TESTING ADS/CFT. John H. Schwarz STRINGS 2003

Transcription

1 TESTING ADS/CFT John H. Schwarz STRINGS 2003 July 6,

2 INTRODUCTION During the past few years 1 Blau et al. constructed a maximally supersymmetric plane-wave background of type IIB string theory as a Penrose limit of the AdS 5 S 5 background. 2 Metsaev showed that IIB strings in this background are described in the GS light-cone gauge formalism by free massive bosons and fermions. 3 Berenstein, Maldacena, and Nastase identified the corresponding sector of SU(N N = 4 SYM and carried out some checks of the AdS/CFT duality in the appropriate limit. There have been many important subsequent developments. 2

3 THE GEOMETRY ds 2 (AdS 5 = R 2 ( cosh 2 ρ dt 2 +dρ 2 +sinh 2 ρ dω 2 3 ds 2 (S 5 = R 2 (cos 2 θ dφ 2 + dθ 2 + sin 2 θ d Ω 2 3 Let r = R sinh ρ, y = R sin θ x + = t/µ, x = µ R 2 (φ t Here µ is an arbitrary mass scale. x has period 2πµ R 2, and so the conjugate (angular momentum is P = J/µ R 2, where J is an integer. Then R, holding r, y, x ± fixed, gives the plane-wave limit ds 2 = 2dx + dx µ 2 (x I 2 (dx dx I dx I 3

4 FREE STRINGS IN THE PLANE WAVE Fourier analysis gives harmonic oscillators [a I m, a J n ] = δ IJ δ mn I, J = 1, 2,..., 8 m, n with frequencies ω n (µα = n 2 + (µα 2, where α = α P. Then H lc = 1 ω n (a I n a I n + fermions. α Since the string action is in light-cone gauge (x + = τ, this is both a spacetime and a world-sheet Hamiltonian for free strings. 4

5 THE DUALITY gy 2 M N R 4 /(α 2 J µ R 2 P gy 2 M 4πg s The Penrose limit R corresponds to J, N with finite λ = g 2 Y M N/J2, which is the loop expansion parameter for BMN operators. The key duality is = D R P + = H lc where D = J + D anom is the dimension and R is the U(1 R-charge. This duality can be tested perturbatively in λ 1/(µα P 2 g 2 = J 2 /N 4πg s (µα P 2 1/J 1/(µ R 2 P 5

6 I will report on two projects, described at Strings 2002 as work in progress, that have now been completed. I In work with Curtis Callan, Hok Kong Lee, Tristan McLoughlin, Ian Swanson, and Xinkai Wu (to appear soon we have studied O(1/ R 2 corrections to the plane-wave limit and computed string theory energy shifts at this order. These were successfully matched to O(1/J corrections to BMN operator dimensions. II In work with Yang-Hui He, Marcus Spradlin, and Anastasia Volovich (hep-th/ we carried out a large µ analysis of the three-string vertex. This is required for comparison with O(g 2 contributions in the dual gauge theory (expanded in powers of λ 1/µ 2. 6

7 PART I O(1/J corrections The AdS 5 S 5 metric has the following expansion in powers of 1/ R 2 ds 2 2 dx + dx +dz 2 +dy 2 ( z 2 +y 2 (dx R2 [ 2y 2 dx dx ( y 4 z 4 (dx ( dx z2 dz y2 dy 2] + O(1/ R 4, where y and z are four-vectors. resulting Hamiltonian is The H lc = δl GS δẋ + = H pp + H int H pp = 1 [ (x A 2 + (p 2 A 2 + (x A 2 ] + i 2 [ψψ ρρ + 2ρΠψ] where Π = γ 1 γ 2 γ 3 γ 4 = γ 5 γ 6 γ 7 γ 8. 7

8 where R 2 H int is 1 [z ( 2 p 2 y + y 2 + 2z 2 ( y 2 p 2 z + z 2 + 2y 2 ] 4 1 { [ (p 8 A 2] 2 + 2(pA 2 (x A 2 + [(x A ] 2 2 } + 1 [ (x A 2] (p Ax A 2 1 { [(ψ ψ + (ρρ ] (ρπψ (ψ ψ ρ ρ 2 +(ρψ (ρ ψ 1 [(ψρ (ψρ + (ψ ρ 2] [ 2 1 ( ψγ jk ψ ργ jk ρ (ρ γ jk Πψ ργ jk Πψ ]} 12 (ψγjk ρ(ργ jk Πρ (j, k j, k i [ ] (p 4 A 2 + (x A (ψψ 2 + (y 2 z 2 ρρ 1 2 (p Ax A (ρψ +ψρ i ( p 2 2 k + y 2 z 2 ρπψ 8

