Noncommutativity and T-duality

Lj. Davidović, B. Nikolić and B. Sazdović Institute of Physics, Belgrade, Serbia

• We will discuss relation between

– Open string parameters G_{μν}^{eff}(G, B) and θ^{μν}(G, B)

– and T-dual background fields ⋆G^{μν}(G, B) and ^{⋆}B^{μν}(G, B)

as functions of the initial background fields: metric tensor G_{μν} and Kalb-Ramond field B_{μν}

• 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 Gravity 2011

• Effective theory

On the solution of boundary conditions the initial theory turns to the effective one with effective metric tensor G_{μν}^{eff}(G, B) and vanishing effective Kalb-Ramond field

• T-duality T-duality in presence of background fields leads T-dual background fields ^{⋆}G^{μν}(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” B_{μν}[x] = b_{μν} + \frac{1}{3}B_{μνρ}x^{ρ} instead of the flat one B_{μν} = b_{μν} = const.

The action

describing the open string propagation in curved background

S = κ ∫_{Σ} d^{2}ξ\sqrt{−g}[\frac{g^{αβ}}{2} G_{μν}(x)+\frac{ϵ^{αβ}}{\sqrt{−g}}B_{μν}(x)]∂_{α}x^{μ}∂_{β}x^{ν},

• x^{μ}(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

• g_{αβ}(ξ) intrinsic world-sheet metric (g = detg_{αβ})

• background fields – G_{μν}(x) space-time metric

– B_{μν}(x) Kalb-Ramond antisymmetric field

Action principle for string

Evolution from the initial to final configuration is such that the action is stationary S = ∫_{τi}^{τf} dτ ∫_{σi}^{σf} dσL(x^{μ}, ẋ^{μ}, x^{′μ}, g_{αβ})

δS = ∫ dτdσ[\frac{∂L}{∂x^{μ}} − ∂_{τ} \frac{∂L}{∂ẋ^{μ}} − ∂_{σ} \frac{∂L}{∂ẋ^{μ}}]δx^{μ}

+ ∫ dτ[\frac{∂L}{∂ẋ^{μ}}δx^{μ}]|_{σ=0}^{σ=π}

From the action principle we get

1) equation of motion ẍ^{μ} = x^{′′μ} − 2B_{νρ}^{μ}ẋ^{ν}x^{′ρ}, B_{μνρ} = ∂_{μ}B_{νρ} + ∂_{ν}B_{ρμ} + ∂_{ρ}B_{μν} is field strength

2) boundary condition γ_{0}^{μ}δx^{μ}|_{σ=0,π} = 0, γ_{0}^{μ} ≡ \frac{∂L}{∂ẋμ} = x^{′μ} − 2(G^{−1}B)_{ν}^{μ}ẋ^{ν}

Noncommutativity and T-duality Gravity 2011

The Boundary conditions

• The closed string fulfills the boundary condition because x^{μ}(0) = x^{μ}(π)

• For the open string we can impose 1) Neumann boundary condition [δx^{μ}]|_{0} , [δx^{μ}]|_{π} are arbitrary i.e. string end-points can move freely γ_{μ}^{0}|_{σ=0} = 0, γ_{μ}^{0}|_{σ=π} = 0

2) Dirichlet boundary condition [δx^{μ}]|_{σ=0} = 0, [δx^{μ}]|_{σ=π} = 0 The edges of the string are fixed

Noncommutativity and effective theory

bosonic open string in flat space-time I

• We impose Neumann boundary conditions

• We treat boundary conditions as constraints

• Constraint γ_{μ}^{0} must be conserved in time – Secundary constraint γ_{μ}^{1} = γ̇_{μ}^{0}

– Infinite set of constraints γ_{μ}^{n} = γ̇_{μ}^{n−1}, (n = 1, 2, ⋯)

– Two σ-dependent constraints Γ^{μ}(σ) ≡ ∑_{n=0}^{∞} \frac{σn}{(n)!}γ_{n}^{μ}|_{σ=0} Γ̄^{μ}(σ) ≡ ∑_{n=0}^{∞} \frac{(σ−π)n}{(n)!}γ_{n}^{μ}|_{σ=π}

• 2π-periodicity x^{μ}(σ) = x^{μ}(σ + 2π) solve constraint at σ = π Γ^{μ}(σ) = 0 → Γ̄^{μ}(σ) = 0

Noncommutativity and effective theory

bosonic open string in flat space-time II

• Solving the constraints

– In canonical formalism {Γ_{μ}(σ), Γ_{ν}(σ̄)} = −κG_{μν}^{E}δ′(σ − σ̄) For G_{μν}^{E} ≠ 0 Γ_{μ}(σ) are SSC

