Integrability in Super Yang-Mills and Strings

Transcription

1 13/9/2006 QFT06, Kyoto Integrability in Super Yang-Mills and Strings Kazuhiro Sakai (Keio University) Thanks for collaborations to: N.Beisert, N.Gromov, V.Kazakov, Y.Satoh, P.Vieira, K.Zarembo 1

2 Contents I. Anomalous dimension in N = 4 super Yang-Mills One-loop: integrable spin-chain All loops: particle model II. Free superstrings on AdS 5 x S5 Classical integrability and the general solutions Quantum strings as integrable particle models 2

3 N = 4 U(N) Super Yang-Mills L = 1 ( Tr (F 4gYM 2 µν ) 2 + 2(D µ Φ i ) 2 ([Φ i, Φ j ]) 2 ) + 2i Ψ/DΨ 2 ΨΓ i [Φ i, Ψ] Global symmetry: SO(4, 2) SU(4) P SU(2, 2 4) Q Φ ab Q +1 Ψ ḃ α P Ψ αb F α β D αβ Φ ab F αβ D αβ Ψ ḋ γ D αβ Ψ γd V F D F DDΦ DF DD Ψ DDΨ 3

4 Conformal Field Theory Correlation function of local operators O 1 (x 1 )O 2 (x 2 ) = δ D1 D 2 B 12 x 12 D 1+D 2 O 1 (x 1 )O 2 (x 2 )O 3 (x 3 ) = C 123 x 12 D 1+D 2 D 3 x23 D 2+D 3 D 1 x31 D 3+D 1 D 2 D i : scaling dimension of the local operator O i 4

5 Single trace operator O = Tr[W A1 W A2 W AJ ] W A {D k Φ, D k Ψ, D k Ψ, D k F } - dominant in the large N limit Scaling dimension: eigenvalue of the Dilatation operator (Beisert 03) ˆD At tree level: ˆD 0 O = dim(o)o [Φ] = 1, [Ψ] = 3, [F ] = 2, [D] = 1 2 5

6 Quantum correction: operator mixing O = ˆDO Ψ Φ Φ Ψ Φ Φ F Ψ Φ Φ F Ψ O ˆD = ˆD 1 n=0 Diagonalize ˆD! λ n ˆD n λ = g 2 YMN ( t Hooft coupling) Hamiltonian of su(2, 2 4) spin chain (Minahan-Zarembo 02) (Beisert-Staudacher 03) 6

7 SU(2) subsector TrZ L X = Φ 1 + iφ 2 ferromagnetic vacuum Z = Φ 5 + iφ 6 Tr(ZZZXZXZZ ) XXX Heisenberg Spin chain L H = ( ) l=1 l l+1 l l+1 7

8 Bethe Ansatz Equation One-magnon States Ψ(p) = L l=1 ψ(l) l ψ(l) = e ipl Schrödinger Eq. H Ψ = E Ψ H = L ( ) E = 2 e ip e ip l=1 l l+1 l l+1 = 4 sin 2 p 2 : Dispersion Relation 8

9 Two-magnon States Ψ(p 1, p 2 ) = 1 l 1 <l 2 L l 1 l 2 ψ(l 1, l 2 ) Schrödinger Eq. H Ψ = E Ψ E = 2 k=1 4 sin 2 p k 2 (Dispersion) ψ(l 1, l 2 ) = e ip 1l 1 +ip 2 l 2 + S(p 2, p 1 )e ip 1l 2 +ip 2 l 1 (Bethe s Ansatz) S(p 1, p 2 ) = eip 1+ip 2 e 2ip e ip 1+ip 2 e 2ip S-matrix 9

10 Factorized scattering ψ(p 2, p 1, p 3,..., p J ) = S(p 1, p 2 )ψ(p 1, p 2, p 3,..., p J ) Periodic boundary condition ψ(p 2,..., p J, p 1 ) = e ip 1L ψ(p 1,..., p J ) Yang equations e ip kl = J S(p k, p l ) l k rapidity variable u = 1 2 cot p 2 Bethe Ansatz Equations ( uk + i 2 u k i 2 ) L = J l k u k u l + i u k u l i (k = 1,..., J) 10

