TBA Equations for Minimal Surfaces in AdS

Transcription

1 Equations for Minimal Surfaces in AdS Katsushi Ito Tokyo Institute of Technology Y. Hatsuda, KI, K. Sakai, Y. Satoh, arxiv: JHEP 004 (200) 08, arxiv: JHEP 009 (200) 064 November 9, University November 9, University

2 Outline Introduction 2 Minimal surface in AdS 3 3 equations and minimal area 4 Remainder Function and Perturbative CFT 5 Outlook November 9, University 2

3 Introduction AdS/CFT Correspondence N = 4 U(N) SYM type IIB Superstrings on AdS 5 S 5 Gluon Scattering Amplitudes [Alday-Maldacena ] gluon scattering Disk amplitude in AdS Wilson Loop in AdS z=0 z=z IR z=0 z=z IR AdS/CFT T-dual vev of Wilson loop = area of minimal surface r=0 y y y 0.5 November 9, University 3

4 Gluon Scattering Amplitudes in N = 4 SYM the BDS conjecture Bern-Dixon-Smirnov, Anastasiou-Bern-Dixon-Kosower Planar L-loop, n-point amplitude (recursive structure) A (L) n (k,,k n ) = A (0) n (k,,k n )M (L) n (ǫ) M n (ǫ) + n (ǫ) IR divergence: Dimensional regularization (D = 4 2ǫ) L= λl M (L) ln M n(ǫ) = A2 ǫ + A 2 ǫ n 6 f(λ) X µ 2 ln f(λ) = 4 g(λ) = 2 For n = 4 i= l= f(l) f (l) l= s i,i+ «2 g(λ) 4 nx i= 0 λl : cusp anomalous dimension l λ l : collinear anomalous dimension 4 = ( s ) 2 log2 + 2π2 t 3 F BDS «µ 2 ln + f(λ) s i,i+ 4 F n (BDS) + C November 9, University 4

5 Test of the BDS conjecture (Weak Coupling) loop calculations (n 5) BDS,... Dual conformal invariance: determines n 5 amplitudes Gluon amplitude/wilson loop duality at weak coupling Drummond-Henn-Korchemsky-Sokatchev Discrepancy in 6-point 2-loop amplitude Bern-Dixon-Kosower-Roiban-Spradlin-Volovich Drummond-Henn-Korchemsky-Sokatchev lnm MHV 6 = lnw(c 6 ) + const., F WL 6 = F BDS 6 + R 6 (u), R 6 0 remainder function R n (u): F n = Fn BDS + R n Non-trivial dependence on the cross-ratios (3n 5) u ij,kl = x2 ij x2 kl x 2, (x 2 ik x2 ij = t [j i] i ) jl For n = 6, u 3,46, u 24,5, u 35,26 are independent cross ratios. November 9, University 5

6 Minimal Surface in AdS AdS 5 : embedding coordinates in R 2,4 : isometry SO(2,4) Y 2 Y Y 2 + Y Y Y 2 4 = Poincaré coordinates: (y µ,r) µ = 0,,2,3 Y µ = yµ r, Y + Y 4 = r, Y Y 4 = r2 + y µ y µ r Action in confomal gauge: S = R2 d 2 z yµ yµ + r r 2π r 2 Euler-Lagrange equation+boundary condition ends at r 0 (AdS boundary) y µ : light-like segments M n exp( S[y,r]) k k 2 y r=0 November 9, University 6

7 Alday-Maldacena s Solution (4pt amplitude) Alday-Maldacena arxiv: AdS 3 constraint: Y 3 = Y 4 = 0 r 2 (y 0 ) 2 + (y ) 2 + (y 2 ) 2 = static gauge: r = r(y,y 2 ), y 0 = y 0 (y,y 2 ) dimesnional regularization D = 3 2ǫ conformal boost: s = t general (s, t) 4-point amplitude (s = t): s = (k + k 2 ) 2, t = (k + k 4 ) 2 y2 0.5 boundary condition: r(±,y 2 ) = r(y, ±) = 0, 0.5 y0 0 y 0 (±,y 2 ) = ±y 2, y 0 (y, ±) = ±y -0.5 y 0 = y y 2, r = ( y 2)( y2 2 ) y r y y S 4 = ln M 4 f(λ) λ November 9, University 7