– Introduce world-sheet parity Ω Ω : σ → −σ, Ωx^{μ}(σ) → x^{μ}(−σ) and new variables q^{μ} = \frac{1}{2}(1 + Ω)x^{μ} q̄^{μ} = \frac{1}{2}(1 − Ω)x^{μ}

– Solve Ω odd parts in terms of Ω even ∗ q̄ = f_{1}(q, p), p̄ = f_{2}(q, p)

∗ x^{μ} = q^{μ} − 2θ^{μν} ∫ dσ_{1}p_{ν}, π_{μ} = p_{μ} • Effective action and background fields

– S^{eff} = κ ∫ d^{2}ξ\frac{1}{2}η^{αβ}G_{μν}^{E}∂_{α}q^{μ}∂_{β}q^{ν}

– G_{μν} → G_{μν}^{eff} = G_{μν}^{E}, B_{μν} → B_{μν}^{eff} = 0

G_{μν}^{E} ≡ [G − 4BG^{−1}B]_{μν} effective metric

• Noncommuatativity

\{X^{μ}(σ), X^{ν}(σ̄)\} = θ^{μν} \{ \begin{matrix}−1 & σ, σ̄ = 0 \\ 1 & σ, σ̄ = π \\ 0 & otherwise \end{matrix} .

θ^{μν} ≡ −\frac{2}{κ}(G_{E}^{−1}BG^{−1})^{μν} non-commutativity parameter

T_{0}-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, B_{μν} = 0, Φ = 0

• – x^{25}(σ + π) = x^{25}(σ) + 2πRm, (m ∈ Z) – m– winding number

• Consequences of compactification:

– Momentum along circle is quantized, p = \frac{n}{R} (n ∈ Z), Lost some states

– New states that wrap around circle arise, winding states Gained some states

Surprising symmetry as stringy property

• Mass square of the states

M^{2} = \frac{n2}{R2} + \frac{m2R2}{α′2}+ contributions from oscilators

• – Complementary behavior of momentum and winding states

– M^{2}(R, n, m) = M^{2}(\frac{α′}{R}, m, n)

– R ⟷ \frac{α′}{R} ≡ R̃, n ⟷ m, R̃ — Dual radius

• T_{0} duality for closed string

Compactification with radius R

is physically indistinguishable from

Compactification with radius R̃ = \frac{α′}{R}

• T_{0} dual coordinate

– Equation of motion ∂_{+}∂_{−}x = 0 ⟹ x = x_{+}(τ + σ) + x_{−}(τ − σ)

– T_{0} dual coordinate x̃ ≡ x_{+}(τ + σ) − x_{−}(τ − σ)

T-duality – nontrivial background I

• – Background fields are independent of the circular

coordinate

– We take all coordinate to be circular → G_{μν}, B_{μν} = const Toroidal duality of all cordinates

• Lagrange multiplier method

⋆S[y, v_{+}, v_{−}] =

κ ∫ d^{2}ξv_{+}^{μ}(B + \frac{1}{2}G)_{μν}v_{−}^{ν} + ∫ d^{2}ξy_{μ}(∂_{+}v_{−}^{μ} − ∂_{−}v_{+}^{μ}),

– y_{μ} – Lagrange multiplier

• Integration over y returns to the original action

∂_{+}v_{−}^{μ} − ∂_{−}v_{+}^{μ} = 0 ⇒ v_{±}^{μ} = ∂_{±}x^{μ}

• Integrating out vector field v_{±}

v_{±}^{μ}(y) = −2[θ^{μν} ∓ \frac{1}{κ}(G_{E}^{−1})^{μν}]∂_{±}y_{ν}

G_{μν}^{E} ≡ [G − 4BG^{−1}B]_{μν}, θ^{μν} ≡ −\frac{2}{κ}(G_{E}^{−1}BG^{−1})^{μν}

are the open string background fields: effective metric and non-commutativity parameter

T-duality–nontrivial background II

• ^{⋆}S[∂_{+}y, ∂_{−}y] = 2 ∫ d^{2}ξ∂_{+}y_{μ}[θ^{μν} + \frac{1}{κ}(G_{E}^{−1})^{μν}]∂_{+}y_{μ} = κ ∫ d^{2}ξ∂_{+}y_{μ}(^{⋆}B + \frac{1}{2}^{⋆}G)_{μν}∂_{+}y_{μ}

