Exact Quantization of a Superparticle in

Transcription

1 21st October, 2010 Talk at SFT and Related Aspects Exact Quantization of a Superparticle in AdS 5 S 5 Tetsuo Horigane Institute of Physics, Univ. of Tokyo(Komaba) Based on arxiv : ( Phys.Rev. D81, , 2010 ) by T.H and Yoichi Kazama 1/25

2 1. Introduction I m going to talk about - 1st quantization - only particle mode 2/25

3 1. Introduction I m going to talk about - 1st quantization - only particle mode Motivation : AdS/CFT correspondence Typical Example : Type IIB superstring on AdS 5 S 5 N =4 SU(N) SY M 3/25

4 1. Introduction I m going to talk about - 1st quantization - only particle mode Motivation : AdS/CFT correspondence Typical Example : Type IIB superstring on AdS 5 S 5 N =4 SU(N) SY M 4/25

5 1. Introduction I m going to talk about - 1st quantization - only particle mode Motivation : AdS/CFT correspondence Typical Example : Type IIB superstring on AdS 5 S 5 Worldsheet theory is notoriously complicated - interacting 2d field theory - RR flux - only global symmetry psu(2,2 4) is available 5/25

6 1. Introduction I m going to talk about - 1st quantization - only particle mode Motivation : AdS/CFT correspondence Typical Example : Type IIB superstring on AdS 5 S 5 Our Strategy Our Hope : Maximal supersymmetry should be rather powerful Make full use of the representation theory of psu(2, 2 4) symmetry in the quantization First stage: Spectrum: Which representations occur? 6/25

7 Our approach : Gauge-fixed Green-Schwarz action in AdS 5 S 5 phase space quantization Merit - only physical d.o.f remains - psu(2,2 4) is geometrically realized ( + modifiaction) - split the spectroscopy problem into two parts, quantization and diagonalization Demerit - Some of the global charges become nonlinear 7/25

8 Our approach : Gauge-fixed Green-Schwarz action in AdS 5 S 5 phase space quantization Merit - only physical d.o.f remains - psu(2,2 4) is geometrically realized ( + modifiaction) - split the spectroscopy problem into two parts, quantization and diagonalization Demerit - Some of the global charges become nonlinear Result ; H AdS 5 S 5 superparticle 1/2BPS[0, l, 0] l=2 -match with the SUGRA (Field theory) result 8/25

9 Plan of Talk 1. Introduction 2. Classical description of a superparticle in AdS 5 S 5 with RR flux 3. Quantization of a superparticle in AdS 5 S 5 with RR flux 4. Complete solution of physical states - Spectrum and wavefunctions - 5. Summary and Prospects 9/25

10 2. Classical description of a superparticle in AdS 5 S 5 with RR flux psu(2, 2 4) algebra: Even part: SO(4, 2) [ ] Commutation relations 15 bosonic generators (4d conformal algebra) P a : translation K a : special conformal J ab : Lorentz rotations D : dilatation (a, b =0,, 3) [J, P ] P, [J, K] K, [J, J] J [D, P ]=P, [D, K] = K, [P, K] D J Similarly for K ±,K,K,J,J ±,J ±,J We will often use the following 4D Light-cone basis SO(6) SU(4) 15 bosocic generators(r-symmetry) [ J i j,j k l] = δ k j J i l δ i l J k j, (J i i = 0) i, j, k, l =1 4 10/25

11 Odd part: 32 supercharges: Q ±i,q ± i,s±i,s ± i, i =1 4 ± ± SU(4) transformation properties Q ±i,s ±i belong to 4, Q ± i,s± i belong to 4 Non-vanishing [Boson,Fermion] type commutators [P, S] Q, [J, Q] Q, [K, Q] S [J, S] S Anti-commutation relations among the supercharges: {Q, Q} P, {S, S} K {Q, S} J so(3,1) D J su(4) 11/25

12 Problem ; Diagonalize the AdS energy E E P + K Eigenoperator of E exists ; [ ] E, Ŝ = 1 2Ŝ [ ] E, ˆQ ˆQ, Ŝ Q ± S = 1 2 ˆQ Our task ; Search all the SuperConformal Primary state ; Ŝ SCP =0 12/25

