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- 1. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks Hamiltonian approach to Dp-brane noncommutativity Bojan Nikoli´ and Branislav Sazdovi´ c c Center for Theoretical Physics Institute of Physics, Belgrade, Serbia Group for gravitation, particles and ﬁelds http://gravity.phy.bg.ac.yu/ Spring School on Strings, Cosmology and Particles, Niˇ , Serbia s 31st March – 4th April 2009 Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity
- 2. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks Outline of the talk Strings and superstrings 1 Conformal symmetry and dilaton ﬁeld 2 Type IIB superstring and non(anti)commutativity 3 Concluding remarks 4 Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity
- 3. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks Basic facts 1 Strings are object with one spatial dimension. During 1 motion string sweeps a two dimensional surface called world sheet, parametrized by timelike parameter τ and spacelike one σ ∈ [0 , π]. There are open and closed strings. Open string endpoints can be forced to move along 2 Dp-branes by appropriate choice of boundary conditions. Dp-brane is an object with p spatial dimensions which satisﬁes Dirichlet boundary conditions. Demanding presence of fermions in theory, we obtain 3 superstring theory. Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity
- 4. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks Basic facts 2 There are two standard approaches to superstring theory: 1 NSR (Neveu-Schwarz-Ramond) (world sheet supersymmetry) and GS (Green-Schwarz) (space-time supersymmetry). New approach has been recently developed - pure spinor 2 formalism, (N. Berkovits, hep-th/0001035). It combines advantages of NSR and GS formalisms and avoid their weaknesses. There are ﬁve consistent superstring theories. 3 Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity
- 5. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks Superstring theories Type I 1 Unoriented open and closed strings, N = 1 supersymmetry, gauge symmetry group SO(32). Type IIA 2 Closed oriented and open strings, N = 2 supersymmetry, nonchiral. Type IIB 3 Closed oriented and open strings, N = 2 supersymmetry, chiral. Two heterotic theories 4 Closed oriented strings, N = 1 supersymmetry, symmetry group either SO(32) or E8 × E8 . Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity
- 6. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks Boundary conditions in canonical formalism As time translation generator Hamiltonian Hc must have well deﬁned functional derivatives with respect to coordinates xµ and their canonically conjugated momenta πµ π (R) (0) δHc = δHc − γµ δxµ 0 . The ﬁrst term is so called regular term. It does not contain τ and σ derivatives of coordinates and momenta variations. The second term has to be zero and we obtain boundary conditions. We obtain the same result in Lagrangian formalism from δS = 0. Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity
- 7. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks Sorts of boundary conditions If the coordinate variations δxµ are arbitrary at the string endpoints, we talk about Neumann boundary conditions (0) (0) = γµ = 0. γµ 0 π If the coordinates are ﬁxed at the string endpoints δxµ = δxµ = 0, 0 π then we have Dirichlet boundary conditions. Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity
- 8. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks Boundary conditions as canonical constraints Boundary conditions are treated as canonical constraints. Then we perform consistency procedure. Let Λ(0) be a constraint, then consistency of constraint demands that it is preserved in time dΛ(n−1) Λ(n) ≡ = Hc , Λ(n−1) ≈ 0 . (n = 1, 2, . . . ) dτ In all cases we will consider here, this is an inﬁnite set of constraints. Using Taylor expansion we can rewrite this set of constraints in compact σ-dependent form ∞ ∞ σ n (n) (σ − π)n (n) ˜ Λ(σ) = Ω (σ = 0) , Λ(σ) = Λ (σ = π) . n! n! n=0 n=0 Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity
