B. Sazdovic - Noncommutativity and T-duality

Noncommutativity and T-duality
Lj. Davidovi´, B. Nikoli´ and B. Sazdovi´
c c c
Institute of Physics, Belgrade, Serbia

• We will discuss relation between

– Open string parameters

Gef f (G, B) and θ µν (G, B)
µν

– and T-dual background fields

Gµν (G, B) and B µν (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 θ µν (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
Gef f (G, B) and vanishing effective Kalb-Ramond field
µν

• T-duality
T-duality in presence of background fields leads T-dual
background fields Gµν (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” Bµν [x] = bµν + 1 Bµνρxρ
3
instead of the flat one Bµν = bµν = const.


1
The action

describing the open string propagation in curved background

2 g αβ αβ
µ ν
S=κ d ξ −g Gµν (x)+ √ Bµν (x) ∂αx ∂β x ,
Σ 2 −g

• xµ(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

• 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

τf σf
S= τi
dτ σi
dσL(xµ, xµ, x µ, gαβ )
˙

∂L ∂L ∂L µ
δS = dτ dσ − ∂τ µ − ∂σ µ δx
∂xµ ∂x
˙ ∂x
˙

∂L µ σ=π
+ dτ δx
∂ xµ
˙ σ=0

From the action principle we get

1) equation of motion

xµ = x
¨ µ
− 2B µ xν x ρ,
νρ ˙

Bµνρ = ∂µBνρ + ∂ν Bρµ + ∂ρBµν is field strength

2) boundary condition
µ µ
γ0 δxµ = 0, γ0 ≡ ∂L
∂ xµ
˙ = x µ − 2(G−1B)µν xν
˙
σ=0,π


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

δxµ , δxµ
0 π

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

0 0
γµ = 0 , γµ =0
σ=0 σ=π

2) Dirichlet boundary condition

δxµ = 0, δxµ =0
σ=0 σ=π

The edges of the string are ﬁxed


4
Noncommutativity and effective theory
bosonic open string in flat space-time I

• We impose Neumann boundary conditions

• We treat boundary conditions as constraints

0
• Constraint γµ must be conserved in time

1
˙0
– Secundary constraint γµ = γµ

n
˙ n−1
– Infinite set of constraints γµ = γµ , (n = 1, 2, · · ·)

– Two σ -dependent constraints

∞ σn µ
Γµ(σ) ≡ n=0 (n)! γn
σ=0

¯ ∞ (σ−π)n µ
Γµ(σ) ≡ n=0 (n)! γn
σ=π

• 2π -periodicity xµ(σ) = xµ(σ + 2π)

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


5
Noncommutativity and eﬀective theory
bosonic open string in ﬂat space-time II

• Solving the constraints

– In canonical formalism

{Γµ(σ), Γν (¯ )} = −κGE δ (σ − σ )
σ µν ¯

For GE = 0
µν Γµ(σ) are SSC

– Introduce world-sheet parity Ω

Ω : σ → −σ , Ωxµ(σ) → xµ(−σ)

and new variables

q µ = 1 (1 + Ω)xµ
2 q µ = 1 (1 − Ω)xµ
¯ 2

– Solve Ω odd parts in terms of Ω even

∗ q = f1(q, p),
¯ p = f2(q, p)
¯

∗ xµ = q µ − 2θ µν dσ1pν , π µ = pµ


5

• Effective action and background fields

– S ef f = κ d2ξ 1 η αβ GE ∂αq µ∂β q ν
2 µν

– Gµν → Gef f = GE ,
µν µν Bµν → Bµνf = 0
ef

GE ≡ [G − 4BG−1B]µν
µν

effective metric

• Noncommuatativity

 −1 σ, σ = 0
¯
{X µ(σ), X ν (¯ )} = θ µν
σ 1 σ, σ = π
¯ .

0 otherwise

θ µν ≡ − κ (G−1BG−1)µν
2
E

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:
n
– Momentum along circle is quantized, p = R (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

n2 m2 R 2
M2 = R2
+ + contributions from oscilators
α2

• – Complementary behavior of
momentum and winding states

– M 2(R, n, m) = M 2( α , m, n)
R

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

• T0 duality for closed string

Compactiﬁcation with radius R
is physically indistinguishable from
˜
Compactiﬁcation 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−] =