8 Minimal surface in AdS 3 We consider the classical string solutions in AdS 3. (z, z): worldsheet coordinates (Euclidean) emmbedding coordinates Y = (Y,Y 0,Y,Y 2 ) in R 2,2 Y Y := Y 2 Y Y 2 + Y 2 2 = action ( Y z = zy, Y z = z Y ) { S = d 2 z Yz Y z + λ( Y Y } + ) equations of motion Virasoro constraints Y z z ( Y z Y z ) Y = 0 Y 2 z = Y 2 z = 0 November 9, University 8

9 Pohlmeyer reduction [Pohlmeyer CMP 46 (976) 207; de Vega-Sanchez PRD 47 (993) 3394] variables Y α(z, z)(real), p(z)(holomorphic), p( z) (anti-hol) e 2α = 2 Y z Y z N i = 2 e 2α ǫ ijkl Y j Y k z Y l z, ǫ ( )02 = +, p = 2 N Y zz p = 2 N Y z z N: normal vector to the surface N Y = N Y z = N Y z = 0, N N = Moving-frame basis q = Y, q 2 = e α Y z, q 3 = e α Yz, q 4 = N. November 9, University 9

10 evolution of moving-frame basis q 0 0 e α 0 z q 2 q 3 = 2e α α z α z 2pe α q 4 0 pe α 0 0 z q q 2 q 3 q 4 = 0 e α α z 0 2 pe α 0 0 α z 2e α 0 0 pe 0 q q 2 q 3 q 4 q q 2 q 3 q 4 Spinor notation: (a, ȧ): spacetime SL(2, R) SL(2, R) ( ) ( ) Y Y Y aȧ = 2 Y + Y = 2 Y Y 0 Y 2 Y 22 Y + Y 0 Y Y 2 ( ) W,aȧ W W α α,aȧ = 2,aȧ = ( (q + q 4 ) aȧ (q 2 ) aȧ 2 (q 3 ) aȧ (q q 4 ) aȧ W 2,aȧ W 22,aȧ ), November 9, University

11 evolution of W α α,aȧ z W α α,aȧ + (Bz L ) α β W β α,aȧ + (Bz R ) β α W α β,aȧ = 0, z W α α,aȧ + (B L z) β α W β α,aȧ + (B R z ) β α W α β,aȧ = 0, Bz L = B z(), Bz R = UB z(i)u, B L z = B z(), B R z = UB z(i)u B z (ζ) = α ( z ( B z (ζ) = α z ( ) 0 e πi 4 U =. e 3πi 4 0 ) ( 0 e α ζ e α p 0 ) ζ ( 0 e α p e α 0 ), ), November 9, University

12 Each entry of the matrix W α α,aȧ is a null vector evolution of ψ W α α,aȧ = ψ L α,aψ Ṙ α,ȧ z ψ L α,a + (BL z ) α β ψ L β,a = 0, zψ L α,a + (BL z ) α β ψ L β,a = 0, z ψα, Ṙ α + (BR z ) β α ψ Ṙ β, α = 0, zψ α, Ṙ α + (BR z ) β α ψ Ṙ β, α = 0. compatibility condition B L z B z L + [BL z,bl z ] = 0, BR z B z R + [BR z,br z ] = 0, = generalized sinh-gordon equation α(z, z) 2e 2α(z, z) + p(z) 2 e 2α(z, z) = 0 November 9, University

13 The left/right linear problem can be promoted to a family of linear problems with the general spectral parameter ζ: ( ) ( ) z + B z (ζ) ψ(z, z;ζ) = 0, z + B z (ζ) ψ(z, z;ζ) = 0, ζ = (left) ζ = i (right) B z (ζ) = A z + ζ Φ z, B z (ζ) = A z + ζφ z = SU(2) Hitchin equations: D z Φ z = D z Φ z = 0, F z z + [Φ z,φ z ] = 0 Area A = 2 d 2 ztrφ z Φ z = 4 d 2 ze 2α November 9, University

