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Balkan Workshop BW2013 Beyond the Standard Models 25 – 29 April, 2013, Vrnjačka Banja, Serbia
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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
Open string noncommutativityT-duality
Weakly curved backgroundClosed string noncommutativity
Outline of the talk
- Open string noncommutativity- T-duality- Weakly curved background- Closed string noncommutativity
Ljubica Davidovic, Bojan Nikolic and Branislav Sazdovic Canonical approach to closed string noncommutativity
Open string noncommutativityT-duality
Weakly curved backgroundClosed 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.
Ljubica Davidovic, Bojan Nikolic and Branislav Sazdovic Canonical approach to closed string noncommutativity
Open string noncommutativityT-duality
Weakly curved backgroundClosed 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)} = 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 Davidovic, Bojan Nikolic and Branislav Sazdovic Canonical approach to closed string noncommutativity
Open string noncommutativityT-duality
Weakly curved backgroundClosed string noncommutativity
Open string noncommutativity
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.
Ljubica Davidovic, Bojan Nikolic and Branislav Sazdovic Canonical approach to closed string noncommutativity
Open string noncommutativityT-duality
Weakly curved backgroundClosed 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 Davidovic, Bojan Nikolic and Branislav Sazdovic Canonical approach to closed string noncommutativity
Open string noncommutativityT-duality
Weakly curved backgroundClosed string noncommutativity
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.
Ljubica Davidovic, Bojan Nikolic and Branislav Sazdovic Canonical approach to closed string noncommutativity
Open string noncommutativityT-duality
Weakly curved backgroundClosed string noncommutativity
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 .
Ljubica Davidovic, Bojan Nikolic and Branislav Sazdovic Canonical approach to closed string noncommutativity
Open string noncommutativityT-duality
Weakly curved backgroundClosed string noncommutativity
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.
Ljubica Davidovic, Bojan Nikolic and Branislav Sazdovic Canonical approach to closed string noncommutativity
Open string noncommutativityT-duality
Weakly curved backgroundClosed string noncommutativity
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).
Ljubica Davidovic, Bojan Nikolic and Branislav Sazdovic Canonical approach to closed string noncommutativity
Open string noncommutativityT-duality
Weakly curved backgroundClosed string noncommutativity
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 .
Ljubica Davidovic, Bojan Nikolic and Branislav Sazdovic Canonical approach to closed string noncommutativity
Open string noncommutativityT-duality
Weakly curved backgroundClosed string noncommutativity
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 .
Ljubica Davidovic, Bojan Nikolic and Branislav Sazdovic Canonical approach to closed string noncommutativity
Open string noncommutativityT-duality
Weakly curved backgroundClosed string noncommutativity
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.
Ljubica Davidovic, Bojan Nikolic and Branislav Sazdovic Canonical approach to closed string noncommutativity
Open string noncommutativityT-duality
Weakly curved backgroundClosed string 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.
Ljubica Davidovic, Bojan Nikolic and Branislav Sazdovic Canonical approach to closed string noncommutativity
Open string noncommutativityT-duality
Weakly curved backgroundClosed 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, andFµνρ =
∂Fµν
∂xρ are fluxes.Ljubica Davidovic, Bojan Nikolic and Branislav Sazdovic Canonical approach to closed string noncommutativity
Open string noncommutativityT-duality
Weakly curved backgroundClosed 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.
Ljubica Davidovic, Bojan Nikolic and Branislav Sazdovic Canonical approach to closed string noncommutativity
Open string noncommutativityT-duality
Weakly curved backgroundClosed string noncommutativity
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. .
Ljubica Davidovic, Bojan Nikolic and Branislav Sazdovic Canonical approach to closed string noncommutativity
Open string noncommutativityT-duality
Weakly curved backgroundClosed string noncommutativity
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.
Ljubica Davidovic, Bojan Nikolic and Branislav Sazdovic Canonical approach to closed string noncommutativity
Open string noncommutativityT-duality
Weakly curved backgroundClosed string noncommutativity
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.
Ljubica Davidovic, Bojan Nikolic and Branislav Sazdovic Canonical approach to closed string noncommutativity
Open string noncommutativityT-duality
Weakly curved backgroundClosed string noncommutativity
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.
Ljubica Davidovic, Bojan Nikolic and Branislav Sazdovic Canonical approach to closed string noncommutativity