0• Fermionic coordinates preserves N=1 SUSY π π α¯α (θ − θ ) = 0 ⇒ (πα1 − πα1 ) ¯ =0 0 0Noncommutativity and T-duality BSW 2011
  16. 16. 13 Type II B theory Neumann b. c., Effective theory and non-commutativityB.Nikoli´ and B. Sazdovi´, Phys. Lett. B666 (2008) 400 c cB. Nikoli´ and B. Sazdovi´, Nucl. Phys. B 836 (2010) 100 c c• Similar method as in bosonic case• Background fields – Ω even corresponds to Type I E Gµν → Gµν 1 α α 1 α −1 α Ψ+µ → (ΨE )µ = Ψ+µ + (BG Ψ−)µ 2 2 αβ αβ αβ −1 αβ Fa → FE = F − (Ψ−G Ψ−) – Ω odd fields vanish Bµν → 0, Ψ− → 0, Fs → 0• Non-commutativity Ω odd fields are source of non-commutativity µ ν µν {x (σ) , x (¯ )} = 2θ θ(σ + σ ) σ ¯ µ α µα {x (σ) , θ (¯ )} = −θ θ(σ + σ ) σ ¯ α ¯β σ 1 αβ {θ (σ) , θ (¯ )} = θ θ(σ + σ ) ¯ 2Noncommutativity and T-duality BSW 2011
  17. 17. 14 Type II B theory Bosonic TIIBb -dulity• Action has global shift symmetry in bosonic direction Similar method produce dual background fields 2 Gµν = α G−1µν , E B µν = α θ µν ψ− = −2G−1µν (ψE )a aµ E ν aµ ψ+ = 2κθ aµ ab ab a Fa = FE + 4(ψE G−1ψE ) E b ab Fs = 2κθ abNoncommutativity and T-duality BSW 2011
  18. 18. 15 Relation between T-duality, effective theory and noncommuatativity Type II B and bosonic duality• T-duality Effective theory Noncommuatativity Bosonic N bc Ω-symm Ω-antisymm 2• Gµν = α G−1µν E Gef f = GE µν µν / B µν = α θ µν / θ µν aµ• ψ− = −2(G−1ψE )aµ E (ψef f )a = (ψE )a µ µ / aµ ψ+ = 2κθ aµ / θ aµ• Fa = FE + 4(ψE G−1ψE ) Fef f = FE ab ab a E b ab ab / Fs = 2κθ ab ab / θ abNoncommutativity and T-duality BSW 2011
  19. 19. 16 Type II B theory Fermionic TIIBf -dulityB. Nikoli´ and B. Sazdovi´ c cFermionic T-duality and momenta noncommutativityhep-th/1103.4520to be published in Phys. Rev. D• Fermionic T-duality — ¯ Duality with respect to fermionic variables θ a, θ a – Suppose that action has a global shift symmetry in ¯ θ α and θ α directions – Similar procedure as in bosonic case produces TIIBf dual background fields: ¯ Bµν = Bµν + (ΨF −1 ¯ −1 Ψ)µν − (ΨF Ψ)νµ ¯ Gµν = Gµν + 2 (ΨF −1 ¯ −1 Ψ)µν + (ΨF Ψ)νµ Ψαµ = 4(F −1 Ψ)αµ , ¯ ¯ −1 Ψµα = −4(ΨF )µα −1 Fαβ = 16(F )αβNoncommutativity and T-duality BSW 2011
  20. 20. 17 Type II B theory Fermionic TIIBf -dulity and Dirichlet boundary conditions I• T-duality Effective theory and Noncommuatativity BOSONIC ←→ Neumann b.c. for xµ . ¯ SUSY b.c. for θ a, θ a FERMIONIC ←→ ? b.c.• DIRICHLET boundary conditions π π π x µ = 0, θ α = 0, ¯α θ =0 0 0 0• Solve constraints – odd variables are independent – trivial solution for coordinates, non-trivial for momenta µ µ ν 1 ¯α 1 α x (σ) = q (σ) , ˜ πµ = pµ −2κBµν q − Ψµ (ηa )α + (¯a )α Ψµ ˜ ˜ η 2 2 α α 1 θ (σ) = θa (σ) , πα = pα − (¯a )α ˜ η 2 ¯α ¯α 1 θ (σ) = θa (σ) , ˜ πα = pα − (ηa )α ¯ ¯ 2 where −1 β β µ ¯β ¯ β ˜µ −1 (ηa )α ≡ 4κ(F )αβ (θa +Ψµ q ) , ˜ (¯a )α ≡ 4κ(θa +Ψµ q )(F )βα ηNoncommutativity and T-duality BSW 2011
