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

Open string noncommutativityT-duality

Weakly curved backgroundClosed string noncommutativity

Canonical approach to closed stringnoncommutativity

Ljubica Davidovic, Bojan Nikolic andBranislav Sazdovic

Institute of Physics, University of Belgrade, Serbia

Balkan Workshop BW2013, 25.-29.04.2013, Vrnjacka banja,Serbia

Ljubica Davidovic, Bojan Nikolic and Branislav Sazdovic Canonical approach to closed string noncommutativity



Outline of the talk

- Open string noncommutativity- T-duality- Weakly curved background- Closed string noncommutativity




Open string noncommutativity

Extended 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 Bµν

Action principle δS = 0

S(x) = κ

∫Σ(ηαβ

2Gµν + εαβBµν)∂αxµ∂βxν ,

gives equations of motion and boundary conditions.





Solution of boundary conditions

xµ = qµ −Θµν

∫ σ

dσ1pν(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 .





Effective action

S(q) = S(x)|bound.cond. = κ

∫d2ξ

12

GEµν∂+xµ∂−xν ,

where

GEµν = (G − 4BG−1B)µν , Θµν = −2

κ(G−1

E BG−1)µν ,

are effective metric and noncommutativity parameter,respectively.




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 .




T-duality

Compactification 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.




T-duality

Canonical T-duality transformations

πµ ∼= κy ′µ ,

⋆πµ ∼= κx ′µ .

There is no closed string noncommutativity for constantGµν and Bµν

{πµ, πν} = 0 =⇒ {yµ, yν} = 0 .




Choice of background fields

Gµν 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.




T-duality along all directions

Generalized Buscher construction has two steps:gauging global symmetry δxµ = λµ which is a symmetryeven Bµν is coordinate dependent

∂αxµ → Dαxµ = ∂αxµ + vµα ,

xµ → ∆xµinv =

∫P dξαDαxµ (this is a new step).




T-dual action and doubled geometry

xµ → Vµ = −κΘµν0 yν + (g−1

E )µν yν [xµ → (yµ, yµ)].⋆Gµν = (G−1

E )µν(∆V ) and ⋆Bµν = κ2Θ

µν(∆V ).T-dual action is of the form

⋆S =κ2

2

∫d2ξ∂+yµΘ

µν− (∆V )∂−yν ,

where

Θµν± (x) = −2

κ

(G−1

E (x)Π±(x)G−1)µν

, Π±µν = Bµν(x)±12

Gµν .




T-dual transformation laws

∂±xµ = −κΘµν± (∆V )∂±yν ∓ 2κΘµν

0±β∓ν (V ) ,

∂±yµ = −2Π∓µν(∆x)∂±xν ∓ β∓µ (x) ,

whereβ±µ (x) = ∓1

6Hµρσ∂∓xρxσ .

It is infinitesimally small and bilinear in xµ. Expression for β±µ

comes from the term∫d2ξvµ

+Bµν(δV )vν− =

∫d2ξβα

µ (V )δvµα .




Transformation laws in canonical form

x ′µ =1κ⋆πµ − κΘµν

0 β0ν (V )− (g−1

E )µνβ1ν (V )

y ′µ =

1κπµ − β0

µ(x) .

These infinitesimal βαµ -terms are improvements in comparison

with flat space. Also they are source of noncommutativity.




Choosing 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) =

∫ σ

σ0

dσ1x ′µ(σ1) , ∆yµ(σ, σ0) =

∫ σ

σ0

dσ1y ′µ(σ1) .

We use information from one background to compute Poissonbrackets in T-dual one.




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, andFµνρ =

∂Fµν

∂xρ are fluxes.Ljubica Davidovic, Bojan Nikolic and Branislav Sazdovic Canonical approach to 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 yµyµ(σ = 2π)− yµ(σ = 0) = 2π⋆Nµ,yµ(τ = 2π)− yµ(τ = 0) = 2π⋆pµ

Fµνρ = ∂Fµν

∂yρ , Fµνρ = ∂Fµν

∂yρare fluxes.




Noncommutativity of y coordinates

yFµν(x) = 1κHµνρxρ, Hµνρ is field strength for Bµν .

{yµ(2π), yν(0)} = −2π2{ ⋆Nµ,⋆Nν} ∼= −2π

κBµνρNρ.

xµ(τ, σ) = xµ0 + pµτ + Nµσ + osc.

yµ(τ, σ) = y0µ + ⋆pµτ + ⋆Nµσ + osc. .




Nongeometric fluxes

⋆Bµν depends on double space coordinates (yµ , yµ)

⋆Bµν = ⋆bµν + Qµνρyρ + Qµνρyρ .

There are two field strengths

Rµνρ = Qµνρ + cycl. , Rµνρ = Qµνρ + cycl. .

Rµνρ and Rµνρ are nongeometric fluxes.




Noncommutativity of x coordinates

xFµν(y , y) =1κ

[Rµνρyρ − (Rµνρ − 4Qµνρ)yρ

]{xµ(2π), xν(0)} = −2π2 {Nµ,Nν}

= −2πκ

[Rµνρ⋆Nρ − (Rµνρ − 4Qµνρ)⋆pρ

],

All Poisson brackets close on winding numbers Nµ, ⋆Nµ

and momenta pµ, ⋆pµ.Coefficients are: geometric flux Hµνρ and nongeometricfluxes Rµνρ and Rµνρ.




Additional relations

If we dualize all directions we have {xµ, yν} = 0.In the case of partial T-dualizationxµ = (x i , xa) → (x i , ya, ya) it holds {x i , ya} = 0.


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

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