14 change of variable dw = p(z)dz gauge transformation ˆα = α log p p 4 ˆψ = gψ, g = e i π 4 σ3 e i π 4 σ2 e 8 log p p σ3 ˆB w (ζ) = ˆB w (ζ) = ˆα w 2 ( w + ˆB w ) ˆψ = 0, ( w + ˆB w ) ˆψ = 0, ˆα ( w 0 i 2 i 0 ( 0 i i 0 ) ( cosh ˆα isinh ˆα ζ isinh ˆα cosh ˆα ) ζ compatibility cond. = sinh-gordon equation ( cosh ˆα isinh ˆα isinh ˆα cosh ˆα w wˆα e 2ˆα + e 2ˆα = 0 ), ). November 9, University

15 solutions with light-like boundary ˆα 0 w p(z): polynomial of degree n 2 (2n-gonal boundary) at large w ( w ζ σ 3 ) ˆψ = 0, ( w ζσ 3 ) ˆψ = 0, have two independent solutions (big and small) ( w ) ( ζ ˆη + = e + wζ 0, ˆη = 0 e w ζ + wζ ( ) w = w e iθ, ζ = ζ e iα, Re w ζ + wζ = w ( ζ )cos(θ + ζ α) cos(θ α) < 0 ˆη + : small ˆη : large cos(θ α) > 0 ˆη : small ˆη + : large ). November 9, University

16 Bases of solutions the Stokes sector Ŵ j : (j 3 2 )π + arg ζ < arg w < (j 2 )π + arg ζ ŝ j : the small solution in Ŵ j (unique up to rescaling) W 2 W ŝ 2k = ( ) k ˆη for w Ŵ 2k ŝ 2k = ( ) kˆη + for w Ŵ 2k W0 W- In Ŵ j one can take ŝ j (large) and ŝ j (small) as the basis of the solution. normalization ŝ j ŝ j+ det(ŝ j,ŝ j+ ) = Stokes data ŝ j+ = ŝ j + b j (ζ)ŝ j, b j (ζ) = ŝ j ŝ j+ November 9, University

17 small solutions in the z-plane the small solution s j = g ŝ j the Stokes sector in the z-plane W j : (2j 3)π n + 2 n arg ζ < arg z < (2j )π n + 2 n arg ζ the differential equation is regular for z <, W j+n = W j monodromy factor µ j : s j+n = µ j s j Ŝ j = (ŝ j,ŝ j+ ), Ŝ j+ = Ŝ j B j+, Ŝ j+n = Ŝ j M j. ( 0 B j = b j = µ 2k = µ 2k = µ, b j+n = µ 2 j b j ) ( µj 0, M j = 0 µ j+ ) November 9, University

18 Two involutions holomorphic involution (Z 2 symmetry) σ 2 [ w + ˆB w (ζ)]σ 2 = [ w + ˆB w ( ζ)],σ 2 [ w + ˆB w (ζ)]σ 2 = [ w + ˆB w ( ζ)] σ 2 ŝ j (w, w;e πi ζ) = iŝ j+ (w, w;ζ) b j (e πi ζ) = b j+ (ζ). antiholomorphic involution w + ˆB w (ζ) = w + ˆB w ( ζ ), w + ˆB w (ζ) = w + ˆB w ( ζ ). ŝ j (w, w; ζ ) = ŝ j (w, w;ζ) b j ( ζ ) = b j (ζ). November 9, University

19 Equations and minimal area Alday-Gaiotto-Maldacena , Hatsuda-Ito-Sakai-Satoh The functional form b j (ζ) (the Riemann-Hilber problem) monodromy factor, constraints from involutions asymptotic form ζ,0 evaluated by the WKB analysis b j (ζ) has nontrivial asymptotics in each angular (Stokes) sector Introduce the Fock-Goncharov coordinates χ j (ζ) cross-ratio simple asymptotics in all angular sectors discontinuities across the Stokes-line = equations Alday-Maldacena-Sever-Vieira Plücker relation for s j +constraints = T-system = Y-system ( equations) November 9, University