• Dual background fields ⋆B^{μν} = \frac{2}{κ}θ^{μν}, ^{⋆}G^{μν} = (\frac{2}{κ})^{2}(G_{E}^{−1})^{μν}

• Turn off background fields – B_{μν} → 0, G_{μν} → (η_{μν}, G_{25,25} = G), κ → \frac{2}{α′} – 2πR = ∫_{0}^{2π} ds = \sqrt{G} ∫_{0}^{2π} dθ = 2π\sqrt{G}

⇒ G = R^{2}, ^{⋆}G = R̃^{2}

– ^{⋆}G = α′^{2}G^{−1} ⇒ RR̃ = α′

– ∂_{±}x = ±∂_{±}y ⇒ y = x̃

Relation between T-duality, effective theory

and noncommuatativity

• T-duality ⋆G^{μν} = α′^{2}G_{E}^{−1μν}, ^{⋆}B^{μν} = α′θ^{μν}

• Effective theory G_{μν}^{eff} = G_{μν}^{E} /

• Noncommuatativity / θ^{μν}

• The same background fields: effective metric

– G_{μν}^{E} ≡ [G − 4BG^{−1}B]_{μν}

and non-commutativity parameter θ^{μν} ≡ −\frac{2}{κ}(G_{E}^{−1}BG^{−1})^{μν}

Type II B theory

• Type IIB theory in pure spinor formulation

S = κ ∫_{Σ} d^{2}ξ [\frac{1}{2}η^{mn}G_{μν} + ε^{mn}B_{μν}] ∂_{m}x^{μ}∂_{n}x^{ν}

+ ∫_{Σ} d^{2}ξ [−π_{α}∂_{−}(θ^{α} + Ψ_{μ}^{α}x^{μ}) + ∂_{+}(θ̄^{α} + Ψ̄_{μ}^{α}x^{μ})π̄_{α} + \frac{1}{2κ}π_{α}F^{αβ}π̄_{β}]

• Variables x^{μ}, θ^{α} and θ̄^{α}

• Background fields – NS-NS G_{μν}, B_{μν}

– NS-R  Ψ_{μ}^{α}, Ψ̄_{μ}^{α}  , gravitinos
– R-R  F^{αβ} ∼ A_{0}, A_{2}, A_{4},  dA_{4}-self dual

Type II B theory

Neumann boundary conditions

• Boundary conditions

[γ_{i}^{(0)}δx^{i} + (π_{α}δθ^{α} + δθ̄^{α}π̄_{α})] |_{0}^{π} = 0

γ_{i}^{(0)} = Π_{+i}^{j}I_{−j} + Π_{−i}^{j}I_{+j} + (π_{α}Ψ_{i}^{α} + Ψ̄_{i}^{α}π̄_{α})

• For bosonic coordinates Neumann boundary conditions

γ_{i}^{(0)}|_{0}^{π} = 0

• Fermionic coordinates preserves N=1 SUSY

(θ^{α} − θ̄^{α})|_{0}^{π} = 0 ⇒ (π_{α1} − π̄_{α1})|_{0}^{π} = 0

Type II B theory

Neumann b. c., Effective theory and non-commutativity

B.Nikolić and B. Sazdović, Phys. Lett. B666 (2008) 400

B. Nikolić and B. Sazdović, Nucl. Phys. B 836 (2010) 100

• Similar method as in bosonic case

• Background fields

– Ω even corresponds to Type I

G_{μν} → G_{μν}^{E}

\frac{1}{2}Ψ_{+μ}^{α} → (Ψ_{E})_{μ}^{α} = \frac{1}{2}Ψ_{+μ}^{α} + (BG^{−1}Ψ_{−})_{μ}^{α}

F_{a}^{αβ} → F_{E}^{αβ} = F^{αβ} − (Ψ_{−}G^{−1}Ψ_{−})^{αβ}

– Ω odd fields vanish B_{μν} → 0, Ψ_{−} → 0, F_{s} → 0

• Non-commutativity Ω odd fields are source of non-commutativity

{x^{μ}(σ) , x^{ν}(σ̄)} = 2θ^{μν}θ(σ + σ̄) {x^{μ}(σ) , θ^{α}(σ̄)} = −θ^{μα}θ(σ + σ̄)

\{θ^{α}(σ) , θ̄^{β}(σ̄)\} = \frac{1}{2}θ^{αβ}θ(σ + σ̄)