11 Local Charges Momentum P = Q 1 = k 1 i ln u k + i 2 u k i 2 Energy E = Q 2 = k ( i u k + i 2 i u k i 2 ) Higher charges Q r = i r 1 k ( ) i (u k + i i 2 )r 1 (u k i 2 )r 1 11

12 Higher loops Difficulties: long-range interaction fluctuation in length Attempts: Ψ 1 Ψ 2 X Y Z closed subsector without change of length su(2) su(1 1) relationship with known models Inozemtsev chain Hubbard model (Beisert-Dippel-Staudacher 04) (Staudacher 04) etc. (Serban-Staudacher 04) (start deviating at 3 loops) (Rej-Serban-Staudacher 05) 12

13 Particle model (Excitation picture) (Berenstein-Maldacena-Nastase 02) Vacuum (Staudacher 04) (Beisert 05) 0 I := ZZZZZZZZZZZZZZZZZZ Asymptotic state X 1 X 2 I := n 1 n 2 n 1 n 2 e ip1n1+ip2n2 ZZZX 1 ZZZ ZZZX 2 ZZZ 13

14 Physical state: total momentum j p j = 0 ( Tr X I = Tr n eipn ZZXZZ n ) = n eipn Tr XZZ = δ(p)tr XZZ One particle state n X I = n eipn ZZXZZ n+1 Z + X I = n eipn ZZZXZZ XZ + I = n eipn ZZXZZZ n Z ± X I = e ip XZ ± I 14

15 One particle state: 8 bosons + 8 fermions Φ i (i=1,...,4), D i Z (i=1,...,4), Ψ αȧ, Ψ a α (a,α=1,...,2) : single excitation of Z Z, F αβ, D i Φ j,... : multiple excitation Vacuum breaks the global symmetry P SU(2, 2 4) P SU(2 2) P SU(2 2) R (8 8) = (2 2) (2 2) Z φ 1 φ 2 ψ 1 ψ 2 φ 1 Φ 11 Φ 12 Ψ 11 Ψ 12 φ 2 Φ 21 Φ 22 Ψ 21 Ψ 22 ψ 1 Ψ 11 Ψ 12 D 11 Z D 12 Z ψ 2 Ψ 21 Ψ 22 D 21 Z D 22 Z 15

16 su(2 2) algebra [R a b, J c ] = δ c b J a 1 2 δa b J c [L α β, J γ ] = δ γ β J α 1 2 δα β J γ {Q α a, S b β} = δ b a Lα β + δ α β Rb a + δ b a δα β C Transformation of one-particle states: (2 2) rep. Q α a φ b I Q α a ψ β I S a α φ b I S a α ψ β I = a δ b a ψα I = b ɛ αβ ɛ ab φ b Z + I = c ɛ ab ɛ αβ ψ β Z I = d δ β α φa Z I C 2 = 1 4 C = 1 2 n particle E 16

17 Central extension of su(2 2) algebra [R a b, J c ] = δ c b J a 1 2 δa b J c [L α β, J γ ] = δ γ β J α 1 2 δα β J γ {Q α a, S b β} = δ b a Lα β + δ α β Rb a + δ b a δα β C {Q α a, Q β b} = ɛ αβ ɛ ab P {S a α, S b β} = ɛ ab ɛ αβ K Transformation of one-particle states: (2 2) rep. Q α a φ b I Q α a ψ β I S a α φ b I S a α ψ β I = a δ b a ψα I = b ɛ αβ ɛ ab φ b Z + I = c ɛ ab ɛ αβ ψ β Z I = d δ β α φa Z I C 2 P K = 1 4 C = 1 2 n particle E 17

18 P X I = α Z + X I α XZ + I = α(e ip 1) XZ + I K X I = β Z X I β XZ I = β(e +ip 1) XZ I E = αβ sin 2 ( p 2 ) 1 αβ : some function of λ From perturbative computation up to 3 loops (small λ ) and BMN energy formula (large λ ), (Minahan-Zarembo 02) (Beisert-Staudacher 03) etc. (Berenstein-Maldacena-Nastase 02) E = 1 + λ π sin2( p) (dispersion relation for a magnon ) (Beisert-Dippel-Staudacher 04) 18