20 Decagon solution (n = 5) polynomial: p(z) = z 3 3Λ 2 z + u = (z z )(z z 2 )(z z 3 ) conditions for the Stokes coefficients b b 2 = µb 4, b 2 b 3 = µ b 5, b 3 b 4 = µ b, b 4 b 5 = µb 2. β 2k = µ b 2k, β j β j+ = β j+3, β 2k = µb 2k β j+5 = β j November 9, University 2

21 the Fock-Goncharov coordinates the WKB line: Re( ) ζ p(z)dz largest the WKB triangulation: quadrant the Fock-Goncharov coordinates χ E = (s s 2 )(s 3 s 4 ) (s 2 s 3 )(s 4 s ) aysmptotic boundary condition ζ 0, ( ) Zi χ γi (ζ) exp ζ + Z i ζ, Z i = p(z)dz γ i 2 χ γ = (s s 2 )(s 3 s 5 ) (s 2 s 3 )(s 5 s ) = β 4, χ γ 2 = χ γ2 = (s 5 s )(s 3 s 4 ) (s s 3 )(s 4 s 5 ) = β 2 3 z 2 z 3 z γ γ November 9, University 2

22 The WKB triangulation jumps discontinuously when argζ crosses the BPS rays (A) (B) (C) (D) (E) 3 arg ζ (A) (B) (C) (D) (E) χ γ β5 β 4 β 4 β3 β3 χ γ2 β 2 β 2 β β β 5 Discontinuities= Integral equations F(ζ) = dx x+ζ 0 x z ζ f(x) F(y + iǫ) F(y iǫ) = 0 4iǫ (x y) 2 + ǫ2f(x) 2πif(y)(ǫ 0) November 9, University 2

23 ln χ γ (ζ) = Z ζ + Z ζ dζ 4πi Zlγ2 ζ + ζ ζ ζ ζ ln( + χγ 2 (ζ )) + dζ 4πi Zl γ2 ζ + ζ ζ ζ ζ ln( + χ γ 2 (ζ )) ln χ γ2 (ζ) = Z 2 ζ + Z 2 ζ + dζ 4πi Zlγ ζ + ζ ζ ζ ζ ln( + χγ (ζ )) dζ 4πi Zl γ ζ + ζ ζ ζ ζ ln( + χ γ (ζ )) We can rewrite the integral equations in the form equations: ǫ (θ) = 2 Z cosh θ ǫ 2 (θ) = 2 Z 2 cosh θ dθ 2π dθ 2π cosh(θ θ + i α) log( + e ǫ 2(θ ) ), cosh(θ θ i α) log( + e ǫ (θ ) ) Z k = Z k e iα k, ζ = e θ+iα k, α = π/2 (α α 2 ) ǫ k (θ) = ǫ γk (θ), ǫ γk (θ) = ǫ γk (θ) (Z 2 -symmetry) χ γk (ζ = e θ+iα k) = e ǫγ k (θ), November 9, University 2

24 Integral equation and equations for general n Gaiotto-Moore-Neitzke log χ γk (ζ) = Z γ k ζ + Z γk ζ γ k,γ 4πi γ l γ dζ ζ ζ + ζ ζ ζ log( + χ γ (ζ )) γ = ±γ, ±γ 2,, ±γ n 2 Z l γ : γ ζ R γ γ γ equations n 3 ǫ k (θ) = 2 Z k cosh θ Z k = Z γk l= 3 2 dθ i γ k,γ l 2π sinh(θ θ + iα k iα l ) log(+e ǫ l(θ ) ) November 9, University 2

25 Homogeneous sine-gordon model This equations are identified with those of the homogeneous sine-gordon model Fernandez-Pouza-Gallas-Hollowood-Miramontes perturbed CFT: generalized parafermions SU(N) 2 /U() N S-matrix Castro-Alvaredo-Fring S ab (θ) = ( ) δ ab [c a tanh 2 (θ + σ ab π ] 2 i) Iab eqs ǫ a (θ) = m a R cosh θ b dθ 2π ii ab sinh(θ θ + σ ab + π 2 i) log(+e ǫ b ). minimal surface HSG model n 2 N 2 Z a m a R γ a,γ b ǫ ab I ab i(α α b ) σ ab + π 2 i November 9, University 2

