B. Sazdović: Canonical Approach to Closed String Non-commutativity
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This document discusses closed string noncommutativity in string theory. It begins by reviewing open string noncommutativity and T-duality. It then considers a weakly curved background with both constant and coordinate-dependent fields. A generalized Buscher transformation is used to derive T-duality along all directions, resulting in a doubled geometry description. This leads to noncommutativity of both winding and momentum modes of the closed string. Nongeometric fluxes are also introduced that contribute to closed string noncommutativity.
![Open string noncommutativity
T-duality
Weakly curved background
Closed string noncommutativity
T-dual action and doubled geometry
xµ → Vµ = −κΘµν
0 yν + (g−1
E )µν ˜yν [xµ → (yµ, ˜yµ)].
⋆Gµν = (G−1
E )µν(∆V) and ⋆Bµν = κ
2 Θµν(∆V).
T-dual action is of the form
⋆
S =
κ2
2
∫
d2
ξ∂+yµΘµν
− (∆V)∂−yν ,
where
Θµν
± (x) = −
2
κ
(
G−1
E (x)Π±(x)G−1
)µν
, Π±µν = Bµν(x)±
1
2
Gµν .
Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity](https://image.slidesharecdn.com/1830-sazdovic-b-130508062433-phpapp02/85/B-Sazdovic-Canonical-Approach-to-Closed-String-Non-commutativity-11-320.jpg)
![Open string noncommutativity
T-duality
Weakly curved background
Closed string noncommutativity
General structure, y
{y′
µ(σ), y′
ν(¯σ)} = F′
µν[x(σ)]δ(σ − ¯σ)
{∆yµ(σ, σ0), ∆yν(¯σ, ¯σ0)} =
∫ σ
σ0
dσ1
∫ ¯σ
¯σ0
dσ2F′
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} = −
Cρ
Fµνρdxρ
= −2πFµνρNρ
,
where Nµ is winding number, xµ(2π) − xµ(0) = 2πNµ, and
Fµνρ =
∂Fµν
∂xρ are fluxes.
Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity](https://image.slidesharecdn.com/1830-sazdovic-b-130508062433-phpapp02/85/B-Sazdovic-Canonical-Approach-to-Closed-String-Non-commutativity-15-320.jpg)
![Open string noncommutativity
T-duality
Weakly curved background
Closed string noncommutativity
General structure, x
{x′µ
(σ), x′ν
(¯σ)} = F′µν
[y(σ), ˜y(σ)]δ(σ − ¯σ)
{∆xµ
(σ, σ0), ∆xν
(¯σ, ¯σ0)} =
∫ σ
σ0
dσ1
∫ ¯σ
¯σ0
dσ2F′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 yµ
yµ(σ = 2π) − yµ(σ = 0) = 2π⋆Nµ,
yµ(τ = 2π) − yµ(τ = 0) = 2π⋆pµ
Fµνρ = ∂Fµν
∂yρ
, ˜Fµνρ = ∂Fµν
∂˜yρ
are fluxes.
Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity](https://image.slidesharecdn.com/1830-sazdovic-b-130508062433-phpapp02/85/B-Sazdovic-Canonical-Approach-to-Closed-String-Non-commutativity-16-320.jpg)
![Open string noncommutativity
T-duality
Weakly curved background
Closed string noncommutativity
Noncommutativity of x coordinates
x Fµν
(y, ˜y) =
1
κ
[
Rµνρ
yρ − (˜Rµνρ
− 4 ˜Qµνρ
)˜yρ
]
{xµ
(2π), xν
(0)} = −2π2
{Nµ
, Nν
}
= −
2π
κ
[
Rµνρ⋆
Nρ − (˜Rµνρ
− 4 ˜Qµνρ
)⋆
pρ
]
,
All Poisson brackets close on winding numbers Nµ, ⋆Nµ
and momenta pµ, ⋆pµ.
Coefficients are: geometric flux Hµνρ and nongeometric
fluxes Rµνρ and ˜Rµνρ.
Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity](https://image.slidesharecdn.com/1830-sazdovic-b-130508062433-phpapp02/85/B-Sazdovic-Canonical-Approach-to-Closed-String-Non-commutativity-19-320.jpg)
B. Sazdović: Canonical Approach to Closed String Non-commutativity
- 1. Open string noncommutativity T-duality Weakly curved background Closed string noncommutativity Canonical approach to closed string noncommutativity Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Institute of Physics, University of Belgrade, Serbia Balkan Workshop BW2013, 25.-29.04.2013, Vrnjaˇcka banja, Serbia Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
- 2. Open string noncommutativity T-duality Weakly curved background Closed string noncommutativity Outline of the talk - Open string noncommutativity - T-duality - Weakly curved background - Closed string noncommutativity Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
- 3. Open string noncommutativity T-duality Weakly curved background Closed string noncommutativity Open string noncommutativity Extended objects (like strings) see space-time geometry different than point-particles - stringy property. The ends of the open string attached to Dp-brane become noncommutative in the presence of Kalb-Ramond field Bµν Action principle δS = 0 S(x) = κ ∫ Σ ( ηαβ 2 Gµν + εαβ Bµν)∂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. Open string noncommutativity T-duality Weakly curved background Closed string noncommutativity Open string noncommutativity Solution of boundary conditions xµ = qµ − Θµν ∫ σ dσ1pν(σ1) , where qµ and pν are effective variables satisfying {qµ(σ), pν(¯σ)} = 2δµ νδs(σ, ¯σ). The coordinate xµ is the linear combination of effective coordinate qµ and effective momenta pν - source of noncommutativity. 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. Open string noncommutativity T-duality Weakly curved background Closed string noncommutativity Open string noncommutativity Effective action S(q) = S(x)|bound.cond. = κ ∫ d2 ξ 1 2 GE µν∂+xµ ∂−xν , where GE µν = (G − 4BG−1 B)µν , Θµν = − 2 κ (G−1 E 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. Open string noncommutativity T-duality Weakly curved background Closed string noncommutativity T-duality It 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 state M2 = n2 R + m2 R2 α′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. Open string noncommutativity T-duality Weakly curved background Closed string noncommutativity T-duality Compactification on circle of radius R is equivalent to compactification on radius ˜R - purely stringy property. Dual action ⋆S has the same form as initial one but with different background fields ⋆ Gµν ∼ (G−1 E )µν , ⋆ Bµν ∼ Θµν , which are essentially parameters of open string noncommutativity. Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
- 8. Open string noncommutativity T-duality Weakly curved background Closed string noncommutativity T-duality Canonical T-duality transformations πµ ∼= κy′ µ , ⋆ πµ ∼= κx′µ . There is no closed string noncommutativity for constant Gµν 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. Open string noncommutativity T-duality Weakly curved background Closed string noncommutativity Choice of background fields Gµν is constant and Bµν = bµν + 1 3Hµνρxρ ≡ bµν + hµν(x). bµν and Hµνρ are constant and Hµνρ is infinitesimaly small. These background fields satisfy space-time equations of motion. Now Bµν is xµ dependent. Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
- 10. Open string noncommutativity T-duality Weakly curved background Closed string noncommutativity T-duality along all directions Generalized Buscher construction has two steps: gauging global symmetry δxµ = λµ which is a symmetry even Bµν is coordinate dependent ∂αxµ → Dαxµ = ∂αxµ + vµ α , xµ → ∆xµ inv = ∫ P dξα Dαxµ (this is a new step). Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