19 su(2 2) S-matrix (Beisert 05) most general ansatz: S 12 φ a 1 φb 2 I = A 12 φ {a 2 φb} 1 I + B 12 φ [a 2 φb] 1 I C 12ɛ ab ɛ αβ ψ α 2 ψβ 1 Z I S 12 ψ α 1 ψβ 2 I = D 12 ψ {α 2 ψβ} 1 I + E 12 ψ [α 2 ψβ] 1 I F 12ɛ αβ ɛ ab φ a 2 φb 1 Z+ I S 12 φ a 1 ψβ 2 I = G 12 ψ β 2 φa 1 I + H 12 φ a 2 ψβ 1 I S 12 ψ α 1 φb 2 I = K 12 ψ α 2 φb 1 I + L 12 φ b 2 ψα 1 I P SU(2 2) R 3 symmetry Coefficients A 12 (p 1, p 2 ),..., L 12 (p 1, p 2 ) are uniquely determined up to an overall factor 19

20 Properties of the S-matrix Unitarity Associativity (Yang-Baxter equation) Not of difference form S 21 S 21 = I S 12 S 13 S 23 = S 23 S 13 S 12 S 12 (u 1 u 2 ) Similarity to the Shastry s R-matrix for the Hubbard model 20

21 The whole S-matrix S psu(2,2 4) = S 0 [ Ssu(2 2) S su(2 2) ] with an overall scalar factor S 0 = 1 + O(λ 3 ) (for the gauge theory) Asymptotic Bethe ansatz equations Impose periodic boundary condition Yang equations: e ip jj = Diagonalize these equations by the nested Bethe ansatz K k j J (finite length ) S(p j, p k ) The S-matrix reproduce long range Several checks up to 3 loops, (Beisert-Kristjansen-Staudacher) (Eden-Jarczak-Sokatchev) psu(2, 2 4) Bethe ansäze (Beisert-Staudacher 05) at most valid up to O(λ J 2 ) 21

22 AdS/CFT Correspondence N = 4 U(N) Super Yang-Mills > plot3d([cos(t)*cos(s),cos(t)*sin(s),sin(t)],t=-pi..pi,s=0..2*pi,sc aling=constrained); IIB Superstrings on x AdS 5 S 5 > sphereplot(2,theta=0..2*pi,phi=0..pi,scaling=constrained); SO(4, 2) SO(6) Page 1 λ = g 2 Y M N R 4 = 4πg s α 2 N g 2 Y M = g s N 4πλ = R4 α 2 22

23 L {}}{ O = Tr( ZZZ ZZZ) > with(plots): > plot3d([cos(t)*cos(s),cos(t)*sin(s),sin(t)],t=-pi..pi,s= plot3d([cosh(t)*cos(s),sinh(t),cosh(t)*sin(s)],t= aling=constrained); Pi,scaling=constrained); E = L x L > with(plots): > sphereplot(2,theta=0..2*pi,phi=0..pi,scaling=constrained with(plots): > plot3d([cosh(t)*cos(s),sinh(t),cosh(t)*sin(s)],t= ,s=0..2* > plot3d([cos(t)*cos(s),cos(t)*sin(s),sin(t)],t=-pi..pi,s= plot3d([cosh(t)*cos(s),sinh(t),cosh(t)*sin(s)],t= Pi,scaling=constrained); aling=constrained); Pi,scaling=constrained); Page 1 O = Tr(Z X Ȳ Z) + x > with(plots): > plot3d([cos(t)*cos(s),cos(t)*sin(s),sin(t)],t=-pi..pi,s=0..2*pi,sc plot3d([cosh(t)*cos(s),sinh(t),cosh(t)*sin(s)],t= ,s=0..2* > sphereplot(2,theta=0..2*pi,phi=0..pi,scaling=constrained plot3d([cos(t)*cos(s),cos(t)*sin(s),sin(t)],t=-pi..pi,s= Pi,scaling=constrained); aling=constrained); Page 1 O = Tr(Z s Z s Z Z) + x > sphereplot(2,theta=0..2*pi,phi=0..pi,scaling=constrained); plot3d([cos(t)*cos(s),cos(t)*sin(s),sin(t)],t=-pi..pi,s=0..2*pi,sc > sphereplot(2,theta=0..2*pi,phi=0..pi,scaling=constrained aling=constrained); Page 1 Page 1 23