26 Regularized area and Free energy cross ratios x± ij x± kl x ± ik x± jl = (s i s j )(s k s l ) (s i s k )(s j s l ) x ± = Y ±Y 0 Y +Y 2 light-cone coordinates at the AdS boundary ζ = left (+), ζ = i right (-) Area (finite+divergent) A = 4 d 2 ze 2α = A Sinh + 4 finite part= free energy F regular 2n-gon solution (CFT limit) d 2 w, A Sinh = 4 d 2 w(e 2α p p) A sinh = F + c n, c n = 7 (n 2)π 2 The constant c n is determined by the superposition of hexagon solutions where zeros of p(z) become far apart from each other. F = π 6 c = π 6n (n 2)(n 3) A Sinh = π 4n (3n2 8n + 4) agrees with regular polygon solutions Alday-Malacena November 9, University 2

27 Remainder Function and Perturbative CFT 6-point gluon scattering amplitudes minimal surface with 6 cusps in AdS 5 SU(4) Hitchin equations with Z 4 -automorphism Stokes data equations for A 3 -integrable (Z 4 -symmetric) field theory [Alday-Gaiotto-Maldacena] Y-function Y a (θ) (a =,2,3) Y = Y 3 Y (θ + iπ 4 )Y (θ iπ 4 ) = + Y 2(θ) Y 2 (θ + iπ 4 )Y 2(θ iπ 4 ) = ( + µy (θ))( + µ Y (θ)) periodicity Y a (θ + 3iπ 2 ) = Y a(θ), reality: Y a (θ) = Y a ( θ) µ = e iφ : monodromy (s i+6 = µ ( )j s i ) November 9, University 2

28 equations (6pt amplitude) cross-ratio U = b 2 b 3 = x2 4 x2 36 x 2 3 x2 46 monodromy b k = Y (, U 2 = b 3 b = x2 25 x2 4 x 2 24 x2 5, U 3 = b b 2 = x2 36 x2 25 x 2, 35 x2 26 (k )πi (2k + )πi ), U k = + Y 2 ( ) 2 4 b b 2 b 3 = b + b 2 + b 3 + µ + µ equations: ǫ(θ) = log Y (θ + iϕ), ǫ(θ) = log Y 2 (θ + iϕ) (Z = Z e iϕ ) ǫ = 2 Z coshθ + K 2 log( + e ǫ ) + K log( + µe ǫ )( + µ e ǫ ) ǫ = 2 2 Z coshθ + 2K log( + e ǫ ) + K 2 log( + µe ǫ )( + µ e ǫ ) 2coshθ K (θ) = K 2 (θ) = 2π coshθ π cosh2θ f g(θ) = dθ f(θ θ )g(θ ): convolution November 9, University 2

29 Remainder function (6pt amplitude) Z Area = 4 d 2 z(e 2α (P P) 4 ) + 4 Z R F = 2π Z d 2 z(p P) 4 4 d ZΣ 2 w + 4 d 2 w 0 Σ 0 = A BDS A BDS like A period A free = 3X Li 2( U k ) Z 2 ( F) 4 Z + k= n dθ 2 Z cosh θ log( + µe ǫ(θ) ))( + µ e ǫ(θ) ) +2 2 Z cosh θ log( + e ǫ(θ)o φ=0 φ=π/4 φ=π/2 φ=3π/ φ=0 φ=π/4 φ=π/2 φ=3π/ Afree 0.2 R Z Z November 9, University 2

30 Thermodynamic Bethe Ansatz Al. Zamoldchikov, NPB 342 ( ) + dim Euclidean QFT on a cylinder with periodic B.C. partiton function Z(R,L) = Tre RH L = Tre LH R L e LRf(R) e LE 0(R) f(r): free energy per unit length at temperature /R E 0 (R): ground state energy L time R L R time E 0 (R) = Rf(R) particle A a (a =,,n) with mass m a (E = m coshθ, p = m sinhθ) purely elastic scattering matrix S ab (θ) = e iδ ab(θ) Bethe wave function Ψ(x,,x N ) = N i= eip ix i Bethe eqs. e ip il j i S ij(θ i θ j ) = ± November 9, University 3