2 µ ν 2 µ µ
κ d ξv+(B + 1 G)µν v− +
2 d ξyµ(∂+v− − ∂−v+),

– yµ – Lagrange multiplier

• Integration over y returns to the original action

µ µ µ
∂+v− − ∂−v+ = 0 ⇒ v± = ∂±xµ

• Integrating out vector field v±

µ
v±(y) = −2[θ µν 1 −1 µν
κ (GE ) ]∂± yν

GE ≡ [G − 4BG−1B]µν ,
µν θ µν ≡ − κ (G−1BG−1)µν
2
E

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


9
T-duality–nontrivial background II

• S[∂+y, ∂−y] = 2 d2ξ∂+yµ[θ µν + κ (G−1)µν ]∂+yµ
1
E

=κ d2ξ∂+yµ( B + 1
2 G)µν ∂+yµ

• Dual background fields

B µν = κ θ µν ,
2
Gµν = ( κ )2(G−1)µν
2
E

• Turn off background fields

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

2π √ 2π √
– 2πR = 0
ds = G 0
dθ = 2π G

⇒ G = R2 , ˜
G = R2
2 ˜
– G = α G−1 ⇒ RR = α

– ∂±x = ±∂±y ⇒ y=x
˜


10
Relation between T-duality, effective theory
and noncommuatativity

• T-duality

2
Gµν = α G−1µν ,
E B µν = α θ µν

• Effective theory

Gef f = GE
µν µν /

• Noncommuatativity

/ θ µν

• The same background fields: effective metric

– GE ≡ [G − 4BG−1B]µν
µν

and non-commutativity parameter

θ µν ≡ − κ (G−1BG−1)µν
2
E


11
Type II B theory

• Type IIB theory in pure spinor formulation
2 1 mn mn µ ν
S=κ d ξ η Gµν + ε Bµν ∂m x ∂n x
Σ 2
2 α α µ ¯α ¯α µ π 1 αβ
+ d ξ −πα ∂− (θ + Ψµ x ) + ∂+ (θ + Ψµ x )¯ α + πα F πβ¯
Σ 2κ

• Variables

¯
xµ, θ α and θ α

• Background ﬁelds

– NS-NS Gµν , Bµν

– NS-R ¯
Ψα, Ψα , gravitinos
µ µ

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


12
Type II B theory
Neumann boundary conditions

• Boundary conditions
π
(0)
γi δx
i ¯α ¯
α
+ παδθ + δ θ πα =0
0

(0) j j α ¯ α¯
γi = Π+i I−j + Π−i I+j + παΨi + Ψi πα

• For bosonic coordinates Neumann boundary conditions
(0) π
γi 0
=0

• Fermionic coordinates preserves N=1 SUSY
π π
α¯α
(θ − θ ) = 0 ⇒ (πα1 − πα1 )
¯ =0
0 0


13
Type II B theory
Neumann b. c., Effective theory and non-commutativity

B.Nikoli´ and B. Sazdovi´, Phys. Lett. B666 (2008) 400
c c

B. Nikoli´ and B. Sazdovi´, Nucl. Phys. B 836 (2010) 100
c c

• Similar method as in bosonic case
• Background fields

– Ω even corresponds to Type I
E
Gµν → Gµν

1 α α 1 α −1 α
Ψ+µ → (ΨE )µ = Ψ+µ + (BG Ψ−)µ
2 2
αβ αβ αβ −1 αβ
Fa → FE = F − (Ψ−G Ψ−)

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

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

µ ν µν
{x (σ) , x (¯ )} = 2θ θ(σ + σ )
σ ¯
µ α µα
{x (σ) , θ (¯ )} = −θ θ(σ + σ )
σ ¯
α ¯β σ 1 αβ
{θ (σ) , θ (¯ )} = θ θ(σ + σ )
¯
2


14
Type II B theory
Bosonic TIIBb -dulity

• Action has global shift symmetry in bosonic direction

Similar method produce dual background ﬁelds

2
Gµν = α G−1µν ,
E B µν = α θ µν

ψ− = −2G−1µν (ψE )a
aµ
E ν
aµ
ψ+ = 2κθ aµ

ab ab a
Fa = FE + 4(ψE G−1ψE )
E
b ab
Fs = 2κθ ab


15
Type II B and bosonic duality

• T-duality Eﬀective theory Noncommuatativity

Bosonic N bc Ω-symm Ω-antisymm

2
• Gµν = α G−1µν
E Gef f = GE
µν µν /

B µν = α θ µν / θ µν

aµ
• ψ− = −2(G−1ψE )aµ
E (ψef f )a = (ψE )a
µ µ /

aµ
ψ+ = 2κθ aµ / θ aµ

• Fa = FE + 4(ψE G−1ψE ) Fef f = FE
ab ab a
E
b ab ab
/

Fs = 2κθ ab
ab
/ θ ab


16
Type II B theory
Fermionic TIIBf -dulity

B. Nikoli´ and B. Sazdovi´
c c
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, θ a

– Suppose that action has a global shift symmetry in
¯
θ α and θ α directions

– Similar procedure as in bosonic case produces
TIIBf dual background ﬁelds:

¯
Bµν = Bµν + (ΨF
−1 ¯ −1
Ψ)µν − (ΨF Ψ)νµ
¯
Gµν = Gµν + 2 (ΨF
−1 ¯ −1
Ψ)µν + (ΨF Ψ)νµ
Ψαµ = 4(F
−1
Ψ)αµ , ¯ ¯ −1
Ψµα = −4(ΨF )µα
−1
Fαβ = 16(F )αβ


17
Type II B theory
Fermionic TIIBf -dulity and Dirichlet boundary conditions I

• T-duality Eﬀective theory and Noncommuatativity

BOSONIC ←→ Neumann b.c. for xµ
. ¯
SUSY b.c. for θ a, θ a

FERMIONIC ←→ ? b.c.