Type II B theory

Bosonic T_{IIBb}-dulity

• Action has global shift symmetry in bosonic direction Similar method produce dual background fields ⋆G^{μν} = α′^{2}G_{E}^{−1μν}, ^{⋆}B^{μν} = α′θ^{μν} ⋆ψ_{−}^{aμ} = −2G_{E}^{−1μν}(ψ_{E})_{ν}^{a} ^{⋆}ψ_{+}^{aμ} = 2κθ^{aμ} ⋆F_{a}^{ab} = F_{E}^{ab} + 4(ψ_{E}^{a}G_{E}^{−1}ψ_{E}^{b}) ^{⋆}F_{s}^{ab} = 2κθ^{ab}

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
• ^{⋆}G^{μν} = α′^{2}G_{E}^{−1μν}  G_{μν}^{eff} = G_{μν}^{E}  /
⋆B^{μν} = α′θ^{μν}  /  θ^{μν}

• ^{⋆}ψ_{−}^{aμ} = −2(G_{E}^{−1}ψ_{E})^{aμ} (ψ_{eff})_{μ}^{a} = (ψ_{E})_{μ}^{a} /

⋆ψ_{+}^{aμ} = 2κθ^{aμ}  /  θ^{aμ}

• ^{⋆}F_{a}^{ab} = F_{E}^{ab} + 4(ψ_{E}^{a}G_{E}^{−1}ψ_{E}^{b}) F_{eff}^{ab} = F_{E}^{ab} /

⋆F_{s}^{ab} = 2κθ^{ab}  /  θ^{ab}

Type II B theory

Fermionic T_{IIBf}-dulity

B. Nikolić and B. Sazdović Fermionic T-duality and momenta noncommutativity hep-th/1103.4520 to be published in Phys. Rev. D

• Fermionic T-duality — Duality with respect to fermionic variables θ^{a}, θ̄^{a}

– Suppose that action has a global shift symmetry in θ^{α} and θ̄^{α} directions

– Similar procedure as in bosonic case produces T_{IIBf} dual background fields:

⋆B_{μν} = B_{μν} + [(Ψ̄F^{−1}Ψ)_{μν} − (Ψ̄F^{−1}Ψ)_{νμ}]

⋆G_{μν} = G_{μν} + 2 [(Ψ̄F^{−1}Ψ)_{μν} + (Ψ̄F^{−1}Ψ)_{νμ}]

⋆Ψ_{αμ} = 4(F^{−1}Ψ)_{αμ} , ^{⋆}Ψ̄_{μα} = −4(Ψ̄F^{−1})_{μα}

⋆F_{αβ} = 16(F^{−1})_{αβ}

Type II B theory

Fermionic T_{IIBf}-dulity and Dirichlet boundary conditions I

• T-duality Effective theory and Noncommuatativity

BOSONIC  ⟷  Neumann b.c. for x^{μ}

. SUSY b.c. for θ^{a}, θ̄^{a}

FERMIONIC  ⟷  ? b.c.

• DIRICHLET boundary conditions

x^{μ}|_{0}^{π} = 0,  θ^{α}|_{0}^{π} = 0,  θ̄^{α}|_{0}^{π} = 0

• Solve constraints – odd variables are independent

– trivial solution for coordinates, non-trivial for momenta

x^{μ}(σ) = q̃^{μ}(σ) , π_{μ} = p̃_{μ}−2κB_{μν}q̃^{′ν}−\frac{1}{2}Ψ̄_{μ}^{α}(η′_{a})_{α}+\frac{1}{2}(η̄′_{a})_{α}Ψ_{μ}^{α}

θ^{α}(σ) = θ_{a}^{α}(σ) , π_{α} = p̃_{α} − \frac{1}{2}(η̄′_{a})_{α}

θ̄^{α}(σ) = θ̄_{a}^{α}(σ) , π̄_{α} = ˜ p̄_{α} − \frac{1}{2}(η′_{a})_{α}

where

(η_{a})_{α} ≡ 4κ(F^{−1})_{αβ}(θ_{a}^{β}+Ψ_{μ}^{β}q̃^{μ}) , (η̄_{a})_{α} ≡ 4κ(θ̄_{a}^{β}+Ψ̄_{μ}^{β}q̃^{μ})(F^{−1})_{βα}

Type II B theory

Non-commutativity relations

• Non-commutativity relations

{P_{μ}(σ), P_{ν}(σ̄)}_{D} = Θ_{μν}Δ(σ + σ̄) , {P_{μ}(σ), P_{α}(σ̄)}_{D} = Θ̄_{μα}Δ(σ + σ̄) , {P_{μ}(σ), P̄_{α}(σ̄)}_{D} = Θ_{αμ}Δ(σ + σ̄) , {P_{α}(σ), P̄_{β}(σ̄)}_{D} = Θ_{αβ}Δ(σ + σ̄) ,

