b. sazdovic - noncommutativity and t-duality

Noncommutativity and T-duality

Lj. Davidovic, B. Nikolic and B. SazdovicInstitute of Physics, Belgrade, Serbia

• We will discuss relation between

– Open string parameters

Geffµν (G, B) and θµν(G, B)

– and T-dual background fields

?Gµν(G, B) and ?Bµν(G, B)

as functions of the initial background fields:

metric tensor Gµν and Kalb-Ramond field Bµν

• 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 θµν(G, B)

Noncommutativity and T-duality BSW 2011

• Effective theory

On the solution of boundary conditions the initial theory

turns to the effective one with effective metric tensor

Geffµν (G, B) and vanishing effective Kalb-Ramond field

• T-duality

T-duality in presence of background fields leads T-dual

background fields ?Gµν(G, B) and ?Bµν(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” Bµν[x] = bµν + 13Bµνρxρ

instead of the flat one Bµν = bµν = const.


1

The action

describing the open string propagation in curved background

S = κ

Z

Σ

d2ξp−ghgαβ

2Gµν(x)+

εαβ

√−gBµν(x)

i∂αx

µ∂βx

ν,

• xµ(ξ), µ = 0, 1, ..., D − 1 the coordinates of the

D-dimentional space-time

• ξα(ξ0 = τ, ξ1 = σ) parametrize 2-dim world-sheet

• gαβ(ξ) intrinsic world-sheet metric (g = detgαβ)

• background fields

– Gµν(x) space-time metric

– Bµν(x) Kalb-Ramond antisymmetric field


2

Action principle for string

Evolution from the initial to final configuration is such that the

action is stationary

S =R τf

τidτR σf

σidσL(xµ, xµ, x′µ, gαβ)

δS =

Zdτdσ

h ∂L∂xµ

− ∂τ

∂L∂xµ

− ∂σ

∂L∂xµ

iδx

µ

+

Zdτh ∂L∂xµ

δxµi

σ=π

σ=0

From the action principle we get

1) equation of motion

xµ = x′′µ − 2Bµνρx

νx′ρ,

Bµνρ = ∂µBνρ + ∂νBρµ + ∂ρBµν is field strength

2) boundary condition

γµ0 δxµ

σ=0,π

= 0, γµ0 ≡ ∂L

∂xµ = x′µ − 2(G−1B)µνx

ν


3

The Boundary conditions

• The closed string fulfills the boundary condition because

xµ(0) = xµ(π)

• For the open string we can impose

1) Neumann boundary condition

hδxµi

0,

hδxµi

π

are arbitrary i.e. string end-points can move freely

γ0µ

σ=0

= 0, γ0µ

σ=π

= 0

2) Dirichlet boundary condition

hδxµi

σ=0= 0,

hδxµi

σ=π= 0

The edges of the string are fixed


4

Noncommutativity and effective theorybosonic open string in flat space-time I

• We impose Neumann boundary conditions

• We treat boundary conditions as constraints

• Constraint γ0µ must be conserved in time

– Secundary constraint γ1µ = γ0

µ

– Infinite set of constraints γnµ = γn−1

µ , (n = 1, 2, · · ·)

– Two σ-dependent constraints

Γµ(σ) ≡P∞n=0

σn

(n)!γµn

σ=0

Γµ(σ) ≡P∞n=0

(σ−π)n

(n)! γµn

σ=π

• 2π-periodicity xµ(σ) = xµ(σ + 2π)

solve constraint at σ = π Γµ(σ) = 0 → Γµ(σ) = 0


5

Noncommutativity and effective theorybosonic open string in flat space-time II

• Solving the constraints

– In canonical formalism

Γµ(σ), Γν(σ) = −κGEµνδ′(σ − σ)

For GEµν 6= 0 Γµ(σ) are SSC

– Introduce world-sheet parity Ω

Ω : σ → −σ, Ωxµ(σ) → xµ(−σ)

and new variables

qµ = 12(1 + Ω)xµ qµ = 1

2(1− Ω)xµ

– Solve Ω odd parts in terms of Ω even

∗ q = f1(q, p), p = f2(q, p)