- 11. Open string noncommutativity T-duality Weakly curved background Closed string noncommutativity T-dual action and doubled geometry xµ → Vµ = −κΘµν 0 yν + (g−1 E )µν ˜yν [xµ → (yµ, ˜yµ)]. ⋆Gµν = (G−1 E )µν(∆V) and ⋆Bµν = κ 2 Θµν(∆V). T-dual action is of the form ⋆ S = κ2 2 ∫ d2 ξ∂+yµΘµν − (∆V)∂−yν , where Θµν ± (x) = − 2 κ ( G−1 E (x)Π±(x)G−1 )µν , Π±µν = Bµν(x)± 1 2 Gµν . Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
- 12. Open string noncommutativity T-duality Weakly curved background Closed string noncommutativity T-dual transformation laws ∂±xµ = −κΘµν ± (∆V)∂±yν ∓ 2κΘµν 0±β∓ ν (V) , ∂±yµ = −2Π∓µν(∆x)∂±xν ∓ β∓ µ (x) , where β± µ (x) = ∓ 1 6 Hµρσ∂∓xρ xσ . It is infinitesimally small and bilinear in xµ. Expression for β± µ comes from the term ∫ d2 ξvµ +Bµν(δV)vν − = ∫ d2 ξβα µ (V)δvµ α . Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
- 13. Open string noncommutativity T-duality Weakly curved background Closed string noncommutativity Transformation laws in canonical form x′µ = 1 κ ⋆ πµ − κΘµν 0 β0 ν (V) − (g−1 E )µν β1 ν (V) y′ µ = 1 κ πµ − β0 µ(x) . These infinitesimal βα µ -terms are improvements in comparison with 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. Open string noncommutativity T-duality Weakly curved background Closed string noncommutativity Choosing evolution parameter ∆xµ (σ, σ0) = xµ (ξ) − xµ (ξ0) = ∫ ( ˙xµ dτ + x′µ dσ) If we take evolution parameter orthogonal to ξ − ξ0 then we have ∆xµ (σ, σ0) = ∫ σ σ0 dσ1x′µ (σ1) , ∆yµ(σ, σ0) = ∫ σ σ0 dσ1y′ µ(σ1) . We use information from one background to compute Poisson brackets in T-dual one. Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
- 15. Open string noncommutativity T-duality Weakly curved background Closed string noncommutativity General structure, y {y′ µ(σ), y′ ν(¯σ)} = F′ µν[x(σ)]δ(σ − ¯σ) {∆yµ(σ, σ0), ∆yν(¯σ, ¯σ0)} = ∫ σ σ0 dσ1 ∫ ¯σ ¯σ0 dσ2F′ 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} = − Cρ Fµνρdxρ = −2πFµνρNρ , where Nµ is winding number, xµ(2π) − xµ(0) = 2πNµ, and Fµνρ = ∂Fµν ∂xρ are fluxes. Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
- 16. Open string noncommutativity T-duality Weakly curved background Closed string noncommutativity General structure, x {x′µ (σ), x′ν (¯σ)} = F′µν [y(σ), ˜y(σ)]δ(σ − ¯σ) {∆xµ (σ, σ0), ∆xν (¯σ, ¯σ0)} = ∫ σ σ0 dσ1 ∫ ¯σ ¯σ0 dσ2F′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 yµ yµ(σ = 2π) − yµ(σ = 0) = 2π⋆Nµ, yµ(τ = 2π) − yµ(τ = 0) = 2π⋆pµ Fµνρ = ∂Fµν ∂yρ , ˜Fµνρ = ∂Fµν ∂˜yρ are fluxes. Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
- 17. Open string noncommutativity T-duality Weakly curved background Closed string noncommutativity Noncommutativity of y coordinates y Fµν(x) = 1 κ Hµνρxρ, Hµνρ is field strength for Bµν. {yµ(2π), yν(0)} = −2π2 { ⋆ Nµ, ⋆ Nν} ∼= − 2π κ BµνρNρ . xµ(τ, σ) = xµ 0 + 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. Open string noncommutativity T-duality Weakly curved background Closed string noncommutativity Nongeometric fluxes ⋆Bµν depends on double space coordinates (yµ , ˜yµ) ⋆ Bµν = ⋆ bµν + Qµνρ yρ + ˜Qµνρ˜yρ . There are two field strengths Rµνρ = 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. Open string noncommutativity T-duality Weakly curved background Closed string noncommutativity Noncommutativity of x coordinates x Fµν (y, ˜y) = 1 κ [ Rµνρ yρ − (˜Rµνρ − 4 ˜Qµνρ )˜yρ ] {xµ (2π), xν (0)} = −2π2 {Nµ , Nν } = − 2π κ [ Rµνρ⋆ Nρ − (˜Rµνρ − 4 ˜Qµνρ )⋆ pρ ] , All Poisson brackets close on winding numbers Nµ, ⋆Nµ and momenta pµ, ⋆pµ. Coefficients are: geometric flux Hµνρ and nongeometric fluxes Rµνρ and ˜Rµνρ. Ljubica Davidovi´c, Bojan Nikoli´c and Branislav Sazdovi´c Canonical approach to closed string noncommutativity
- 20. Open string noncommutativity T-duality Weakly curved background Closed string noncommutativity Additional relations If we dualize all directions we have {xµ, yν} = 0. In the case of partial T-dualization xµ = (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