D-Branes and AdS/CFT Junaid Saif Khan Supervised by: Dr. Babar A. - - PowerPoint PPT Presentation

Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References

D-Branes and AdS/CFT

Junaid Saif Khan

Supervised by: Dr. Babar A. Qureshi

MS mid-year presentation

LUMS Lahore University of Management Sciences , Pakistan

Department of Physics, 2018

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References

Plan

1

Relativistic Strings Point Particle : Action and Quantization Classical Strings Quantum Strings

2

Superstrings Action Quantization and Spectrum

3

D-Branes Gauge Fields on D-branes Parallel Dp-branes Strings and D-brane Charges

4

T-dualities T-dualities T-duality and Closed Strings T-duality and Open Strings

5

Proposed Work

6

References

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Point Particle : Action and Quantization Classical Strings Quantum Strings

Point Particle : Action

Action in terms of proper time : SPP = −mc τf

τi

• −ηµν

dxµ dτ dxν dτ dτ. (1) → Lorentz invariant and reparametrization invariant. Equation of Motion : dpµ dτ = 0. (2) → Point particle moves with constant momentum along the worldline. Classically equivalent action : SPP = 1 2

• dτ
• e−1 ˙

xµ ˙ xνηµν − m2e

• .

(3) This has couple of advantages : → Works even for massless particle. → No annoying square root.

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Point Particle : Action and Quantization Classical Strings Quantum Strings

Point Particle : BRST Quantization

BRST Quantization : Action in terms of momentum and velocity : SPP =

• dτ
• ˙

xµpνηµν − e 2 (pµpνηµν + m2)

• .

(4) → Gauge invariant action. Global fermionic symmetry arises once the gauge is fixed. → By doing this, B-field is introduced corresponding to this ghost-antighosts term ap- pears in the action. SBRST =

• dτ
• ˙

xµpµ + ιb ˙ b − 1 2

• pµpµ + m2

. (5) → The purpose of introducing these new fields is to produce the BRST charge Ω against which the action is invariant. Ω = bαGα, (6) where bα are grassmann numbers generated by fermionic fields.

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Point Particle : Action and Quantization Classical Strings Quantum Strings

Point Particle : BRST Quantization

From Legendre transformation we get : G = 1 2

• pµpµ + m2

(7) Hence, the action of Ω is nilpotent i.e., {Ω, Ω} = 0. → SBRST is invariant under BRST transformation (δΩ) and under BRST operator Ω. (See write-up) → More states than physical ones due to BRST operator. Why? Our goal is to produce the physical states which is done by the cohomology of Ω group which is defined as : HBRST = KerΩ ImΩ , (8) KerΩ : gives the states which annihilate under Ω, the physical ones. ImΩ : gives the states which are obtained by acting Ω on any arbitrary state |χ, all states.

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Point Particle : Action and Quantization Classical Strings Quantum Strings

Classical Strings : The Nambu-Goto Action

Nambu-Goto action : SNG = − T0 c τf

τi

dτ σ1 dσ ˙ X.X ′2 − ˙ X 2X ′ 2. (9) → In terms of induced metric (like in point particle case) : SNG = − T0 c

• dτdσ
• −det(Γαβ),

(10) where Γαβ = ηµν ∂xµ ∂ξα ∂xν ∂ξβ (induced metric) (11) Boundary conditions from EOM : δX µ(τ, σ∗) = 0 (Dirichlet boundary condition), (12) Πσ

µ(τ, σ∗) = 0

(Neumann boundary condition). (13) where Πσ

µ = ∂L/∂X µ′.

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Point Particle : Action and Quantization Classical Strings Quantum Strings

Classical Strings : The Polyakov Action

Polyakov action : SP = − T0 2

• d2σ
• −det(gαβ)gαβ∂αX µ∂βX νGµν.

(14) → Choose Gµν = ηµν. → Symmetries : Global Poincare (Translational and Lorentzian), Local gauge (Repa- rametrization and Local Weyl). Equations of Motion : ∂αX µ∂βXµ − 1 2 gαβglm∂lX µ∂mX ν = 0 and ∂α

• −ggαβ∂βX µ
• = 0. (15)

Boundary conditions (for open string) : gασ∂αX µ = 0 (Neumann boundary condition), (16) δX µ = 0 (Dirichlet boundary condition). (17) Note : Local gauge symmetries demonstrate the repetition in degrees of freedom so we would fixed them by quantization.