∗ xµ = qµ − 2θµνR

dσ1pν, πµ = pµ


5

• Effective action and background fields

– Seff = κR

d2ξ12η

αβGEµν∂αqµ∂βqν

– Gµν → Geffµν = GE

µν, Bµν → Beffµν = 0

GEµν ≡ [G− 4BG−1B]µν

effective metric

• Noncommuatativity

Xµ(σ), Xν(σ) = θµν

8<:−1 σ, σ = 0

1 σ, σ = π

0 otherwise

.

θµν ≡ −2κ(G

−1E BG−1)µν

non-commutativity parameter


6

T0-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, Bµν = 0, Φ = 0

• – x25(σ + π) = x25(σ) + 2πRm, (m ∈ Z)

– m– winding number

• Consequences of compactification:

– Momentum along circle is quantized, p = nR (n ∈ Z),

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

Gained some states


7

Surprising symmetry as stringy property

• Mass square of the states

M2 = n2

R2 + m2R2

α′2+ contributions from oscilators

• – Complementary behavior of

momentum and winding states

– M2(R, n, m) = M2(α′R , m, n)

– R ←→ α′R ≡ R, n ←→ m, R — Dual radius

• T0 duality for closed string

Compactification with radius R

is physically indistinguishable fromCompactification with radius R = α′

R

• T0 dual coordinate

– Equation of motion

∂+∂−x = 0 =⇒ x = x+(τ + σ) + x−(τ − σ)

– T0 dual coordinate

x ≡ x+(τ + σ)− x−(τ − σ)


8

T-duality – nontrivial background I

• – Background fields are independent of the circular

coordinate

– We take all coordinate to be circular

→ Gµν, Bµν = const

Toroidal duality of all cordinates

• Lagrange multiplier method

?S[y, v+, v−] =

κ

Zd

2ξv

µ+(B + 1

2G)µνvν− +

Zd

2ξyµ(∂+v

µ−− ∂−v

µ+),

– yµ – Lagrange multiplier

• Integration over y returns to the original action

∂+vµ− − ∂−vµ

+ = 0 ⇒ vµ± = ∂±xµ

• Integrating out vector field v±

vµ±(y) = −2[θµν ∓ 1

κ(G−1E )µν]∂±yν

GEµν ≡ [G− 4BG−1B]µν, θµν ≡ −2

κ(G−1E BG−1)µν

are the open string background fields:

