B. Nikolic/ B. Sazdovic: Hamiltonian Approach to Dp-brane Noncommutativity

Strings and superstrings
Conformal symmetry and dilaton ﬁeld
Type IIB superstring and non(anti)commutativity
Concluding remarks

Hamiltonian approach to Dp-brane
noncommutativity

Bojan Nikoli´ and Branislav Sazdovi´
c c

Center for Theoretical Physics
Institute of Physics, Belgrade, Serbia
Group for gravitation, particles and ﬁelds
http://gravity.phy.bg.ac.yu/

Spring School on Strings, Cosmology and Particles, Niˇ , Serbia
s
31st March – 4th April 2009

Bojan Nikoli´
c Hamiltonian approach to Dp-brane noncommutativity

Concluding remarks

Outline of the talk

1

2

3

Concluding remarks
4

Bojan Nikoli´

Concluding remarks

Basic facts 1

Strings are object with one spatial dimension. During
1

motion string sweeps a two dimensional surface called
world sheet, parametrized by timelike parameter τ and
spacelike one σ ∈ [0 , π]. There are open and closed
strings.
Open string endpoints can be forced to move along
2

Dp-branes by appropriate choice of boundary conditions.
Dp-brane is an object with p spatial dimensions which
satisﬁes Dirichlet boundary conditions.
Demanding presence of fermions in theory, we obtain
3

superstring theory.

Bojan Nikoli´

Concluding remarks

Basic facts 2

There are two standard approaches to superstring theory:
1

NSR (Neveu-Schwarz-Ramond) (world sheet
supersymmetry) and GS (Green-Schwarz) (space-time
supersymmetry).
New approach has been recently developed - pure spinor
2

formalism, (N. Berkovits, hep-th/0001035). It combines
advantages of NSR and GS formalisms and avoid their
weaknesses.
There are ﬁve consistent superstring theories.
3

Bojan Nikoli´

Concluding remarks

Superstring theories

Type I
1

Unoriented open and closed strings, N = 1 supersymmetry,
gauge symmetry group SO(32).
Type IIA
2

Closed oriented and open strings, N = 2 supersymmetry,
nonchiral.
Type IIB
3

Closed oriented and open strings, N = 2 supersymmetry,
chiral.
Two heterotic theories
4

Closed oriented strings, N = 1 supersymmetry, symmetry
group either SO(32) or E8 × E8 .

Bojan Nikoli´

Concluding remarks

Boundary conditions in canonical formalism

As time translation generator Hamiltonian Hc must have
well deﬁned functional derivatives with respect to
coordinates xµ and their canonically conjugated momenta
πµ
π
(R) (0)
δHc = δHc − γµ δxµ 0 .
The ﬁrst term is so called regular term. It does not contain
τ and σ derivatives of coordinates and momenta variations.
The second term has to be zero and we obtain boundary
conditions.
We obtain the same result in Lagrangian formalism from
δS = 0.

Bojan Nikoli´

Concluding remarks

Sorts of boundary conditions

If the coordinate variations δxµ are arbitrary at the string
endpoints, we talk about Neumann boundary conditions
(0) (0)
= γµ = 0.
γµ 0 π

If the coordinates are ﬁxed at the string endpoints

δxµ = δxµ = 0,
0 π

then we have Dirichlet boundary conditions.

Bojan Nikoli´

Concluding remarks

Boundary conditions as canonical constraints

Boundary conditions are treated as canonical constraints.
Then we perform consistency procedure. Let Λ(0) be a
constraint, then consistency of constraint demands that it
is preserved in time
dΛ(n−1)
Λ(n) ≡ = Hc , Λ(n−1) ≈ 0 . (n = 1, 2, . . . )
dτ
In all cases we will consider here, this is an inﬁnite set of
constraints. Using Taylor expansion we can rewrite this set
of constraints in compact σ-dependent form
∞ ∞
σ n (n) (σ − π)n (n)
˜
Λ(σ) = Ω (σ = 0) , Λ(σ) = Λ (σ = π) .
n! n!
n=0 n=0

Bojan Nikoli´

Concluding remarks

Bosonic string with dilaton

Action which describes dynamics of bosonic string in the
presence of gravitational Gµν (x), antisymmetric NS-NS
ﬁeld Bµν (x) and dilaton ﬁeld Φ(x) is of the form

εαβ
√ 1 αβ
d2 ξ −g g Gµν + √ Bµν ∂α xµ ∂β xν + ΦR(2)
S=κ ,
2 −g
Σ

where ξ α = (τ , σ) parameterizes world sheet Σ with
intrinsic metric gαβ . With R(2) we denote scalar curvature
with respect to the metric gαβ .