24 Sigma model on R S 3 λ S = dσdτ [ a X 0 a X 0 + a X i a X i + Λ (X i X i 1)] 4π Equations of Motion (i = 1,..., 4) + X i + ( + X j X j )X i = 0, + X 0 = 0 Gauge: X 0 = κτ : energy of the string λ 2π ) κ = λ ( = 2π 0 dσ τ X 0 = λκ Virasoro Constraints ( ± X i ) 2 = ( ± X 0 ) 2 = κ 2 24

25 SU(2) Principal Chiral Field Model g SU(2) X S 3 g = Right current ( X1 + ix 2 X 3 + ix 4 X 3 + ix 4 X 1 ix 2 j = g 1 dg ) d j j j = 0, d j = 0 Virasoro constraints 1 2 Trj2 ± = κ2 25

26 Lax Connection a(x) = 1 1 x 2 j + x 1 x 2 j x : spectral parameter d j j j = 0 d a(x) a(x) a(x) = 0 d j = 0 Lax pair [L(x), M(x)] = 0 L(x) = σ a σ (x) = σ 1 ( j+ 2 1 x j ) 1 + x M(x) = τ a τ (x) = τ 1 ( j+ 2 1 x + j ) 1 + x 26

27 Auxiliary Linear Problem { L(x)Ψ(x; τ, σ) = 0 M(x)Ψ(x; τ, σ) = 0 { σ Ψ = a σ Ψ τ Ψ = a τ Ψ Ψ(x; τ, σ) = P exp σ 0 a σ dσ Monodromy matrix Ψ(x; τ, σ + 2π) = Ω(x; τ, σ)ψ(x; τ, σ) Ω(x; τ, σ) = P exp 2π 0 a σ dσ 27

28 τ Monodromy Matrix σ ( τ, σ + 2π) ( τ, σ) Ω(x; τ, σ) Ω(x; τ, σ) = U 1 Ω(x; τ, σ)u (τ, σ + 2π) (τ, σ) Ω(x; τ, σ) Ω(x) ( e ip(x) 0 0 e ip(x) ) p(x) : quasi-momentum T (x) := Tr Ω(x) = 2 cos p(x) (transfer matrix eigenvalue) 28

29 T (x) (Kazakov-Marshakov-Minahan-Zarembo 04) +2 x 2 +p(x) x p(x) x = 1 x = +1 29

30 Virasoro Constraints 1 2 Trj2 ± = κ2 p(x) πκ x 1 (x ±1) Single-valuedness dp = 2πZ dp = 0, dp = 2πˆn a  a ˆB a ˆn a : mode number ˆB a Ĉ a  a ˆB a 30

31 Explicit form of general finite gap solution (Dorey-Vicedo 06) θ(2π + 0 ω X 1 + ix 2 = C + b dq D) 1 θ( b dq + D) θ(2π + 0 ω X 3 + ix 4 = C b dq D) 2 θ( b dq + D) ( exp i ( dq 0 + exp i dq ) ) θ( z) = ( ) i m z + πi(π m) m m Z g exp : Riemann theta function dq = σdp + τ dq p : quasi-momentum q : quasi-energy ω j : normalized holomorphic differentials b j = B j B g+1 : closed B-cycles ( A i ω j = δ ij ) D, C 1, C 2 : constants 31

32 Finite gap solution on Yang-Mills side Traditional thermodynamic limit ( up + i 2 ) L = J u p u q + i u p i 2 q=1 q p u p u q i L, u k O(1) u i Strings 32

33 Novel thermodynamic limit ( up + i 2 ) L = J u p u q + i u p i 2 q=1 q p u p u q i L, J, u k Lu k Log of both sides 1 u p + 2πn p = 2 L J q p 1 u p u q n p Z : mode number 33

34 u Resolvent J 1 G(u) = 1 L q=1 u u q ( up + i 2 ) u G(u) = C dvρ(v) u v C 1 u p i 2 C 2 BAE 1 u + 2πn a = 2/G(u) for u C a 34

35 Quasi-momenta p 1 (u) = p 2 (u) = G(u) 1 2u BAE 1 u + 2πn a = 2/G(u) p 1 (u + i0) = p 2 (u i0) + 2πn a (u C a ) p 1 (u) ( ) up up + i 2 i 2 u p p i 2 i 2 p 2 (u) hyper-elliptic curve 35

36 Comparison with Yang-Mills side Frolov-Tseytlin limit: L λ with rescaling u = λ 4π x Interior cuts Poles (x = ±1) ( 1 < x < 1) degenerate into the origin u = 0 36

