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B. Sazdović: Canonical Approach to Closed String Non-commutativity

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Balkan Workshop BW2013
Beyond the Standard Models
25 – 29 April, 2013, Vrnjačka Banja, Serbia

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B. Sazdović: Canonical Approach to Closed String Non-commutativity

  1. 1. Open string noncommutativityT-dualityWeakly curved backgroundClosed string noncommutativityCanonical approach to closed stringnoncommutativityLjubica Davidovi´c, Bojan Nikoli´c andBranislav Sazdovi´cInstitute of Physics, University of Belgrade, SerbiaBalkan Workshop BW2013, 25.-29.04.2013, Vrnjaˇcka banja,SerbiaLjubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
  2. 2. Open string noncommutativityT-dualityWeakly curved backgroundClosed string noncommutativityOutline of the talk- Open string noncommutativity- T-duality- Weakly curved background- Closed string noncommutativityLjubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
  3. 3. Open string noncommutativityT-dualityWeakly curved backgroundClosed string noncommutativityOpen string noncommutativityExtended objects (like strings) see space-time geometrydifferent than point-particles - stringy property.The ends of the open string attached to Dp-brane becomenoncommutative in the presence of Kalb-Ramond field BµνAction principle δS = 0S(x) = κ∫Σ(ηαβ2Gµν + εαβBµν)∂αxµ∂βxν,gives equations of motion and boundary conditions.Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
  4. 4. Open string noncommutativityT-dualityWeakly curved backgroundClosed string noncommutativityOpen string noncommutativitySolution of boundary conditionsxµ= qµ− Θµν∫ σdσ1pν(σ1) ,where qµ and pν are effective variables satisfying{qµ(σ), pν(¯σ)} = 2δµνδs(σ, ¯σ).The coordinate xµ is the linear combination of effectivecoordinate qµ and effective momenta pν - source ofnoncommutativity.Pure stringy property{xµ(0), xν(0)} = −2Θµν, {xµ(π), xν(π)} = 2Θµν.Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
  5. 5. Open string noncommutativityT-dualityWeakly curved backgroundClosed string noncommutativityOpen string noncommutativityEffective actionS(q) = S(x)|bound.cond. = κ∫d2ξ12GEµν∂+xµ∂−xν,whereGEµν = (G − 4BG−1B)µν , Θµν= −2κ(G−1E BG−1)µν,are effective metric and noncommutativity parameter,respectively.Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
  6. 6. Open string noncommutativityT-dualityWeakly curved backgroundClosed string noncommutativityT-dualityIt relates string theories with different backgrounds.Compactification on a circle has two consequences:momentum becomes quantized - p = n/R (n ∈ Z) ,the new state arises (winding states N)x(π) − x(0) = 2πRN .Mass squared of any stateM2=n2R+ m2 R2α′2+ oscillators ,is invariant under n ↔ m and R ↔ ˜R ≡ α′/R .Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
  7. 7. Open string noncommutativityT-dualityWeakly curved backgroundClosed string noncommutativityT-dualityCompactification on circle of radius R is equivalent tocompactification on radius ˜R - purely stringy property.Dual action ⋆S has the same form as initial one but withdifferent background fields⋆Gµν∼ (G−1E )µν, ⋆Bµν∼ Θµν,which are essentially parameters of open stringnoncommutativity.Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
  8. 8. Open string noncommutativityT-dualityWeakly curved backgroundClosed string noncommutativityT-dualityCanonical T-duality transformationsπµ∼= κy′µ , ⋆πµ ∼= κx′µ.There is no closed string noncommutativity for constantGµν and Bµν{πµ, πν} = 0 =⇒ {yµ, yν} = 0 .Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
  9. 9. Open string noncommutativityT-dualityWeakly curved backgroundClosed string noncommutativityChoice of background fieldsGµν is constant and Bµν = bµν + 13Hµνρxρ ≡ bµν + hµν(x).bµν and Hµνρ are constant and Hµνρ is infinitesimaly small.These background fields satisfy space-time equations ofmotion.Now Bµν is xµ dependent.Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
  10. 10. Open string noncommutativityT-dualityWeakly curved backgroundClosed string noncommutativityT-duality along all directionsGeneralized Buscher construction has two steps:gauging global symmetry δxµ= λµwhich is a symmetryeven Bµν is coordinate dependent∂αxµ→ Dαxµ= ∂αxµ+ vµα ,xµ→ ∆xµinv =∫PdξαDαxµ(this is a new step).Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