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Point Particle : Action and Quantization Classical Strings Quantum Strings

Classical Strings : Light-cone quantization

Light-cone quantization :

⋆ Classical EOM. ⋆ Fix the gauge symmetries. ⋆ Find out the complete classical solution. ⋆ See the spectrum of both open and closed strings.

After fixing all gauge symmetries, EOM modified (see write-up) and we get following solution : X µ(τ, σ) = xµ + vµτ + X µ

R (τ − σ) + X µ L (τ + σ).

(18) → For closed strings : σ ∈ [0, 2π] while XL and XR are independent. → For open strings : σ ∈ [0, π] while XL = XR. Express Eq. 18 in terms of Fourier modes : For closed strings : X µ(τ, σ) = xµ + vµτ + ι

• α′

2

• n=0

1 n

• αµ

n e−ιn(τ+σ) + ˜

αµ

n e−ιn(τ−σ)

• .

(19) For Open strings : X µ(τ, σ) = xµ + vµτ + ι

• 2α

′

n=0

1 n αµ

n e−ιnτ cos nσ.

(20)

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Point Particle : Action and Quantization Classical Strings Quantum Strings

Classical Strings : Light-cone quantization

→ The conserved worldsheet momentum which is the string spacetime momentum is given by (see write-up for derivation) : pµ = l 2π vµ α′ ⇒ vµ = 2πα

′

l pµ. (21) → Plugging above two solutions in the Virasoro Constraints and using the relation −PµPµ = M2, we get mass-shell conditions : For open strings : M2 = 1 2α

′

• n=0

αi

−nαi n.

(22) For closed strings : M2 = 2 α

′

• n=0

αi

−nαi n.

(23) Result : One can conclude the mass of a string from its oscillations.

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Point Particle : Action and Quantization Classical Strings Quantum Strings

Quantum Strings : Quantization and Spectrum

For relativistic quantum strings :

⋆ Classical fields convert to quantum operators. ⋆ Parameters (appear in classical solutions) become creation and annihilation operators. ⋆ Classical solutions become solutions of the operator equation. ⋆ One can read the spectrum of the quantum relativistic strings by applying creation ope- rators to the vacuum state.

1) All the classical fields as operators by imposing a canonical quantization condition : [X i(τ, σ), X j(τ, σ′)] = 0 (24) [Πi(τ, σ), Πj(τ, σ′)] = 0 (25) [X i(τ, σ), Πj(τ, σ′)] = ιδijδ(σ − σ′). (26) In conclusion, all oscillation modes xi, pi, αi

n, ˜

αi

n become operators. When we plug-

in these mode expansions into above relations, we get : [xi, pj] = ιδij. (27) [αi

n, αj m] = [˜

αi

n, ˜

αj

m] = nδijδn+m,0.

(28)

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Point Particle : Action and Quantization Classical Strings Quantum Strings

Quantum Strings : Closed strings Spectrum

2) For n > 0, creation and annihilation operators are related as : (ai

−n)† =

1 √n αi

−n

, (˜ ai

−n)† =

1 √n ˜ αi

−n

(creation operators). (29) ai

n =

1 √n αi

n

, ˜ ai

n =

1 √n ˜ αi

n

(annihilation operators). (30) 3) We have determined the Heisenberg equations and its solutions can be characte- rized by the Eq. (19) and Eq. (20) and the constants in these equations satisfy the commutation relations. Essentially we have done the quantization of the string. 4) Work out the spectrum : → Mass shell condition for closed strings : M2 = 2 α′

D−1

• i=2
• n=0

n(Ni

n + ˜

Ni

n) − D − 2

6α′ . (31)

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Point Particle : Action and Quantization Classical Strings Quantum Strings

Quantum Strings : Closed strings Spectrum

→ For ground state : Ni

m = ˜

Ni

m = 0 for all i, m.

M2 = − D − 2 6α′ . (32) We find the tachyon for any spacetime dimension more than 2. Hence, the ground state is tachyonic for closed strings. → First excited state : αi

−1 ˜

αj

−1 |0, pµ .

(33) Corresponding to n = 1, we get : M2 = 26 − D 6α′ ⇒ D = 26. (for massless particle) (34) → First excited state can be further decompose into trace part, symmetric traceless part and anti-symmetric traceless part which produces scaler field Φ, graviton field Gµν and Kalb-Ramon or B-field Bµν respectively. → Higher excitations are all massive.