{P_{α}(σ), P_{β}(σ̄)}_{D} = {P̄_{α}(σ), P̄_{β}(σ̄)}_{D} = 0 ,

where the noncommutativity parameters are

Θ_{μν} = 2κ ^{⋆}B_{μν} , Θ̄_{μα} = \frac{κ}{2} ^{⋆}Ψ̄_{μα}

Θ_{αμ} = −\frac{κ}{2} ^{⋆}Ψ_{αμ} , Θ_{αβ} = −\frac{κ}{8} ^{⋆}F_{βα} ,

and

P_{A}(σ) = ∫_{0}^{σ} dσ_{1}π_{A}(σ_{1}) A = {μ, a, ā}

Relation between T-duality, effective theory and noncommuatativity

Type II B and fermionic duality

• T-duality  {Γ_{a}, Γ_{b}}  {P_{a}, P_{b}}
• ^{⋆}G_{μν}  ^{⋆}G_{μν}  /
⋆B_{μν}  /  θ_{μν} = 2κ^{⋆}B_{μν}
• ^{⋆}ψ_{aμ}  \frac{1}{2}^{⋆}ψ_{aμ}  θ_{aμ} = −\frac{κ}{2}^{⋆}ψ_{aμ}
⋆ψ̄_{μa}  \frac{1}{2}^{⋆}ψ̄_{μa}  θ̄_{μa} = \frac{κ}{2}^{⋆}ψ̄_{μa}
• ^{⋆}F_{ab}  − \frac{1}{8}^{⋆}F_{ab}  θ_{ab} = −\frac{κ}{8}^{⋆}F_{ab}

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

R_{μν} − \frac{1}{4}B_{μρσ}B_{ν}^{ρσ} = 0 ,

D_{ρ}B_{μν}^{ρ} = 0

– B_{μνρ} = ∂_{μ}B_{νρ} + ∂_{ν}B_{ρμ} + ∂_{ρ}B_{μν} is a field strength – R_{μν} and D_{μ} Ricci tensor and covariant derivative

• We will consider the following particular solution

G_{μν} = const, B_{μν}[x] = b_{μν} + \frac{1}{3}B_{μνρ}x^{ρ},

– b_{μν} is constant

– B_{μνρ} is constant and infinitesimally small

• – We will work up to the first order in B_{μνρ}

– Ricci tensor R_{μν} is an infinitesimal of the second order

and as such is neglected

Noncommutativity and T-duality Gravity 2011

T-duality of weakly curved background (T_{wcb})

Lj. Davidović and B. Sazdović T-duality in the weakly curved background in preparation

• More complicated procedure then in flat background ⋆G^{μν} = α′^{2}G_{E}^{−1μν}(^{⋆}x), ^{⋆}B^{μν} = α′θ^{μν}(^{⋆}x) ⋆x is T_{wcb} of x and ỹ is T_{0} dual of y ⋆x = g^{−1}(2by + ỹ)

Effective theory and non-commutativity in

weakly curved background

Lj. Davidović and B. Sazdović Phys. Rev. D 83 (2011) 066014

Lj. Davidović and B. Sazdović, Non-commutativity parameters depend not only on the effective coordinate but on its T-dual as well hep-th/1106.1064 to be published in JHEP

• Similar procedure but much more complicated calculation

• Effective background fields

G_{μν}^{eff}(u) = G_{μν}^{E}(u), B_{μν}^{eff} = −\frac{κ}{2}(gθ(u)g)_{μν}

u = q + 2bq̃

• Non-commutativity parameter

– Nontrivial both at string endpoints and at string interior

– Depends on the σ-integral of the effective momenta P_{μ}(σ) = ∫_{0}^{σ} dηp_{μ}(η)

which is in fact T_{0}-dual of the effective coordinate, P_{μ} = κg_{μν}q̃^{ν}.

Relation between T_{wcb}-duality, effective theory and noncommuatativity

• T_{wcb}-duality  Effective theory  Noncommuatativity

Dual background fields

⋆G^{μν} = α′^{2}G_{E}^{−1μν}(^{⋆}x) G_{μν}^{eff} = G_{μν}^{E}(u) /

⋆B^{μν} = α′θ^{μν}(^{⋆}x)  /  θ^{μν}(v)

Dual variables

⋆x = g^{−1}(2by+ỹ) u = q+2bq̃ v = q−\frac{3}{π}bQ̃_{cm}