This document summarizes a talk on Hamiltonian approaches to Dp-brane noncommutativity in string theory. It discusses:
1) Basic facts about strings and superstrings, including open and closed strings, Dp-branes, and the 5 consistent superstring theories.
2) Boundary conditions for strings ending on Dp-branes and their treatment as canonical constraints. This leads to noncommutativity on the Dp-brane.
3) Details of the Type IIB superstring theory in 10 dimensions and the model considered, which involves graviton, gravitinos, and R-R fields but no dilatinos or dilaton.
Measures of Central Tendency: Mean, Median and Mode
B. Nikolic/ B. Sazdovic: Hamiltonian Approach to Dp-brane Noncommutativity
1. 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
2. Strings and superstrings
Conformal symmetry and dilaton field
Type IIB superstring and non(anti)commutativity
Concluding remarks
Outline of the talk
Strings and superstrings
1
Conformal symmetry and dilaton field
2
Type IIB superstring and non(anti)commutativity
3
Concluding remarks
4
Bojan Nikoli´
c Hamiltonian approach to Dp-brane noncommutativity
3. Strings and superstrings
Conformal symmetry and dilaton field
Type IIB superstring and non(anti)commutativity
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´
c Hamiltonian approach to Dp-brane noncommutativity
4. Strings and superstrings
Conformal symmetry and dilaton field
Type IIB superstring and non(anti)commutativity
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´
c Hamiltonian approach to Dp-brane noncommutativity
5. Strings and superstrings
Conformal symmetry and dilaton field
Type IIB superstring and non(anti)commutativity
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´
c Hamiltonian approach to Dp-brane noncommutativity
6. Strings and superstrings
Conformal symmetry and dilaton field
Type IIB superstring and non(anti)commutativity
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´
c Hamiltonian approach to Dp-brane noncommutativity
7. Strings and superstrings
Conformal symmetry and dilaton field
Type IIB superstring and non(anti)commutativity
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´
c Hamiltonian approach to Dp-brane noncommutativity
8. Strings and superstrings
Conformal symmetry and dilaton field
Type IIB superstring and non(anti)commutativity
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´
c Hamiltonian approach to Dp-brane noncommutativity
9. Strings and superstrings
Conformal symmetry and dilaton field
Type IIB superstring and non(anti)commutativity
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´
c Hamiltonian approach to Dp-brane noncommutativity
10. Strings and superstrings
Conformal symmetry and dilaton field
Type IIB superstring and non(anti)commutativity
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´
c Hamiltonian approach to Dp-brane noncommutativity
11. Strings and superstrings
Conformal symmetry and dilaton field
Type IIB superstring and non(anti)commutativity
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´
c Hamiltonian approach to Dp-brane noncommutativity
12. Strings and superstrings
Conformal symmetry and dilaton field
Type IIB superstring and non(anti)commutativity
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´
c Hamiltonian approach to Dp-brane noncommutativity
13. Strings and superstrings
Conformal symmetry and dilaton field
Type IIB superstring and non(anti)commutativity
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´
c Hamiltonian approach to Dp-brane noncommutativity
14. Strings and superstrings
Conformal symmetry and dilaton field
Type IIB superstring and non(anti)commutativity
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´
c Hamiltonian approach to Dp-brane noncommutativity
15. Strings and superstrings
Conformal symmetry and dilaton field
Type IIB superstring and non(anti)commutativity
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´
c Hamiltonian approach to Dp-brane noncommutativity
16. Strings and superstrings
Conformal symmetry and dilaton field
Type IIB superstring and non(anti)commutativity
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´
c Hamiltonian approach to Dp-brane noncommutativity
17. Strings and superstrings
Conformal symmetry and dilaton field
Type IIB superstring and non(anti)commutativity
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´
c Hamiltonian approach to Dp-brane noncommutativity
18. Strings and superstrings
Conformal symmetry and dilaton field
Type IIB superstring and non(anti)commutativity
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´
c Hamiltonian approach to Dp-brane noncommutativity
19. Strings and superstrings
Conformal symmetry and dilaton field
Type IIB superstring and non(anti)commutativity
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´
c Hamiltonian approach to Dp-brane noncommutativity
20. Strings and superstrings
Conformal symmetry and dilaton field
Type IIB superstring and non(anti)commutativity
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´
c Hamiltonian approach to Dp-brane noncommutativity
22. Strings and superstrings
Conformal symmetry and dilaton field
Type IIB superstring and non(anti)commutativity
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´
c Hamiltonian approach to Dp-brane noncommutativity
23. Strings and superstrings
Conformal symmetry and dilaton field
Type IIB superstring and non(anti)commutativity
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´
c Hamiltonian approach to Dp-brane noncommutativity
24. Strings and superstrings
Conformal symmetry and dilaton field
Type IIB superstring and non(anti)commutativity
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´
c Hamiltonian approach to Dp-brane noncommutativity