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Point Particle : Action and Quantization Classical Strings Quantum Strings

Quantum Strings : Open strings Spectrum

→ Mass shell condition for open strings : M2 = 1 α′

D−1

• i=2
• n=0

nNi

n − D − 2

24α′ . (35) → For ground state : Ni

m = 0 for all i, m.

M2 = − D − 2 24α′ . (36) We again find the tachyon for any spacetime dimension more than 2. Hence, the ground state is also tachyonic for open strings. → First excited state : αi

−1 |0, pµ .

(37) Corresponding to n = 1, we get : M2 = 1 α′ − D − 2 24α′ = 26 − D 24α′ ⇒ D = 26. (for massless particle) (38) First excited state produces massless vector field Aµ. → Higher excitations are all massive.

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Action Quantization and Spectrum

Superstrings : Action

Nambu-Goto action : S = 1 4πα′

• dτ

π dσ

• ˙

X j ˙ X j − X j′X j′ + Sψ. (39) Sψ is the Dirac action for a fermion lives on 2D world is defined as : Sψ = 1 2π

• dτ

π dσ

• ψj

1(∂τ + ∂σ)ψj 1 + ψj 2(∂τ + ∂σ)ψj 2

• .

(40) Polyakove action : SP = 1 4πα′

• d2σ
• −ggαβGµν
• ∂αX µ∂βX ν + ιψ

µγβ∂αψν

• ,

(41) where γµ are Dirac spinors (real) which are defined as : γ1 = −1 1

• ,

γ2 = 1 1

• .

(42) Equation of motions : (∂τ + ∂σ)ψj

1 = 0

, (∂τ − ∂σ)ψj

2 = 0.

(43) Boundary Conditions : ψj

1(τ, σ∗)δψj 1(τ, σ∗) − ψj 2(τ, σ∗)δψj 2(τ, σ∗) = 0.

(44)

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Action Quantization and Spectrum

Superstrings : Action

→ ψj

1 and ψj 2 are right-moving and left-moving respectively :

ψj

1(τ, σ) = Ψj 1(τ − σ)

; ψj

2(τ, σ) = Ψj 2(τ + σ).

(45) Let’s define the fermionic field Ψj all over the interval σ ∈ [−π, π] irrespective of just at the end points, the boundary conditions for open strings become modified as : Open strings : Ψj

1(τ, σ)

• σ=0

= Ψj

2(τ, σ)

• σ=0

. (46) (Ramond) Ψj

1(τ, σ)

• σ=π

= +Ψj

2(τ, σ)

• σ=π

. (47) Ψj

1(τ, σ) =

√ α′

n∈Z

sj

ne−ιn(τ−σ)

, Ψj

2(τ, σ) =

√ α′

n∈Z

sj

ne−ιn(τ+σ).

(48) (Neveu-Schwarz) Ψj

1(τ, σ)

• σ=π

= −Ψj

2(τ, σ)

• σ=π

. (49) Ψj

1(τ, σ) =

√ α′

• n∈Z+ 1

2

bj

ne−ιn(τ−σ)

, Ψj

2(τ, σ) =

√ α′

• n∈Z+ 1

2

bj

ne−ιn(τ+σ).

(50)

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Action Quantization and Spectrum

Superstrings : Action

Closed strings : right-moving and left-moving are independent periodic function of 2π. Therefore, for right movers we have : Ψj

1(τ, σ) =

√ α′

n∈Z

sj

ne−ιn(τ−σ)

(Ramond sector), (51) Ψj

1(τ, σ) =

√ α′

• n∈Z+ 1

2

bj

ne−ιn(τ−σ)

(Neveu-Schwarz sector). (52) Similarly, we have two possibilities for left movers Ψj

• 2. Therefore, we have four

sectors : (R,R), (R,NS), (NS,NS), (NS,R).

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Action Quantization and Spectrum

Open Superstrings

For open strings, expansion operators satisfy the following anticommutation rela- tion : For R sector :

• sj

n, sk p

• = δn+p,0δjk,

(53) For NS sector :

• bj

n, bk p

• = δn+p,0δjk.

(54) Spectrum of NS sector : M2 = 1 α′

n=0

αj

−nαj n +

• r∈Z+ 1

2

rbj

−rbj r − (D − 2)

16

• .