  11. 11. Open string noncommutativityT-dualityWeakly curved backgroundClosed string noncommutativityT-dual action and doubled geometryxµ → Vµ = −κΘµν0 yν + (g−1E )µν ˜yν [xµ → (yµ, ˜yµ)].⋆Gµν = (G−1E )µν(∆V) and ⋆Bµν = κ2 Θµν(∆V).T-dual action is of the form⋆S =κ22∫d2ξ∂+yµΘµν− (∆V)∂−yν ,whereΘµν± (x) = −2κ(G−1E (x)Π±(x)G−1)µν, Π±µν = Bµν(x)±12Gµν .Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
  12. 12. Open string noncommutativityT-dualityWeakly curved backgroundClosed string noncommutativityT-dual transformation laws∂±xµ= −κΘµν± (∆V)∂±yν ∓ 2κΘµν0±β∓ν (V) ,∂±yµ = −2Π∓µν(∆x)∂±xν∓ β∓µ (x) ,whereβ±µ (x) = ∓16Hµρσ∂∓xρxσ.It is infinitesimally small and bilinear in xµ. Expression for β±µcomes from the term∫d2ξvµ+Bµν(δV)vν− =∫d2ξβαµ (V)δvµα .Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
  13. 13. Open string noncommutativityT-dualityWeakly curved backgroundClosed string noncommutativityTransformation laws in canonical formx′µ=1κ⋆πµ− κΘµν0 β0ν (V) − (g−1E )µνβ1ν (V)y′µ =1κπµ − β0µ(x) .These infinitesimal βαµ -terms are improvements in comparisonwith flat space. Also they are source of noncommutativity.Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
  14. 14. Open string noncommutativityT-dualityWeakly curved backgroundClosed string noncommutativityChoosing evolution parameter∆xµ(σ, σ0) = xµ(ξ) − xµ(ξ0) =∫( ˙xµdτ + x′µdσ)If we take evolution parameter orthogonal to ξ − ξ0 then wehave∆xµ(σ, σ0) =∫ σσ0dσ1x′µ(σ1) , ∆yµ(σ, σ0) =∫ σσ0dσ1y′µ(σ1) .We use information from one background to compute Poissonbrackets in T-dual one.Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
  15. 15. Open string noncommutativityT-dualityWeakly curved backgroundClosed string noncommutativityGeneral structure, y{y′µ(σ), y′ν(¯σ)} = F′µν[x(σ)]δ(σ − ¯σ){∆yµ(σ, σ0), ∆yν(¯σ, ¯σ0)} =∫ σσ0dσ1∫ ¯σ¯σ0dσ2F′µν[x(σ1)]δ(σ1 − σ2){yµ(σ), yν(¯σ)} = −[Fµν(σ) − Fµν(¯σ)]θ(σ − ¯σ) .Putting σ = 2π and ¯σ = 0, we get{yµ(2π), yν(0)} = −2π2{⋆Nµ, ⋆Nν} = −CρFµνρdxρ= −2πFµνρNρ,where Nµ is winding number, xµ(2π) − xµ(0) = 2πNµ, andFµνρ =∂Fµν∂xρ are fluxes.Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
  16. 16. Open string noncommutativityT-dualityWeakly curved backgroundClosed string noncommutativityGeneral structure, x{x′µ(σ), x′ν(¯σ)} = F′µν[y(σ), ˜y(σ)]δ(σ − ¯σ){∆xµ(σ, σ0), ∆xν(¯σ, ¯σ0)} =∫ σσ0dσ1∫ ¯σ¯σ0dσ2F′µν[y(σ1), ˜y(σ1)]δ(σ1−σ2){xµ(σ), xν(¯σ)} = −[Fµν(σ) − Fµν(¯σ)]θ(σ − ¯σ) .Putting σ = 2π and ¯σ = 0, we get{xµ(2π), xν(0)} = −2π2{Nµ, Nν} = −2π(Fµνρ⋆Nρ+ ˜Fµνρ⋆pρ),where ⋆Nµ, and ⋆p are winding number and momenta for yµyµ(σ = 2π) − yµ(σ = 0) = 2π⋆Nµ,yµ(τ = 2π) − yµ(τ = 0) = 2π⋆pµFµνρ = ∂Fµν∂yρ, ˜Fµνρ = ∂Fµν∂˜yρare fluxes.Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
  17. 17. Open string noncommutativityT-dualityWeakly curved backgroundClosed string noncommutativityNoncommutativity of y coordinatesy Fµν(x) = 1κ Hµνρxρ, Hµνρ is field strength for Bµν.{yµ(2π), yν(0)} = −2π2{ ⋆Nµ, ⋆Nν} ∼= −2πκBµνρNρ.xµ(τ, σ) = xµ0 + pµτ + Nµσ + osc.yµ(τ, σ) = y0µ + ⋆pµτ + ⋆Nµσ + osc. .Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
  18. 18. Open string noncommutativityT-dualityWeakly curved backgroundClosed string noncommutativityNongeometric fluxes⋆Bµν depends on double space coordinates (yµ , ˜yµ)⋆Bµν= ⋆bµν+ Qµνρyρ + ˜Qµνρ˜yρ .There are two field strengthsRµνρ= Qµνρ+ cycl. , ˜Rµνρ= ˜Qµνρ+ cycl. .Rµνρ and ˜Rµνρ are nongeometric fluxes.Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
  19. 19. Open string noncommutativityT-dualityWeakly curved backgroundClosed string noncommutativityNoncommutativity of x coordinatesx Fµν(y, ˜y) =1κ[Rµνρyρ − (˜Rµνρ− 4 ˜Qµνρ)˜yρ]{xµ(2π), xν(0)} = −2π2{Nµ, Nν}= −2πκ[Rµνρ⋆Nρ − (˜Rµνρ− 4 ˜Qµνρ)⋆pρ],All Poisson brackets close on winding numbers Nµ, ⋆Nµand momenta pµ, ⋆pµ.Coefficients are: geometric flux Hµνρ and nongeometricfluxes Rµνρ and ˜Rµνρ.Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
  20. 20. Open string noncommutativityT-dualityWeakly curved backgroundClosed string noncommutativityAdditional relationsIf we dualize all directions we have {xµ, yν} = 0.In the case of partial T-dualizationxµ = (xi, xa) → (xi, ya, ˜ya) it holds {xi, ya} ̸= 0.Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity

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