  21. 21. 18 Type II B theory Non-commutativity relations• Non-commutativity relations {Pµ(σ), Pν (¯ )}D = Θµν ∆(σ + σ ) , σ ¯ σ ¯ {Pµ(σ), Pα(¯ )}D = Θµα∆(σ + σ ) , ¯ ¯ σ Pµ(σ), Pα(¯ ) D = Θαµ∆(σ + σ ) , ¯ ¯ σ Pα(σ), Pβ (¯ ) D = Θαβ ∆(σ + σ ) , ¯ ¯ ¯ σ {Pα(σ), Pβ (¯ )}D = Pα(σ), Pβ (¯ ) D = 0 , σ where the noncommutativity parameters are Θµν = 2κ Bµν , ¯ µα = κ Ψµα Θ ¯ 2 κ κ Θαµ = − Ψαµ , Θαβ = − Fβα , 2 8 and σ PA(σ) = dσ1πA(σ1) A = {µ, a, a} ¯ 0Noncommutativity and T-duality BSW 2011
  22. 22. 19 Relation between T-duality, effective theory and noncommuatativity Type II B and fermionic duality• T-duality {Γa, Γb} {Pa, Pb}• Gµν Gµν / Bµν / θµν = 2κ Bµν 1• ψaµ 2 ψaµ θaµ = − κ ψaµ 2 ¯ ψµa 1 ¯ ψµa ¯ θµa = κ ¯ ψµa 2 2 1• Fab − 8 Fab θab = − κ Fab 8Noncommutativity and T-duality BSW 2011
  23. 23. 20 Bosonic string in weakly curved background• The consistency of the theory – Quantum world-sheet conformal invariance – produce conditions on background fields space-time equations of motion 1 ρσ Rµν − Bµρσ Bν = 0 , 4 ρ DρB µν = 0 – Bµνρ = ∂µBνρ + ∂ν Bρµ + ∂ρBµν is a field strength – Rµν and Dµ Ricci tensor and covariant derivative• We will consider the following particular solution 1 ρ Gµν = const, Bµν [x] = bµν + Bµνρx , 3 – bµν is constant – Bµνρ is constant and infinitesimally small• – We will work up to the first order in Bµνρ – Ricci tensor Rµν is an infinitesimal of the second order and as such is neglectedNoncommutativity and T-duality BSW 2011
  24. 24. 21 T-duality of weakly curved background (Twcb)Lj. Davidovi´ and B. Sazdovi´ c cT-duality in the weakly curved backgroundin preparation• More complicated procedure then in flat background 2 Gµν = α G−1µν ( x), E B µν = α θ µν ( x) x is Twcb of x and y is T0 dual of y ˜ x = g −1(2by + y ) ˜Noncommutativity and T-duality BSW 2011
  25. 25. 22 Effective theory and non-commutativity in weakly curved backgroundLj. Davidovi´ and B. Sazdovi´ c cPhys. Rev. D 83 (2011) 066014Lj. Davidovi´ and B. Sazdovi´, c cNon-commutativity parameters depend not only on the effectivecoordinate but on its T-dual as wellhep-th/1106.1064to be published in JHEP• Similar procedure but much more complicated calculation• Effective background fields Gef f (u) = GE (u), µν µν Bµνf = − κ (gθ(u)g)µν ef 2 u = q + 2b˜ q• Non-commutativity parameter – Nontrivial both at string endpoints and at string interior – Depends on the σ -integral of the effective momenta σ Pµ(σ) = 0 dηpµ(η) which is in fact T0-dual of the effective coordinate, Pµ = κgµν q ν . ˜Noncommutativity and T-duality BSW 2011
  26. 26. 23 Relation between Twcb-duality, effective theory and noncommuatativity• Twcb-duality Effective theory Noncommuatativity Dual background fields 2 Gµν = α G−1µν ( x) E Gef f = GE (u) µν µν / B µν = α θ µν ( x) / θ µν (v) Dual variables 3 ˜ x = g −1(2by+ y ) ˜ u = q+2b˜ q v = q− π bQcmNoncommutativity and T-duality BSW 2011

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