37 T (u) (Kazakov-Marshakov-Minahan-Zarembo 04) +2 u 2 +p(u) u p(u) u = 0 37

38 Classical Superstring on AdS 5 x S5 Classical IIB Superstrings on = > plot3d([cos(t)*cos(s),cos(t)*sin(s),sin(t)],t=-pi..pi,s=0..2*pi,sc aling=constrained); x Coset sigma-model on PSU(2, 2 4) Sp(1, 1) Sp(2) AdS 5 S 5 > sphereplot(2,theta=0..2*pi,phi=0..pi,scaling=constrained); Page 1 X i (σ, τ ) ψ α (σ, τ ) g(σ, τ ) PSU(2, 2 4) J = g 1 dg decomposition w.r.t. Z 4 -grading J = H + Q 1 + P + Q 2 38

39 (Metsaev-Tseytlin 98) (Roiban-Siegel 02) Sigma-Model Action λ S σ = ( 1 2 2π strp P 1 2 strq 1 Q 2 + Λ strp ) Lax Connection A(z) = H + ( 1 2 z z 2) P (Bena-Polchinski-Roiban 03) + ( 1 2 z z 2) ( P Λ) + z 1 Q 1 + z Q 2 { Bianchi Identity Equation of Motion Flatness Condition dj J J = 0 da(z) A(z) A(z) = 0 39

40 Monodromy Matrix Ω(z) = P exp 2π 0 dσa(z) P exp 2π 0 dσa(1) γ Ω(z) Physical quantity: Conjugacy class of Ω(z) ( Generating functions of conserved charges) Eigenvalues of the Monodromy Matrix Ω diag (z) = u(z)ω(z)u(z) 1 = diag(e i p 1, e i p 2, e i p 3, e i p 4 e i ˆp 1, e i ˆp 2, e i ˆp 3, e i ˆp 4 ) p i (z), ˆp i (z) : quasi-momenta 40

41 Spectral Curve for the Sigma-Model ˆp 1 ˆp /x 2 x 2 p 1 x 1 C 1 1/ C 1 1/x 1 1/Ĉ2 Ĉ 2 p 2 p ˆp 3 Ĉ 1 1/Ĉ1 ˆp 4 p 4 Distribution of cuts with mode numbers  a Ĉ a ˆB a ˆB a  a and fillings Ĉ a (Beisert-Kazakov-K.S.-Zarembo C 05) ˆB a ˆB a determines a classical solution 41

42 Giant Magnons (Hofman-Maldacena 06) (Okamura & Suzuki s talk) ϕ HM limit: p J E J = fixed λ = fixed p = fixed ( ) BMN limit: λ n = pj = fixed O(3)-sigma model Sine-Gordon model 42

43 Non compact sector and log S scaling for S J > with(plots): > plot3d([cosh(t)*cos(s),sinh(t),cosh(t)*sin(s)],t= ,s=0..2* > plot3d([cos(t)*cos(s),cos(t)*sin(s),sin(t)],t=-pi..pi,s= Pi,scaling=constrained); aling=constrained); O = Tr(D s 1 ZD s 2 Z) > with(plots): > plot3d([cosh(t)*cos(s),sinh(t),cosh(t)*sin(s)],t= Pi,scaling=constrained); x twist-two operator log S scaling is universal folded string > plot3d([cos(t)*cos(s),cos(t)*sin(s),sin(t)],t=-pi..pi,s=0..2*pi,sc > sphereplot(2,theta=0..2*pi,phi=0..pi,scaling=constrained aling=constrained); Page 1 (Gubser-Klebanov-Polyakov 02) > plot3d([cos(t)*cos(s),cos(t)*sin(s),sin(t)],t=-pi..pi,s= aling=constrained); O = Tr(D s 1 ZD s 2 Z D s J Z) x > sphereplot(2,theta=0..2*pi,phi=0..pi,scaling=constrained); Page 1 S cλ log S (one-loop) > sphereplot(2,theta=0..2*pi,phi=0..pi,scaling=constrained Page 1 S c λ log S (classical level) (Callan-Gross 73) (Korchemsky 95) etc. (K.S.-Satoh 06) 43