Bojan Nikoli´

Concluding remarks

Beta functions 1

Quantum conformal invariance is determind by the beta
functions
1
βµν ≡ Rµν − Bµρσ Bν ρσ + 2Dµ aν ,
G
4
βµν ≡ Dρ B ρ µν − 2aρ B ρ µν ,
B

D − 26 1
β Φ ≡ 2πκ − R − Bµρσ B µρσ − Dµ aµ + 4a2 .
6 24
Theory is conformal invariant on the quantum level under
the following conditions βµν = βµν = 0 and β Φ = 0 (or
G B

β Φ = c = const.). This is a consequence of the relation

D ν βµν + (4π)2 κDµ β Φ = 0 .
G

Bojan Nikoli´

Concluding remarks

Beta functions 2

We will consider one particular solution of these equations

Gµν (x) = Gµν = const , Bµν (x) = Bµν = const ,

Φ(x) = Φ0 + aµ xµ , (aµ = const) .
Sigma model becomes conformal ﬁeld theory with central
charge c. There are two possibilities: c = 0, or c = 0 plus
adding of Liouville term

βΦ √ 1
d2 ξ −gR(2) R(2) , ∆ = gαβ ∇α ∂β ,
SL = −
2(4π)2 κ ∆
Σ

which annihilates conformal anomaly and restores
quantum conformal invariance.
Bojan Nikoli´

Concluding remarks

Beta functions 3

Conformal gauge condition gαβ = e2F ηαβ .
1 1 1
There are three cases: (1) a2 = α , a2 = α ; (2) a2 = α ,
˜
2 = 1 ; (3) a2 = 1 , a2 = 1 . Limit α → ∞ gives the results
˜ α˜
a α α
for the case without Liouville term.
Constant α is chosen in such a way that Liouville term
eliminates anomaly term

βΦ
1
= .
(4πκ)2
α

a2 ≡ (Gµνf )aµ aν , Gef f = Gµν − 4(BG−1 B)µν .
˜ µν
ef

Bojan Nikoli´

Concluding remarks

Choice of boundary conditions

We split xµ (µ = 0, 1, 2, . . . D − 1) into Dp-brane
coordinates xi (i = 0, 1, . . . p) and orthogonal ones xa
(a = p + 1, p + 2, . . . , D − 1).
For xi and F we choose Neumann boundary conditions,
and for xa Dirichlet ones.
Bµν → Bij , aµ → ai .

Bojan Nikoli´

Concluding remarks

Solution of boundary conditions 1

Initial variables are expressed in terms of their Ω even
parts (effective variables), where Ω is world-sheet parity
transformation Ω : σ → −σ.
Initial coordinates depend both on effective coordinates
and effective momenta.
Presence of momenta causes noncommutativity.

Bojan Nikoli´

Concluding remarks

Solution of boundary conditions 2

In all three cases coordinate ⋆ F = F + α ai xi is
2
commutative (in limit α → ∞ the role of commutative
coordinate takes ai xi ), while other ones are
noncommutative.
Case (1) - all constraints originating from boundary
conditions are of the second class (Dirac constraints do not
appear). The number of Dp-brane dimensions is
unchanged.
Case (2) - one Dirac constraint of the ﬁrst class appears.
The number of Dp-brane dimensions decreases.
Case (3) - two constraints originating from boundary
conditions are of the ﬁrst class. The number of Dp-brane
dimensions decreases.
Bojan Nikoli´

Concluding remarks

Type IIB superstring theory in D = 10

NS-NS sector: graviton Gµν , Neveu-Schwarz ﬁeld Bµν and
dilaton Φ.
α ¯α
NS-R sector: two gravitinos, ψµ i ψµ , and two dilatinos, λα
¯
and λα . Spinors are of the same chirality.
R-R sector: scalar C0 , two rank antisymmetric tensor Cµν
and four rank antisymmetric tensor Cµνρσ with selfdual ﬁeld
strength.