(55) → For ground state : M2 = − (D − 2) 16α′ . (56) Hence, the ground state is again tachyonic for D > 2. → For 1st excited state, apply bj

−1/2 on the ground state :

bj

−1/2 |0 ,

(57) corresponding to this value of n = 1, we get : M2 = (10 − D) 16α′ ⇒ D = 10. (massless particle) (58)

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Action Quantization and Spectrum

Open Superstrings

Spectrum of R sector : M2 = 1 α′

• n≥1
• αj

−nαj n + nsj −nsj n

• .

(59) This implies following results : 1) There are no tachyonic states in Ramond sector 2) All the Ramond ground states are massless. 3) All excited states are massive. 4) There would be equal number of bosonic and fermionic states in the excited states which is a signal of supersymmetry on the worldsheet. Conclusion : Ramond sector has world-sheet supersymmetry.

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Action Quantization and Spectrum

Closed Superstrings

Closed superstrings have four sectors : (R,R), (R,NS), (NS,R), (NS,NS). (60) → Define an operator called (−1)F : (−1)F =

• +1

(for bosonic state) −1 (for fermionic state) → Truncate NS sector into NS+ and NS− in order to get supersymmetry in states. Reason? → NS+ sector contains the massless states and throws away the tachyonic states while NS− sector contains a tachyon. → As R sector contains same numnber of states but just for the convention truncate R sector into R+ contains bosonic states and R− contains fermionic states. Closed strings can be made from open strings. Above discussion expresses from the R− and NS+ sectors, open superstrings completely characterized and has a supersymmetric spectrum. Right-moving and left-moving parts are defined as : right sector : NS+ R+

• ,

left sector : NS+ R−

• .

(61)

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Action Quantization and Spectrum

Closed Superstrings : IIA Superstring

IIA superstring : By combining multiplicatively these two sectors lead to IIA closed superstring theory which has the following sectors : type IIA superstring : (R−, R+), (R−, NS+), (NS+, R+), (NS+, NS+). (62) → The type IIA superstring has no tachyons and its massless states are obtained by combining the massless states of the various sectors. These sectors produce following fields : IIA spectrum : Gµν, Bµν, Φ, Aµ, Aµνρ. (63) Here Aµ is a Maxwell field and Aµνρ is three-index antisymmetric gauge field pro- duced by massless (R−,R+) bosons while the first three fields (graviton, Kalb- Ramond and scaler) are produced by massless (NS+,NS+) bosons.

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Action Quantization and Spectrum

Closed Superstrings : IIB Superstring

Type IIB superstring arises when we take the same type of Ramond sectors : right sector : NS+ R−

• ,

left sector : NS+ R−

• .

(64) which has the following sectors : type IIB superstring : (R−, R−), (R−, NS+), (NS+, R−), (NS+, NS+). (65) These produce following fields when apply to the ground state : IIB spectrum : Gµν, Bµν, Φ, χ, Aµν, Aµνρσ. (66) Due to the massless R-R bosons in the type IIB theory, antisymmetric gauge field Aµνρσ with four indices, scalar field χ, a Kalb-Ramond field Aµν arises and the first three fields are produced by massless (NS+,NS+) bosons.

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Gauge Fields on D-branes Parallel Dp-branes Strings and D-brane Charges

Gauge Fields on D-branes

D-branes provide surfaces to which open strings attach to. → Dp brane is a brane which lives in p spatial dimension. To locate these ‘hyperpla- nes’ we need (d − p) linear conditions. Example? → Split spacetime coordinates (xµ) into two groups : coordinates tangential to the brane and the coordinates normal to the brane. → The location of the brane is specified by the coordinates normal to the brane. Tangential coordinates of string’s endpoints must satisfy Neumann boundary condi- tion : X j′(σ, τ)|σ=0 = X j′(σ, τ)|σ=π = 0, j = 0, 1, ..., p. (67) Normal coordinates satisfies Dirichlet boundary conditions : X a(σ, τ)|σ=0 = X a(σ, τ)|σ=π = xa, a = p + 1, ..., d. (68)

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Gauge Fields on D-branes Parallel Dp-branes Strings and D-brane Charges

Gauge Fields on D-branes

In summary, X 0, X 1, ..., X p

• NM coordinates

X p+1, X p+2, ..., X d

• DIR coordinates

. (69) In terms of light-cone coordinates these coordinates are written as : X +, X −, X i

• NM

X a

• DIR

, i = 2, ..., p a = p + 1, ...d. (70) There are fields corresponding to each coordinate that live on the Dp-brane.