44 Towards quantization of strings on AdS x S 5 5 Green-Schwarz, pure spinor,... several difficulties in conventional approach Green-Schwarz string as an integrable particle model? Conformal gauge vs Uniform gauge worldsheet Lorentz symmetry relativistic ( crossing symmetry) non-relativistic global symmetry (no good subsector) unbroken Virasoro constraint??? broken easy 44

45 Relativistic particle model O(4)-sigma model (SU(2) principal chiral field model) Zamolodchikovs S-matrix (Zamolodchikov-Zamolodchikov 77) p 2 p 1 = δ(p 1 p 1 )δ(p 2 p 2 )Ŝa b a b (θ 1 θ 2 ) p 1 p 2 p = (m cosh πθ, m sinh πθ) Ŝ a b a b (θ) = ) ( b a a b ) ( a b a σ 1 (θ) + σ 2 (θ) + σ 3 (θ) b a b a b 45

46 Unitarity Crossing Symmetry Ŝ c 1c 2 b 1 b 2 (θ)ŝ b 1b 2 a 1 a 2 ( θ) = Î c 1c 2 a 1 a 2 Associativity ( ) Ŝ a b a b iπ θ = Ŝ a b a b (θ) Ŝ b 1b 2 c 1 c 2 (θ)ŝ c 1b 3 a 1 c 3 (θ + θ )Ŝ c 2c 3 a 2 a 3 (θ ) = Ŝ c 1c 2 a 1 a 2 (θ)ŝ b 1c 3 c 1 a 3 (θ + θ )Ŝ b 2b 3 c 2 c 3 (θ ) θ θ+θ θ+θ θ θ θ 46

47 Ŝ(θ) is constrained up to an overall factor (CDD ambiguity) Ŝ a b a b (θ) = S 0 (θ) 2 [ b a ) ( a b a b i θ a b i i θ ) ( b a a b ] Minimal Solution S 0 (θ) = i Γ( θ 2i )Γ( θ 2i ) Γ( θ 2i )Γ( 1 2 θ 2i ) 47

48 Yang equations e iµp(θ α) = α β Ŝ(θ α θ β ) : Matrix equation Can be diagonarized by nested Bethe ansatz e iµ sinh πθ α = S 2 0 (θ α θ β ) θ α u j + i/2 θ β α j α u j i/2 1 = u j θ β i/2 u j u i + i u β j θ β + i/2 u i j j u i i 1 = v k θ β i/2 v k v l + i v β k θ β + i/2 v l k k v l i k θ α v k + i/2 θ α v k i/2 48

49 The above BAEs reproduce Classical solution (Gromov-Kazakov-K.S.-Vieira 06) - macroscopic number of particles (1-cut for θ ) - macroscopic number of magnons - double scaling limit - eliminate θ Single theta-cut solution Classical solution (KMMZ) z = x + 1 x 49

50 The above BAEs reproduce AFS string Bethe ansatz equations (Arutyunov-Frolov-Staudacher 04) (Gromov-Kazakov 06) - macroscopic number of particles (1-cut for θ ) - macroscopic number of magnons - double scaling limit - eliminate θ BDS all-loop Bethe Ansatz equations + dressing factor (Beisert-Dippel-Staudacher 04) introduced to repair the 3-loop discrepancy 50

51 Non-relativistic particle model AFS s dressing factor overall factor of the string S-matrix S psu(2,2 4) = S 0 [ Ssu(2 2) S su(2 2) ] small λ S 0 = 1 + O(λ 3 ) (SYM perturbation) =? (classical string) large λ ( S 0 = exp constrained by symmetry 2i λ θ 0 (p 1, p 2 ) n=0 θ n (p 1, p 2 ) ( 1 λ ) n ) (Arutyunov-Frolov-Staudacher 04) (worldsheet 1-loop) θ 1 (p 1, p 2 ) (Hernández-López 06) (Janik 06) (crossing symmetry) θ n (p 1, p 2 ) (Beisert-Hernández-López 06) 51

52 Summary The spectral problem of the dilatation operator is fully solved at one-loop Magnon picture may solve the problem even at all loops for infinitely long operators General solutions of classical strings on the AdS background are available Integrable formulation of quantum strings is in progress 52

53 Prospects Mismatch between the both sides Finite length corrections, wrapping effects Related subjects Integrability in Open spin chain N < 4 theories Plane-wave matrix model, LLM background 53