Bojan Nikoli´

Concluding remarks

Model

We consider model without dilatinos and dilaton.
Action for type IIB superstring theory in pure spinor
formulation (ghosts free) is

1 ab
d2 ξ η Gµν + εab Bµν ∂a xµ ∂b xν
SIIB = κ
2
Σ

d2 ξ −πα (∂τ − ∂σ )(θ α + ψµ xµ )
α
+
Σ
1
¯ ¯α
d2 ξ (∂τ + ∂σ )(θ α + ψµ xµ )¯α + πα F αβ πβ
+ ¯
π
2κ
Σ

where xµ are space-time coordinates (µ = 0, 1, 2, . . . 9),
¯
and θ α and θ α are same chirality spinors.
Bojan Nikoli´

Concluding remarks

R-R sector 1

10
ik
F Γαβ . Γαβ = (Γ[µ1 ...µk ] )αβ
αβ
=
F
k! (k) (k) (k)
k=0

Bispinor F αβ satisﬁes chirality condition Γ11 F = −F Γ11 ,
and consequently, F(k) (k odd) survive.
Because of duality relation, independent tensors are F(1) ,
F(3) and selfdual part of F(5) .
Massless Dirac equation for F gives, F(k) = dC(k−1) .
Type IIB contains only potentials C0 , Cµν and Cµνρσ .

Bojan Nikoli´

Concluding remarks

R-R sector 2

1
Fs = (F αβ + F βα ) −→ C0 , Cµνρσ ,
αβ
2
1
Fa = (F αβ − F βα ) −→ Cµν .
αβ
2

Bojan Nikoli´

Concluding remarks

Boundary conditions

For coordinates xµ we choose Neumann boundary
conditions.
In order to preserve N = 1 SUSY from initial N = 2, for
fermionic coordinates we choose
π π
¯
(θ α − θ α ) = 0 =⇒ (πα − πα )
¯ = 0.
0
0

Bojan Nikoli´

Concluding remarks

Solution of boundary conditions

Solving boundary conditions, we get
σ σ
Θµα
xµ (σ) = q µ − 2Θµν dσ1 pν + dσ1 (pα + pα ) ,
¯
2
0 0

σ σ
Θαβ
θ α (σ) = η α − Θµα dσ1 pµ − dσ1 (pβ + pβ ) ,
¯
4
0 0

σ σ
Θαβ
¯
θ α (σ) = η α − Θµα
¯ dσ1 pµ − dσ1 (pβ + pβ ) ,
¯
4
0 0
where
1α 1
¯ ¯
ηα ≡ (θ + Ωθ α) , η α ≡ (Ωθ α + θ α ) ,
¯
2 2
≡ πα + Ω¯α , pα ≡ Ωπα + πα .
¯ ¯
pα π
Bojan Nikoli´

Concluding remarks

Noncommutativity

Using the solution, we have

{xµ (σ) , xν (¯ )} = Θµν ∆(σ + σ ) ,
¯
σ

1
{xµ (σ) , θ α (¯ )} = − Θµα ∆(σ + σ ) ,
¯
σ
2

1
¯σ
{θ α (σ) , θ β (¯ )} = Θαβ ∆(σ + σ ) .
¯
4

Bojan Nikoli´

Concluding remarks

Type I theory as effective one

Putting the solution of boundary conditions into initial
Lagrangian, we obtain effective one
κ ef f ab
Lef f = G η ∂a q µ ∂b q ν − πα (∂τ − ∂σ ) η α + (Ψef f )α q µ
2 µν µ

1 αβ
+ (∂τ + ∂σ ) η α + (Ψef f )α q µ πα +
¯ ¯ ¯
πα Fef f πβ .
µ
2κ
Transition from the initial L to effective Lef f Lagrangian is
¯
realized by changing the variables xµ , θ α and θ α with q µ ,
α and η α ,and changing the background ﬁelds
¯
η
→ Gµν − 4Bµρ Gρλ Bλν , ψ+µ → ψ+µ + 2Bµρ Gρν ψ−ν ,
α α α
Gµν
β
αβ
→ Fa − ψ−µ Gµν ψ−ν ,
αβ α
Fa
α αβ
→ 0, ψ−µ → 0 , Fs → 0 .
Bµν
Bojan Nikoli´

Concluding remarks

Conclusions

(1) Initial coordinates depend both on effective ones and
effective momenta. Presence of mometa causes
non(anti)commutativity of the coordinates.
(2) The number of Dp-brane dimensions depends on the
relations between background fields. For particular relations
between them, first class constraints appear and decrease the
number of Dp-brane dimensions.
(3) Effective theory (initial one on the solution of boundary
conditions) is Ω even.
(4) Effective background fields depend both on Ω even fields
and Ω even combinations of the Ω odd fields.

Bojan Nikoli´

B. Nikolic/ B. Sazdovic: Hamiltonian Approach to Dp-brane Noncommutativity

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