→ Dimensions of Dp-brane is p + 1. (Why?)

Fields on Dp-branes : → The ground states are tachyonic therefore there would be a tachyonic field.

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Gauge Fields on D-branes Parallel Dp-branes Strings and D-brane Charges

Gauge Fields on D-branes

→ The excited states have two cases : the first in which the oscillator arises from a coordinate tangent to the brane and the other in which oscillator arises from a coordinate normal to the brane. → From first, excited states are photon states and the associated field is a Maxwell gauge field. → From second, we get a massless scaler field. (For explanation, see write-up.) As a summary, we have two very important results :

1

A Dp-brane has a tachyonic field.

2

A Dp-brane has a Maxwell field on its world-volume.

3

A Dp-brane has a massless scalar for each direction normal to it.

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Gauge Fields on D-branes Parallel Dp-branes Strings and D-brane Charges

Parallel Dp-branes and Open Strings

Mass-shell condition : M2 = xa

2 − xa 1

2πα′ 2 + 1 α′

• − 1 +

∞

• n=1

p

• i=2

nai†

n ai n + ∞

• j=1

d

• a=p+1

jai†

j ai j

• .

(71) → For ground state : M2 = −1 α′ + xa

2 − xa 1

2πα′ 2 . (72) If the two branes are separated by |xa

2 − xa 1| = 2π

√ α′ or more, the ground state is no more tachyonic. Open string interactions : Open strings in the presence of two parallel Dp-branes has four sectors. Consequently, we now have four massless gauge fields.

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Gauge Fields on D-branes Parallel Dp-branes Strings and D-brane Charges

Four sectors of a string on two parallel D-branes are shown :

FIGURE – Parallel D2-branes.

D-branes interact as following :

FIGURE – Interaction of D-branes.

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Gauge Fields on D-branes Parallel Dp-branes Strings and D-brane Charges

Strings Charges

Point particle interacts with Maxwell field, carries electric charge. Similarly strings incorporates with Kalb–Ramond field carry a string charge. String charge can be envisioned as a current flowing along the string for stretched strings. A charged point particle couples to a Maxwell field has interaction (Lagrangian) : LP = q

• Aµ

dxµ dτ dτ. (73) Similarly, for strings we have : −

• dτdσ ∂X µ

∂τ ∂X ν ∂σ Bµν(X(τ, σ)). (74) where Bµν is Kalb-Ramond field. It is called an electric coupling because it’s gene- ralization of the electric coupling of a point particle to a Maxwell field. Hence, string carries electric Kalb-Ramond charge.

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Gauge Fields on D-branes Parallel Dp-branes Strings and D-brane Charges

D-brane Charges

Now let’s move on to branes. A Dp-brane has a world volume which is p + 1 dimensional. Hence the interaction becomes : −

• dτdσ1...dσk

∂X µ ∂τ ∂X µ1 ∂σ1 ... ∂X µk ∂σk Yµµ1...µk (X(τ, σ1, ...σk)) (75) Relation betweeen electric charge and electric current for point particle : ∂F µν ∂xν = jµ. (76) The generalization of Eq. 76 for the strings : 1 κ2 ∂Z µνρ ∂xρ = jµν, (77) where κ is just a constant which balances the dimensions of the equation. From above one can conclude that like point particles, strings and branes also carry charges.

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References T-dualities T-duality and Closed Strings T-duality and Open Strings

T-duality

T-duality is a way to compactify the dimensions offer by string theory. This leads to a symmetry : If a spatial dimension is curled up into a circle of radius R then the scenario is physically equivalent to a scenario where the the radius is ∝ 1

R .

If the space is being compactified into a torus then this called toroidal and the symmetry is called T-duality.

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References T-dualities T-duality and Closed Strings T-duality and Open Strings

T-duality and Closed Strings

Generalized mode expansion : X µ(τ, σ) = xµ 2 + ˜ xµ 2

• +
• −ι
• α′

2 (αµ

0 +˜

αµ

0 )τ+

• α′

2 (αµ

0 −˜

αµ

0 )σ

• +oscillating terms.

(78) The corresponding spacetime momentum of the string is given as : pµ = 1 √ 2α′ (αµ

0 + ˜

αµ

0 ).