13 Action ( ) stantially. S = 1 2e dt Gauge-fixed Green-Schwarz action in AdS 5 S 5 ( e 2φ ( ẋ + ẋ +ẋ x ) +( φ) 2 }{{} AdS metric in Poincaré coord.(z = e φ ) (Metsaev, Thorn, Tseytlin hep-th/ ) +e A 0 ea 0 2ẋ+ i [ ] e 2φ (θ i θi + θ θi i ) + (η i η i + η i η i ) 2i η i (γ A ) i je A 0 ηj 1 [ ] ) 4 (ẋ+ ) 2 (η 2 ) 2 ( η 4G i (γ A ) i j η j ) 2 A B (η(v A )η)(η(v B )η) ( ( ) G A B ẏ A ẏ B }{{} SO(6) invariant metric tensor of S 5 + ] ) +4G A B η i (V A ) i jη j ẏ B ] where i x +,x, x, x, φ ; AdS 5 coordinate y A ; S 5 coordinate θ i, θ i, η i, η i ; total 16 fermionic coordinate (V A ) i j ; SO(6) Killing vector on S 5 13/25

14 Action ( ) stantially. S = 1 2e dt Gauge-fixed Green-Schwarz action in AdS 5 S 5 ( e 2φ ( ẋ + ẋ +ẋ x ) +( φ) 2 }{{} AdS metric in Poincaré coord.(z = e φ ) (Metsaev, Thorn, Tseytlin hep-th/ ) +e A 0 ea 0 G A B ẏ A ẏ B SO(6) invariant metric tensor of S 5 2ẋ+ i [ ] e 2φ (θ i θi + θ θi i ) + (η i η i + η i η i ) 2i η i (γ A ) i je A 0 ηj 1 [ ] ) 4 (ẋ+ ) 2 (η 2 ) 2 ( η 4G i (γ A ) i j η j ) 2 A B (η(v A )η)(η(v B )η) ( ( ) }{{} + ] ) +4G A B η i (V A ) i jη j ẏ B ] Go to the phase space to quantize at equal time 1. Introduce the momentum and do the constraint analysis 2. Construct the global charges in terms of phase space variables 14/25

15 3. Quantization of a superparticle in AdS 5 S 5 Equal Time Commutator between physical variables ( i{ } D [, ], x + = τ ) [x, P x ] = [ x, P x ]=[x,p ]=[φ,p φ ]=i, [y A,P B ]=iδ A B {S i,s j } = { S i, S j } = δ i j (S i i P θ i, S i P e φ η i ) Hilbert space (i =1,, 4) - H particle f(x, φ,y)g(s i, S j ) f, g (S i, S j = 0) - quantum mechanical norm ( part) ; dφ 0 AdS 5 dp dx 1 dx 2 (f 2 f 1) - f(x, φ,y) should be normalizable under this norm 15/25

16 4. Complete solution of physical states - Spectrum and wavefunctions 4.1 Set up of the problem Our task ; Solve Ŝ±i f, g = Ŝ ± i f, g =0 complicated 16 simultaneous 1st order differential equations ( N S = S i S i,n S = S i S i,s S = S i S i,z = e φ ) Ŝ +i = i ( ) P z S i + i xs i + i 2 P [2P x S i z S i + 1 z ( S i (N S 1) 2li k S k )] Ŝ i = i [ ] 2 2zP x S i 2 S i (S S) S i (z z + N S + 1) + 2l i ks k P x + ix Q +i i + i xq i P S [2P i 2 x S i z S i + 1 ( S i (N P i z S 1) 2li S )] x k k i P S i 16/25

17 4. Complete solution of physical states - Spectrum and wavefunctions 4.1 Set up of the problem Our task ; Solve Ŝ±i f, g = Ŝ ± i f, g =0 complicated 16 simultaneous 1st order differential equations Plan of our solution n( N S = S i S i,n S = S i S i,s S = S i S i,z = e φ ) 1. Restrict the candidates for SCP state - Restrict the allowed form of, f(y A ) f(y A )g(s i, S j ) 2. Solve for each candidate and determine AdS 5 part 17/25

18 4.2 Allowed highest weight unitary representations for SU(4) part Orbital part ( f(y A ) ) Orbital part wave function lives in L 2 (S 5 ). So it can be expanded by scalar spherical harmonic function. They are symmetric traceless represenation of SO(6) and their SU(4) Dynkin labels are [0, l, 0] (each integer l appear once) This fact can be understood algebraically as will be shown below. 18/25