31 thermodynamic limit L, N a, N a /L:fixed ρ (r) a (θ):denisty of roots ρ a (h) :denisty of holes ρ = ρ (r) + ρ (h) total energy n E = m a coshθρ (r) a (θ)dθ entropy S = n a= a= [ρ a lnρ a ρ (r) a lnρ (r) a ρ a (h) lnρ a (h) ] minimizing free energy F = E TS with constraint dθρ (r) a = N a /L = equation n m a R cosh θ = ǫ a (θ) + ϕ ab log( + e ǫ b ), b= ρ (r) a ρ a = e ǫa + e ǫa ϕ ab = i d dθ log S ab(θ) November 9, University 3

32 Analytic solution in the IR and UV limits r = m R (m :lowest mass gap) IR (large mass ) limit r ǫ a ˆm a r coshθ ( ˆm a = m a /m ) free massive field theory (or other CFT) UV (small mass) limit r 0 ǫ a :const ǫ a = b N ab ln( + e ǫ b) CFT E 0 (R) = 6R c(r) π L(x) = 2 { 6 r n c(r) = π 2 a= ˆm ak ( ˆm a r) r 6 n π 2 a= L( +e ) = c ǫa CFT r 0 ( ) x 0 dt ln t t + ln( t) t : Rogers dilogarithm function November 9, University 3

33 small mass expansion: perturbed CFT Klassen-Melzer, NPB338 (990) 485 S = S CFT + λ d 2 wǫ(w, w) ǫ(w, w) has conformal weight (D ǫ,d ǫ ) partition function Z = exp ( λ d 2 wǫ(w, w) ) 0 R F = lim L L log Z = RE(R) 4 (mr)2 E(R) = E 0 R 2 ( λ) n ( ) 2π 2+2n(Dǫ ) d 2 z 2 d 2 z n n! R n= V (0)ǫ()ǫ(z 2 ) ǫ(z n )V ( ) 0,connected z i 2(Dε ) i November 9, University 3

34 Remainder function for 6-point amplitude Z 4 parafermion (c = 2(N 2) N+2 = ) deformed by enegy operator ǫ(z, z) with conformal dims. ( 3, 3 ) (2πλ) 2 = [ m πγ( 3 4 )] 8 3 γ( Γ(x) 6 ) (γ(x) = Γ( x) ) Fateev PLB 324 R = 3 4 k= Li 2( Y 2 ( (2k+)πi 2 )) Z 2 F C 8 3 F = π 2 = π2 6 + φ2 [ ] 8 π γ( 3 4 ) 3 γ( 6 )γ( 3 ) 3π + Z 2 C 8 3 γ( 3 + φ 2π )γ( 3 φ 3π ) Z Y 2 (θ) = Ỹ (0) (θ,φ) + Ỹ () 2 (θ,φ) Z Ỹ (0) 2 = + 2cos( 2φ () ), Ỹ 2 = y () (φ)cos 3 4(ϕ + iθ) 3 y () (φ) cos φ cos 2φ 3 November 9, University 3

35 Numerical results vs CFT perturbation pert large Z R φ=0 φ=0(pert) 0.7 φ=π/4 φ=π/4(pert) φ=π/ φ=π/2(pert) φ=3π/4 φ=3π/4(pert) Z R Z November 9, University 3

36 Outlook AdS 3 case: 2n-point amplitudes SU(n 2) 2 /U() n 3 parafermions AdS 5 case: m-point amplitudes: SU(m 4) 4 U() m 5 [SU(4) ] m 4 SU(4) m 4 Y-system is different from that of HSG model. (new integrable field theory?) CFT perturbation of generalized parafermions = analytical (power series) results for the remainder function m = 6 SU(2) 4 /U() (Z 4 parafermion) form factor Maldacena-Zhiboedov Hitchin eqs. / CFT correspondence (ODE/IM correspondence Dorey-Tateo, Bazhanov-Lukyanov-Zamolodchikov) November 9, University 3