1. 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
2. • 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.
Noncommutativity and T-duality BSW 2011
3. 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
Noncommutativity and T-duality BSW 2011
4. 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
Noncommutativity and T-duality BSW 2011
5. 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
Noncommutativity and T-duality BSW 2011
6. 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
Noncommutativity and T-duality BSW 2011
7. 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
Noncommutativity and T-duality BSW 2011
8. 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
Noncommutativity and T-duality BSW 2011
9. 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
Noncommutativity and T-duality BSW 2011
10. 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)
˜
Noncommutativity and T-duality BSW 2011
11. 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
Noncommutativity and T-duality BSW 2011
13. 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
Noncommutativity and T-duality BSW 2011
14. 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
Noncommutativity and T-duality BSW 2011
16. 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
Noncommutativity and T-duality BSW 2011
17. 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
Noncommutativity and T-duality BSW 2011
18. 15
Relation between T-duality, effective theory
and noncommuatativity
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
Noncommutativity and T-duality BSW 2011
19. 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
Noncommutativity and T-duality BSW 2011
20. 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
Noncommutativity and T-duality BSW 2011
21. 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
Noncommutativity and T-duality BSW 2011
22. 19
Relation between T-duality, effective theory
and noncommuatativity
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
Noncommutativity and T-duality BSW 2011
23. 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
Noncommutativity and T-duality BSW 2011
24. 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 )
˜
Noncommutativity and T-duality BSW 2011
25. 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 .
˜
Noncommutativity and T-duality BSW 2011
26. 23
Relation between Twcb-duality, effective theory
and noncommuatativity
• 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
Noncommutativity and T-duality BSW 2011