• DIRICHLET boundary conditions
π π π
x
µ
= 0, θ
α
= 0, ¯α
θ =0
0 0 0
• Solve constraints
– odd variables are independent
– trivial solution for coordinates, non-trivial for momenta
µ µ ν 1 ¯α 1 α
x (σ) = q (σ) ,
˜ πµ = pµ −2κBµν q − Ψµ (ηa )α + (¯a )α Ψµ
˜ ˜ η
2 2

α α 1
θ (σ) = θa (σ) , πα = pα − (¯a )α
˜ η
2
¯α ¯α 1
θ (σ) = θa (σ) , ˜
πα = pα − (ηa )α
¯ ¯
2
where
−1 β β µ ¯β ¯ β ˜µ −1
(ηa )α ≡ 4κ(F )αβ (θa +Ψµ q ) ,
˜ (¯a )α ≡ 4κ(θa +Ψµ q )(F )βα
η


18
Type II B theory
Non-commutativity relations

• Non-commutativity relations

{Pµ(σ), Pν (¯ )}D = Θµν ∆(σ + σ ) ,
σ ¯
σ ¯
{Pµ(σ), Pα(¯ )}D = Θµα∆(σ + σ ) ,
¯
¯ σ
Pµ(σ), Pα(¯ ) D = Θαµ∆(σ + σ ) ,
¯
¯ σ
Pα(σ), Pβ (¯ ) D = Θαβ ∆(σ + σ ) ,
¯
¯ ¯ σ
{Pα(σ), Pβ (¯ )}D = Pα(σ), Pβ (¯ ) D = 0 ,
σ

where the noncommutativity parameters are

Θµν = 2κ Bµν , ¯ µα = κ Ψµα
Θ ¯
2
κ κ
Θαµ = − Ψαµ , Θαβ = − Fβα ,
2 8
and
σ
PA(σ) = dσ1πA(σ1) A = {µ, a, a}
¯
0


19
Type II B and fermionic duality

• T-duality {Γa, Γb} {Pa, Pb}

• Gµν Gµν /

Bµν / θµν = 2κ Bµν

1
• ψaµ 2 ψaµ θaµ = − κ ψaµ
2

¯
ψµa 1 ¯
ψµa ¯
θµa = κ ¯
ψµa
2 2

1
• Fab − 8 Fab θab = − κ Fab
8


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
1 ρσ
Rµν − Bµρσ Bν = 0 ,
4
ρ
DρB µν = 0
– Bµνρ = ∂µBνρ + ∂ν Bρµ + ∂ρBµν is a field strength

– Rµν and Dµ Ricci tensor and covariant derivative

• We will consider the following particular solution

1 ρ
Gµν = const, Bµν [x] = bµν + Bµνρx ,
3
– 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. Davidovi´ and B. Sazdovi´
c c
T-duality in the weakly curved background
in preparation

• More complicated procedure then in ﬂat background

2
Gµν = α G−1µν ( x),
E B µν = α θ µν ( x)

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

x = g −1(2by + y )
˜


22
Effective theory and non-commutativity in
weakly curved background
Lj. Davidovi´ and B. Sazdovi´
c c
Phys. Rev. D 83 (2011) 066014

Lj. Davidovi´ and B. Sazdovi´,
c c
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

Gef f (u) = GE (u),
µν µν Bµνf = − κ (gθ(u)g)µν
ef
2

u = q + 2b˜
q

• Non-commutativity parameter

– Nontrivial both at string endpoints and at string interior

– Depends on the σ -integral of the effective momenta
σ
Pµ(σ) = 0 dηpµ(η)

which is in fact T0-dual of the effective coordinate,
Pµ = κgµν q ν .
˜


23
Relation between Twcb-duality, effective theory

• Twcb-duality Effective theory Noncommuatativity

Dual background fields

2
Gµν = α G−1µν ( x)
E Gef f = GE (u)
µν µν /

B µν = α θ µν ( x) / θ µν (v)

Dual variables

3 ˜
x = g −1(2by+ y )
˜ u = q+2b˜
q v = q− π bQcm


B. Sazdovic - Noncommutativity and T-duality

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