effective metric and non-commutativity parameter


9

T-duality–nontrivial background II

• ?S[∂+y, ∂−y] = 2R

d2ξ∂+yµ[θµν + 1

κ(G−1E )µν]∂+yµ

= κR

d2ξ∂+yµ(?B + 1

2

?G)µν∂+yµ

• Dual background fields

?Bµν = 2κθµν, ?Gµν = (2

κ)2(G−1

E )µν

• Turn off background fields

– Bµν → 0, Gµν → (ηµν, G25,25 = G), κ → 2α′

– 2πR =R 2π

0ds =

√GR 2π

0dθ = 2π

√G

⇒ G = R2, ?G = R2

– ?G = α′2G−1 ⇒ RR = α′

– ∂±x = ±∂±y ⇒ y = x


10

Relation between T-duality, effective theoryand noncommuatativity

• T-duality

?Gµν = α′2G−1µνE , ?Bµν = α′θµν

• Effective theory

Geffµν = GE

µν /

• Noncommuatativity

/ θµν

• The same background fields: effective metric

– GEµν ≡ [G− 4BG−1B]µν

and non-commutativity parameter

θµν ≡ −2κ(G

−1E BG−1)µν


11

Type II B theory

• Type IIB theory in pure spinor formulation

S = κ

Z

Σd2ξ

1

2η

mnGµν + ε

mnBµν

∂mx

µ∂nx

ν

+

Z

Σd2ξ

−πα∂−(θ

α+ Ψ

αµx

µ) + ∂+(θ

α+ Ψ

αµx

µ)πα +

1

2κπαF

αβπβ

• Variables

xµ, θα and θα

• Background fields

– NS-NS Gµν, Bµν

– NS-R Ψαµ, Ψα

µ , gravitinos

– R-R F αβ ∼ A0, A2, A4, dA4-self dual


12

Type II B theoryNeumann boundary conditions

• Boundary conditions

hγ

(0)i δx

i+παδθ

α+ δθ

απα

i π

0= 0

γ(0)i = Π+i

jI−j + Π−i

jI+j +

παΨ

αi + Ψ

αi πα

• For bosonic coordinates Neumann boundary conditions

γ(0)i

π0

= 0

• Fermionic coordinates preserves N=1 SUSY

(θα − θ

α)π

0= 0 ⇒ (πα1

− πα1)π0

= 0


13

Type II B theoryNeumann b. c., Effective theory and non-commutativity

B.Nikolic and B. Sazdovic, Phys. Lett. B666 (2008) 400

B. Nikolic and B. Sazdovic, Nucl. Phys. B 836 (2010) 100

• Similar method as in bosonic case

• Background fields

– Ω even corresponds to Type I

Gµν → GEµν

1

2Ψ

α+µ → (ΨE)

αµ =

1

2Ψ

α+µ + (BG

−1Ψ−)

αµ

Fαβa → F

αβE = F

αβ − (Ψ−G−1

Ψ−)αβ

– Ω odd fields vanish Bµν → 0, Ψ− → 0, Fs → 0

• Non-commutativity

Ω odd fields are source of non-commutativity

xµ(σ) , x

ν(σ) = 2θ

µνθ(σ + σ)

xµ(σ) , θ

α(σ) = −θ

µαθ(σ + σ)

θα(σ) , θ

β(σ) =

1

2θ

αβθ(σ + σ)


14

Type II B theoryBosonic TIIBb-dulity

• Action has global shift symmetry in bosonic direction

Similar method produce dual background fields

?Gµν = α′2G−1µνE , ?Bµν = α′θµν

?ψaµ− = −2G−1µν

E (ψE)aν

?ψaµ+ = 2κθaµ

?F aba = F ab

E + 4(ψaEG−1

E ψbE) ?F ab

s = 2κθab


15

Relation between T-duality, effective theoryand noncommuatativity

Type II B and bosonic duality

• T-duality Effective theory Noncommuatativity

Bosonic N bc Ω-symm Ω-antisymm

• ?Gµν = α′2G−1µνE Geff

µν = GEµν /

?Bµν = α′θµν / θµν

• ?ψaµ− = −2(G−1

E ψE)aµ (ψeff)aµ = (ψE)a

µ /

?ψaµ+ = 2κθaµ / θaµ

• ?F aba = F ab

E + 4(ψaEG−1

E ψbE) F ab

eff = F abE /

?F abs = 2κθab / θab


16

Type II B theoryFermionic TIIBf -dulity

B. Nikolic and B. Sazdovic

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 θa, θa

– Suppose that action has a global shift symmetry in

θα and θα directions

– Similar procedure as in bosonic case produces

TIIBf dual background fields:

?Bµν = Bµν +

h(ΨF

−1Ψ)µν − (ΨF

−1Ψ)νµ

i

?Gµν = Gµν + 2

h(ΨF

−1Ψ)µν + (ΨF

−1Ψ)νµ

i

?Ψαµ = 4(F

−1Ψ)αµ ,

?Ψµα = −4(ΨF

−1)µα

?Fαβ = 16(F

−1)αβ


17

Type II B theoryFermionic TIIBf -dulity and Dirichlet boundary conditions I

• T-duality Effective theory and Noncommuatativity

BOSONIC ←→ Neumann b.c. for xµ

. SUSY b.c. for θa, θa

FERMIONIC ←→ ? b.c.