- 9. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks Bosonic string with dilaton Action which describes dynamics of bosonic string in the presence of gravitational Gµν (x), antisymmetric NS-NS ﬁeld Bµν (x) and dilaton ﬁeld Φ(x) is of the form εαβ √ 1 αβ d2 ξ −g g Gµν + √ Bµν ∂α xµ ∂β xν + ΦR(2) S=κ , 2 −g Σ where ξ α = (τ , σ) parameterizes world sheet Σ with intrinsic metric gαβ . With R(2) we denote scalar curvature with respect to the metric gαβ . Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity
- 10. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks Beta functions 1 Quantum conformal invariance is determind by the beta functions 1 βµν ≡ Rµν − Bµρσ Bν ρσ + 2Dµ aν , G 4 βµν ≡ Dρ B ρ µν − 2aρ B ρ µν , B D − 26 1 β Φ ≡ 2πκ − R − Bµρσ B µρσ − Dµ aµ + 4a2 . 6 24 Theory is conformal invariant on the quantum level under the following conditions βµν = βµν = 0 and β Φ = 0 (or G B β Φ = c = const.). This is a consequence of the relation D ν βµν + (4π)2 κDµ β Φ = 0 . G Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity
- 11. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks Beta functions 2 We will consider one particular solution of these equations Gµν (x) = Gµν = const , Bµν (x) = Bµν = const , Φ(x) = Φ0 + aµ xµ , (aµ = const) . Sigma model becomes conformal ﬁeld theory with central charge c. There are two possibilities: c = 0, or c = 0 plus adding of Liouville term βΦ √ 1 d2 ξ −gR(2) R(2) , ∆ = gαβ ∇α ∂β , SL = − 2(4π)2 κ ∆ Σ which annihilates conformal anomaly and restores quantum conformal invariance. Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity
- 12. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks Beta functions 3 Conformal gauge condition gαβ = e2F ηαβ . 1 1 1 There are three cases: (1) a2 = α , a2 = α ; (2) a2 = α , ˜ 2 = 1 ; (3) a2 = 1 , a2 = 1 . Limit α → ∞ gives the results ˜ α˜ a α α for the case without Liouville term. Constant α is chosen in such a way that Liouville term eliminates anomaly term βΦ 1 = . (4πκ)2 α a2 ≡ (Gµνf )aµ aν , Gef f = Gµν − 4(BG−1 B)µν . ˜ µν ef Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity
- 13. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks Choice of boundary conditions We split xµ (µ = 0, 1, 2, . . . D − 1) into Dp-brane coordinates xi (i = 0, 1, . . . p) and orthogonal ones xa (a = p + 1, p + 2, . . . , D − 1). For xi and F we choose Neumann boundary conditions, and for xa Dirichlet ones. Bµν → Bij , aµ → ai . Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity
- 14. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks Solution of boundary conditions 1 Initial variables are expressed in terms of their Ω even parts (effective variables), where Ω is world-sheet parity transformation Ω : σ → −σ. Initial coordinates depend both on effective coordinates and effective momenta. Presence of momenta causes noncommutativity. Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity
- 15. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks Solution of boundary conditions 2 In all three cases coordinate ⋆ F = F + α ai xi is 2 commutative (in limit α → ∞ the role of commutative coordinate takes ai xi ), while other ones are noncommutative. Case (1) - all constraints originating from boundary conditions are of the second class (Dirac constraints do not appear). The number of Dp-brane dimensions is unchanged. Case (2) - one Dirac constraint of the ﬁrst class appears. The number of Dp-brane dimensions decreases. Case (3) - two constraints originating from boundary conditions are of the ﬁrst class. The number of Dp-brane dimensions decreases. Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity
- 16. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks Type IIB superstring theory in D = 10 NS-NS sector: graviton Gµν , Neveu-Schwarz ﬁeld Bµν and dilaton Φ. α ¯α NS-R sector: two gravitinos, ψµ i ψµ , and two dilatinos, λα ¯ and λα . Spinors are of the same chirality. R-R sector: scalar C0 , two rank antisymmetric tensor Cµν and four rank antisymmetric tensor Cµνρσ with selfdual ﬁeld strength. Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity
- 17. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks Model We consider model without dilatinos and dilaton. Action for type IIB superstring theory in pure spinor formulation (ghosts free) is 1 ab d2 ξ η Gµν + εab Bµν ∂a xµ ∂b xν SIIB = κ 2 Σ d2 ξ −πα (∂τ − ∂σ )(θ α + ψµ xµ ) α + Σ 1 ¯ ¯α d2 ξ (∂τ + ∂σ )(θ α + ψµ xµ )¯α + πα F αβ πβ + ¯ π 2κ Σ where xµ are space-time coordinates (µ = 0, 1, 2, . . . 9), ¯ and θ α and θ α are same chirality spinors. Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity
- 18. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks R-R sector 1 10 ik F Γαβ . Γαβ = (Γ[µ1 ...µk ] )αβ αβ = F k! (k) (k) (k) k=0 Bispinor F αβ satisﬁes chirality condition Γ11 F = −F Γ11 , and consequently, F(k) (k odd) survive. Because of duality relation, independent tensors are F(1) , F(3) and selfdual part of F(5) . Massless Dirac equation for F gives, F(k) = dC(k−1) . Type IIB contains only potentials C0 , Cµν and Cµνρσ . Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity
- 19. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks R-R sector 2 1 Fs = (F αβ + F βα ) −→ C0 , Cµνρσ , αβ 2 1 Fa = (F αβ − F βα ) −→ Cµν . αβ 2 Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity
- 20. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks Boundary conditions For coordinates xµ we choose Neumann boundary conditions. In order to preserve N = 1 SUSY from initial N = 2, for fermionic coordinates we choose π π ¯ (θ α − θ α ) = 0 =⇒ (πα − πα ) ¯ = 0. 0 0 Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity
- 21. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks Solution of boundary conditions Solving boundary conditions, we get σ σ Θµα xµ (σ) = q µ − 2Θµν dσ1 pν + dσ1 (pα + pα ) , ¯ 2 0 0 σ σ Θαβ θ α (σ) = η α − Θµα dσ1 pµ − dσ1 (pβ + pβ ) , ¯ 4 0 0 σ σ Θαβ ¯ θ α (σ) = η α − Θµα ¯ dσ1 pµ − dσ1 (pβ + pβ ) , ¯ 4 0 0 where 1α 1 ¯ ¯ ηα ≡ (θ + Ωθ α) , η α ≡ (Ωθ α + θ α ) , ¯ 2 2 ≡ πα + Ω¯α , pα ≡ Ωπα + πα . ¯ ¯ pα π Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity
- 22. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks Noncommutativity Using the solution, we have {xµ (σ) , xν (¯ )} = Θµν ∆(σ + σ ) , ¯ σ 1 {xµ (σ) , θ α (¯ )} = − Θµα ∆(σ + σ ) , ¯ σ 2 1 ¯σ {θ α (σ) , θ β (¯ )} = Θαβ ∆(σ + σ ) . ¯ 4 Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity
- 23. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks Type I theory as effective one Putting the solution of boundary conditions into initial Lagrangian, we obtain effective one κ ef f ab Lef f = G η ∂a q µ ∂b q ν − πα (∂τ − ∂σ ) η α + (Ψef f )α q µ 2 µν µ 1 αβ + (∂τ + ∂σ ) η α + (Ψef f )α q µ πα + ¯ ¯ ¯ πα Fef f πβ . µ 2κ Transition from the initial L to effective Lef f Lagrangian is ¯ realized by changing the variables xµ , θ α and θ α with q µ , α and η α ,and changing the background ﬁelds ¯ η → Gµν − 4Bµρ Gρλ Bλν , ψ+µ → ψ+µ + 2Bµρ Gρν ψ−ν , α α α Gµν β αβ → Fa − ψ−µ Gµν ψ−ν , αβ α Fa α αβ → 0, ψ−µ → 0 , Fs → 0 . Bµν Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity
- 24. Strings and superstrings Conformal symmetry and dilaton ﬁeld Type IIB superstring and non(anti)commutativity Concluding remarks Conclusions (1) Initial coordinates depend both on effective ones and effective momenta. Presence of mometa causes non(anti)commutativity of the coordinates. (2) The number of Dp-brane dimensions depends on the relations between background ﬁelds. For particular relations between them, ﬁrst class constraints appear and decrease the number of Dp-brane dimensions. (3) Effective theory (initial one on the solution of boundary conditions) is Ω even. (4) Effective background ﬁelds depend both on Ω even ﬁelds and Ω even combinations of the Ω odd ﬁelds. Bojan Nikoli´ c Hamiltonian approach to Dp-brane noncommutativity

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