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 1 / 37

Relativistic Strings Superstrings D-Branes T-dualities Proposed Work References Plan Relativistic Strings 1 Point Particle : Action and Quantization Classical Strings Quantum Strings Superstrings 2 Action Quantization and Spectrum D-Branes 3 Gauge Fields on D-branes Parallel Dp-branes Strings and D-brane Charges T-dualities 4 T-dualities T-duality and Closed Strings T-duality and Open Strings Proposed Work 5 References 6 2 / 37

Relativistic Strings Superstrings Point Particle : Action and Quantization D-Branes Classical Strings T-dualities Quantum Strings Proposed Work References Point Particle : Action Action in terms of proper time : � � τ f dx µ dx ν S PP = − mc − η µν d τ d τ. (1) d τ τ i → Lorentz invariant and reparametrization invariant. Equation of Motion : dp µ = 0 . (2) d τ → Point particle moves with constant momentum along the worldline. Classically equivalent action : � � � S PP = 1 e − 1 ˙ x µ ˙ x ν η µν − m 2 e d τ . (3) 2 This has couple of advantages : → Works even for massless particle. → No annoying square root. 3 / 37

Relativistic Strings Superstrings Point Particle : Action and Quantization D-Branes Classical Strings T-dualities Quantum Strings Proposed Work References Point Particle : BRST Quantization BRST Quantization : Action in terms of momentum and velocity : � � � x µ p ν η µν − e 2 ( p µ p ν η µν + m 2 ) S PP = d τ ˙ . (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. � � p µ p µ + m 2 �� b − 1 � x µ p µ + ι b ˙ ˙ S BRST = d τ . (5) 2 → The purpose of introducing these new fields is to produce the BRST charge Ω against which the action is invariant. Ω = b α G α , (6) where b α are grassmann numbers generated by fermionic fields. 4 / 37

Relativistic Strings Superstrings Point Particle : Action and Quantization D-Branes Classical Strings T-dualities Quantum Strings Proposed Work References Point Particle : BRST Quantization From Legendre transformation we get : � p µ p µ + m 2 � G = 1 (7) 2 Hence, the action of Ω is nilpotent i.e., { Ω , Ω } = 0. → S BRST is invariant under BRST transformation ( δ Ω ) 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 : H BRST = Ker Ω Im Ω , (8) Ker Ω : gives the states which annihilate under Ω , the physical ones. Im Ω : gives the states which are obtained by acting Ω on any arbitrary state | χ � , all states. 5 / 37

Relativistic Strings Superstrings Point Particle : Action and Quantization D-Branes Classical Strings T-dualities Quantum Strings Proposed Work References Classical Strings : The Nambu-Goto Action Nambu-Goto action : � τ f � σ 1 �� ˙ X . X ′ � 2 − ˙ S NG = − T 0 X 2 X ′ 2 . d τ d σ (9) c τ i 0 → In terms of induced metric (like in point particle case) : � � S NG = − T 0 d τ d σ − det (Γ αβ ) , (10) c where ∂ x µ ∂ x ν Γ αβ = η µν (induced metric) (11) ∂ξ α ∂ξ β Boundary conditions from EOM : δ X µ ( τ, σ ∗ ) = 0 (Dirichlet boundary condition) , (12) Π σ µ ( τ, σ ∗ ) = 0 (Neumann boundary condition) . (13) µ = ∂ L /∂ X µ ′ . where Π σ 6 / 37

Relativistic Strings Superstrings Point Particle : Action and Quantization D-Branes Classical Strings T-dualities Quantum Strings Proposed Work References Classical Strings : The Polyakov Action Polyakov action : � � S P = − T 0 d 2 σ − det ( g αβ ) g αβ ∂ α X µ ∂ β X ν G µν . (14) 2 → Choose G µν = η µν . → Symmetries : Global Poincare (Translational and Lorentzian), Local gauge (Repa- rametrization and Local Weyl). Equations of Motion : �� � ∂ α X µ ∂ β X µ − 1 2 g αβ g lm ∂ l X µ ∂ m X ν = 0 − gg αβ ∂ β X µ and ∂ α = 0 . (15) Boundary conditions (for open string) : g ασ ∂ α X µ = 0 (Neumann boundary condition) , (16) δ X µ = 0 (Dirichlet boundary condition) . (17) Note : Local gauge symmetries demonstrate the repetition in degrees of freedom so we would fixed them by quantization. 7 / 37