19 There exists a quadratic product relation 4 satisfied by l i j ( ) L i j l i kl k j 1 4ˆl 2 δ i j 2li j =0 l i j =(V A ) i j / y A An example of the power of the quadratic relation ( ): L 2 1 λ 1, λ 2, λ 3 =0 (λ 1 +2λ 2 + λ 3 + 2) }{{} 0 λ 1 =0 l 2 1 E 1 λ 1, λ 2, λ 3 =0 Result: The only allowed HWS are 0, l, 0, l =0, 1, 2,... ˆl2 0, l, 0 = l(l + 4) 0, l, 0 l 19/25

20 Lessons L Product relation among the generators Restriction on the highest weight module { } In our case {J i kj k j 1 4 δi j(j m nj m n) (4 N 2 )Ji j} SCP =0 N =(S) 2 +( S) 2 Ω l = S 1 S 2 S 1 S 2 0, l, 0 [0,l+2, 0], vac = 0, 0, 0 fvac = S 1 S 1 S 2 S 2 S 3 S 3 S 4 S 4 0, 0, 0, 20/25

21 4.3 Solution of the superconformal primary condition - comlpete Our original task ; Construct all the solution of Ŝ f, g =0 Fix the part ; AdS 5 Φ l (z, P x,p x,p ) Ω l for each to satisfy Ŝ (Φ(z, P x,p x,p ) Ω l )=0 ( where z = e φ ) On Ω l, the supercharge operators effectively simplify substantially: Below we use the indices i =(α, ˆα), α =1, 2, ˆα =3, 4. 21/25

22 We now impose the remaining superconformal primary conditions one by one to determine the form of Φ. (1) First, consider 0= 2 Ŝ +ˆα Ψ =(S +ˆα + Q ˆα ) Ψ = i [ ( ) 2 2 P x + P S ˆα P P x ( z l +1 z ) +2P z S ˆα ] Ψ From the coefficient of S ˆα and S ˆα, we get two simple differential equations ( ) (i) P x + P Φ =0 ( P x (ii) z l +1 ) +2P z Φ =0 z They determine the dependence on P x and z as ( 1) Φ = f(p )ψ ( ψ = exp P xp x P z 2 P ) z l+1 22/25

23 (2) Next consider the following condition 0= 2 Ŝ ˆα Ψ =(S ˆα Q +ˆα ) Ψ This gives a simple differential equation with respect to P : (S ˆα Q +ˆα ) Ψ = i P [(1 + z 2 )P ( l + 1 ) 2 P ] xp x + P S ˆα Ψ =0 P P This determines f(p ) as f(p )=ce P P l+(1/2), c = constant (3) The remaining conditions 0 = 2 Ŝ + α Ψ = (S+ α Q α ) Ψ and 0= 2 Ŝ α Ψ =( S α + Q+ α ) Ψ are satisfied automatically. 23/25

24 Thus we found all the unitary superconformal primary states in the form 5 Ψ l = C l exp ( P xp x P (z 2 + 1)P ) z l+1 P l+(1/2) S 1 S 1 S 2 S 2 0 0, l, 0, l =0, 1, 2,... Properties of Ψ l Quantum numbers of Ψ l : [ ] AdS energy E Ψ l = E l Ψ l, E l = l +2 SU(2) L SU(2) R spins J 3 L,R Ψ l =0 SU(4) Casimir Ĵ 2 Ψ l =(l + 2)(l + 6) Ψ l Dimension of the representation (up to the action of ˆP s) 2 8 dim [0, l, 0] = 64 3 (l + 1)(l + 2)2 (l + 3) These are precisely the 1 BPS superconformal multiplets of 1-particle states realized in supregravity 2 Single trace operators Tr (φ {I 1φ I 2 φ Il+2} ) and its descendants in SYM. 24/25

25 5. Summary Result Quantize the superparticle in Exhaust all the SCP states, agreed with the SUGRA result AdS 5 S 5 from first principle Our Message Construction of the quantum superconformal generators is important and useful for spectroscopy psu(2, 2 4)!"#$%"&'%(%)#(*++,-#.(/0+12+#% On going project!$),-(%)#(#3&$%#4'#(,5($,/#(%62#(,5(/0+12+#%( String case need to introduce some good regularization to define and calculate the charge algebra 25/25