(79) Compactification : 1) Assume the bosonic string which is propagating in a 26 spacetime dimensions. 2) Compactify one of the spatial dimension (X 25) in a circle of radius R. The expansion

• f this dimension become modify while the remaining dimensions remain unaffec-

ted. 3) Quantized momentum along the compactified direction : p25 = K R , K ∈ Z. (80) Here K is called the Kaluza-Klein excitation number.

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References T-dualities T-duality and Closed Strings T-duality and Open Strings

T-duality and Closed Strings

Consequences : i) Before compactification : continuous momentum spectrum. ii) After compactification : quantized momentum spectrum and winding states. Mass shell formula : M2 = −pµpµ = K 2 R2 + W 2R2 α′2 + 2 α′

• NR + NL − 2
• ,

(81) where NR and NL represent the total number of levels on the right and left moving sides respectively. According to T-duality : R → R = α′ R . (82) leaves Eq. 81 invariant provided W and K are exchangeable. Conclusion : ◮ As R → ∞, the winding states get massive and the momentum states form a continuum. ◮ As R → 0, the momentum states get massive and the winding states form a conti- nuum.

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References T-dualities T-duality and Closed Strings T-duality and Open Strings

T-duality and Open Strings

Generalized mode expansion : X µ(τ, σ) = xµ + pµτ + ι

• n=0

1 n αµ

n e−ιnτ cos nσ.

(83) and the string spacetime momentum is related as : pµ = l 2π vµ α′ . (84) → It is convenient to write the above equation in terms left and right moving parts such as X µ

L (τ + σ) = xµ + x′µ

2 + pµ(τ + σ) 2 + ι 2

• n=0

1 n αµ

n e−ιn(τ+σ),

(85) and X µ

R (τ − σ) = xµ − x′µ

2 + pµ(τ − σ) 2 + ι 2

• n=0

1 n αµ

n e−ιn(τ−σ).

(86) Compactification and T-duality : compactified and then applying a T-duality trans- formation which is X µ

L → X µ L

and X µ

R → −X µ R .

(87)

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References T-dualities T-duality and Closed Strings T-duality and Open Strings

T-duality and Open Strings

→ Under this transformation, dual string coordinates in the 25 direction such as

• X 25(τ, σ) = X 25

L

− X 25

R = x′25 + p25σ +

• n=0

1 n α25

n e−ιnτ sin nσ.

(88) Here the word ‘dual’ implies that we moved from the system R → R = α′/R. Consequences : i) The oscillator terms die out at the endpoints σ = 0, π implies that the endpoints do not move in the X 25 direction. This means that T-duality maps Neumann boundary condition into Dirichlet boundary condition and vice versa in the relevant directions.

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References T-dualities T-duality and Closed Strings T-duality and Open Strings

T-duality and Open Strings

Dual string coordinates : Compacted dual string coordinates at the endpoints are

• X 25(τ, σ)|σ=0 = x′

and

• X 25(τ, σ)|σ=π = x′ + 2πK

R. (89) → The string winds the dual circle K times. Since, the end points of the string are fixed by the Dirichlet boundary conditions therefore the winding mode is stable. Hence, this string cannot unwind without breaking. Conclusion : ◮ After compactification we have momentum but no winding modes with Neumann boundary conditions. ◮ Once the T-duality transformation is done we have winding modes but no momen- tum in the dual circular direction with Dirichlet boundary conditions. ◮ From Eq. 89, one can conclude that the end points of dual open string are attached to D-brane that can wrap around the circle.

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Proposed Work

Derive AdS/CFT correspondence by studying the low energy action of open strings

• n a stack of N D3-branes.

The D-brane acts as a source in supergravity. At low energies and small string coupling, the gravity theory in the bulk and the gauge theory on the brane describe the same physics giving rise to AdS/CFT. After this derivation, we will use the AdS/CFT to study various correlators on the boundary gauge theory.

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Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References

References

J.W. van Holten, arXiv :hep-th/0201124.

• B. Zwiebach, A First Course in String Theory, (Cambridge University Press, Cam-

bridge, 2004). Polchinski, J. G. String theory, (Cambridge University Press, Cambridge, 2007) David Tong, arXiv :hep-th/0908.0333. Katrin Becker, Melanie Becker and John H. Schwarz, String Theory and M-Theory A Modern Introduction, (Cambridge University Press, Cambridge, 2006). C.V. Johnson, D-Branes, (Cambridge University Press, Cambridge, 2002).

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Thank You.

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