Relativistic Strings Superstrings Point Particle : Action and Quantization D-Branes Classical Strings T-dualities Quantum Strings Proposed Work References 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 ) = x µ + v µ τ + X µ R ( τ − σ ) + X µ L ( τ + σ ) . (18) → For closed strings : σ ∈ [ 0 , 2 π ] while X L and X R are independent. → For open strings : σ ∈ [ 0 , π ] while X L = X R . Express Eq. 18 in terms of Fourier modes : For closed strings : � � � � α ′ 1 X µ ( τ, σ ) = x µ + v µ τ + ι n e − ι n ( τ + σ ) + ˜ α µ α µ n e − ι n ( τ − σ ) . (19) 2 n n � = 0 For Open strings : � ′ � 1 X µ ( τ, σ ) = x µ + v µ τ + ι n e − ι n τ cos n σ. n α µ 2 α (20) n � = 0 8 / 37

Relativistic Strings Superstrings Point Particle : Action and Quantization D-Branes Classical Strings T-dualities Quantum Strings Proposed Work References Classical Strings : Light-cone quantization → The conserved worldsheet momentum which is the string spacetime momentum is given by (see write-up for derivation) : ′ l v µ ⇒ v µ = 2 πα p µ = p µ . (21) α ′ 2 π l → Plugging above two solutions in the Virasoro Constraints and using the relation − P µ P µ = M 2 , we get mass-shell conditions : � 1 M 2 = α i − n α i n . For open strings : (22) 2 α ′ n � = 0 � M 2 = 2 α i − n α i For closed strings : n . (23) ′ α n � = 0 Result : One can conclude the mass of a string from its oscillations. 9 / 37

Relativistic Strings Superstrings Point Particle : Action and Quantization D-Branes Classical Strings T-dualities Quantum Strings Proposed Work References 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) [ Π i ( τ, σ ) , Π j ( τ, σ ′ )] = 0 (25) [ X i ( τ, σ ) , Π j ( τ, σ ′ )] = ιδ ij δ ( σ − σ ′ ) . (26) In conclusion, all oscillation modes x i , p i , α i α i n , ˜ n become operators. When we plug- in these mode expansions into above relations, we get : [ x i , p j ] = ιδ ij . (27) [ α i n , α j α i α j m ] = n δ ij δ n + m , 0 . m ] = [˜ n , ˜ (28) 10 / 37

Relativistic Strings Superstrings Point Particle : Action and Quantization D-Branes Classical Strings T-dualities Quantum Strings Proposed Work References Quantum Strings : Closed strings Spectrum 2 ) For n > 0, creation and annihilation operators are related as : 1 1 − n ) † = − n ) † = ( a i √ n α i a i α i (˜ , √ n ˜ (creation operators) . (29) − n − n 1 1 a i √ n α i a i α i ˜ √ n ˜ n = , n = (annihilation operators) . (30) n n 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 : D − 1 � � M 2 = 2 n ) − D − 2 n + ˜ n ( N i N i 6 α ′ . (31) α ′ i = 2 n � = 0 11 / 37

Relativistic Strings Superstrings Point Particle : Action and Quantization D-Branes Classical Strings T-dualities Quantum Strings Proposed Work References Quantum Strings : Closed strings Spectrum → For ground state : N i m = ˜ N i m = 0 for all i , m . M 2 = − D − 2 6 α ′ . (32) We find the tachyon for any spacetime dimension more than 2. Hence, the ground state is tachyonic for closed strings. → First excited state : α i α j − 1 | 0 , p µ � . − 1 ˜ (33) Corresponding to n = 1, we get : M 2 = 26 − D ⇒ D = 26 . (for massless particle) (34) 6 α ′ → 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 G µν and Kalb-Ramon or B-field B µν respectively. → Higher excitations are all massive. 12 / 37