• DIRICHLET boundary conditions

xµπ

0= 0, θ

απ

0= 0, θ

απ

0= 0

• Solve constraints

– odd variables are independent– trivial solution for coordinates, non-trivial for momenta

xµ(σ) = q

µ(σ) , πµ = pµ−2κBµν q

′ν−1

2Ψ

αµ(η

′a)α+

1

2(η′a)αΨ

αµ

θα(σ) = θ

αa (σ) , πα = pα −

1

2(η′a)α

θα(σ) = θ

αa (σ) , πα = ˜pα −

1

2(η′a)α

where

(ηa)α ≡ 4κ(F−1

)αβ(θβa+Ψ

βµq

µ) , (ηa)α ≡ 4κ(θ

βa+Ψ

βµq

µ)(F

−1)βα


18

Type II B theoryNon-commutativity relations

• Non-commutativity relations

Pµ(σ),Pν(σ)D = Θµν∆(σ + σ) ,

Pµ(σ),Pα(σ)D = Θµα∆(σ + σ) ,Pµ(σ), Pα(σ)

D= Θαµ∆(σ + σ) ,Pα(σ), Pβ(σ)

D

= Θαβ∆(σ + σ) ,

Pα(σ),Pβ(σ)D =Pα(σ), Pβ(σ)

D

= 0 ,

where the noncommutativity parameters are

Θµν = 2κ?Bµν , Θµα =

κ

2

?Ψµα

Θαµ = −κ

2

?Ψαµ , Θαβ = −κ

8

?Fβα ,

and

PA(σ) =

Z σ

0

dσ1πA(σ1) A = µ, a, a


19

Relation between T-duality, effective theoryand noncommuatativityType II B and fermionic duality

• T-duality Γa, Γb Pa,Pb

• ?Gµν?Gµν /

?Bµν / θµν = 2κ?Bµν

• ?ψaµ12

?ψaµ θaµ = −κ2

?ψaµ

?ψµa12

?ψµa θµa = κ2

?ψµa

• ?Fab − 18

?Fab θab = −κ8

?Fab


20

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

Rµν −1

4BµρσB

ρσν = 0 ,

DρBρµν = 0

– Bµνρ = ∂µBνρ + ∂νBρµ + ∂ρBµν is a field strength

– Rµν and Dµ Ricci tensor and covariant derivative

• We will consider the following particular solution

Gµν = const, Bµν[x] = bµν +1

3Bµνρx

ρ,

– bµν is constant

– Bµνρ is constant and infinitesimally small

• – We will work up to the first order in Bµνρ

– Ricci tensor Rµν is an infinitesimal of the second order

and as such is neglected


21

T-duality of weakly curved background (Twcb)

Lj. Davidovic and B. Sazdovic

T-duality in the weakly curved background

in preparation

• More complicated procedure then in flat background

?Gµν = α′2G−1µνE (?x), ?Bµν = α′θµν(?x)

?x is Twcb of x and y is T0 dual of y

?x = g−1(2by + y)


22

Effective theory and non-commutativity inweakly curved background

Lj. Davidovic and B. SazdovicPhys. Rev. D 83 (2011) 066014

Lj. Davidovic and B. Sazdovic,

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

Geffµν (u) = GE

µν(u), Beffµν = −κ

2(gθ(u)g)µν

u = q + 2bq

• Non-commutativity parameter

– Nontrivial both at string endpoints and at string interior

– Depends on the σ-integral of the effective momenta

Pµ(σ) =R σ

0dηpµ(η)

which is in fact T0-dual of the effective coordinate,

Pµ = κgµνqν.


23

Relation between Twcb-duality, effective theoryand noncommuatativity

• Twcb-duality Effective theory Noncommuatativity

Dual background fields

?Gµν = α′2G−1µνE (?x) Geff

µν = GEµν(u) /

?Bµν = α′θµν(?x) / θµν(v)

Dual variables

?x = g−1(2by+y) u = q+2bq v = q− 3πbQcm


b. sazdovic - noncommutativity and t-duality

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