9 + i ( 4 (z j z k ψγ jk ψ ργ jk ρ i ( 4 (y j y k ψγ j k ψ ργ j k ρ i ( 8 (z k y k + z k y k ψγ kk ψ ργ kk ρ (p ky k + z k p k ψγ kk ρ (p jz k (ψγ jk Πψ + ργ jk Πρ 1 ( 4 (p j y k ψγ j k Πψ + ργ j k Πρ 1 ( 4 (p ky k + z k p k ψγ kk Πψ + ργ kk Πρ i 2 (p kp k z k y k ψγ kk Πρ. 9

10 The energy shifts are then computed using first-order perturbation theory. Restricting to the space spanned by two-impurity states of the form A nb n J gives a matrix with 256 eigenvalues. These should then be compared with the O(1/J contributions to the dimensions of the corresponding two-impurity single-trace BMN operators. A supermultiplet of BMN operators consists of 256 operators. If the lowest one has R-charge ( R 0 and dimension D 0, there are of them at level 8 L L with D = D 0 + L/2 and R = R 0 + L/2, L = 0,..., 8. ( = D R and R L/2 are constant. 10

11 After a lengthy analysis, we find that to first order in g 2 Y M N n (R, L = 2 + g2 Y M N π 2 sin g2 Y M N R 2 n 2 ( nπ R + 3 L/2. ( 1 6 L R + O(R 2 This agrees precisely with our string theory analysis in all cases. We conjecture that the following formula is exact in the planar approximation: n (R, L = g2 Y M N π 2 sin 2 ( nπ R + 3 L/2 11

12 PART II O(g 2 corrections Light-cone gauge string field theory was worked out for bosonic strings in by Mandelstam, Kaku, Kikkawa, Cremmer and Gervais. In it was generalized to superstrings by Green, Brink, and JHS. The three-string vertex was generalized to include the mass parameter µ in 2002 by Spradlin and Volovich. Type IIB superstring field: Φ(x +, x, x I (σ, θ a (σ 1 Transcribe first-quantized operators to second-quantized ones: H 2 = i dx D 8 x(σd 8 θ(σ ΦH lc Φ, etc. 2 Add interactions: H = H 2 + H , Q = Q 2 + Q etc. 12

13 THREE STRING VERTEX H 3 is given in the multi-fock-space description by V 3 = GE a E b 0, where E a = exp 1 3 N rs 2 mna (r m a (s n. r,s=1 m,n= E b is a similar fermionic expression and G is polynomial in the oscillators. The Neumann matrices N rs mn arise as products and inverses of various infinite matrices. The α r = α P (r may be scaled so that α 1 = y, α 2 = 1 y, α 3 = 1. This makes the value of µ meaningful. All the quantities that we will compute are functions of two variables: 0 y 1 and 0 µ <. 13

14 Fourier analysis of the overlaps with string 3 gives (m, n = 1, 2,... mn = 2 π ( 1m+n+1 y sin mπy mn n 2 m 2 y 2 A (1 mn = 2 π ( 1m (1 y sin mπy mn n 2 m 2 (1 y 2 A (2 A (3 mn = δ mn We also define C mn = mδ mn B m = 2 sin mπy π ( 1m+1 y(1 ym 3/2 (C r mn = m 2 + (µα r 2 δ mn (U r mn = [C 1 (C r µα r ] mn 3 (Γ + mn = (A (r U r A (rt mn r=1 14

15 N rs All the components of mn are expressed simply in terms of (Γ 1 + mn and Y m (µ, y = (Γ 1 + B m, F (µ, y = log[1 µy(1 yb T Γ 1 + B]. Last summer I derived a formula for (Γ 1 + mn in terms of Y m and F m δ mn + y(1 y(ω m µ(ω n µy m Y n 2ω m 2(ω m + ω n e F This was also obtained by Pankiewicz. Thus we only need to find Y m and F. 15

16 Using various identities satisfied by the infinite matrices A (r mn and the infinite vector B m, one can derive the differential equation Y m µ = [ 1 F 2 µ ( 1 µ ω m µ ] ωm 2 Y m. This has the solution Y m = m ω m exp [ 1 2 µ 0 F µ ( 1 µ ] dµ Y m (0, y, ω m where Y m (0, y is a known flat-space expression. Thus, if we knew F (µ, y, we could deduce all the Neumann coefficients. 16

17 Since Y m m 2µ B m for large µ, the preceding formula for Y m (µ, y implies that 0 (m 2 + µ 2 3/2 F (µ, ydµ = G(m, y, where G(z, y = 2τ 0 z + 2 z 2log ( Γ(1 + z Γ(1 + zyγ(1 + z(1 y. and τ 0 = y log y + (1 ylog (1 y. The inverse integral transform, which does not seem to exist in the mathematical literature, is F (µ, y = iµ2 π π 0 cos θ G( iµcos θ, ydθ. 17

18 Using this, one can show that for our specific choice of G(m, y F (µ, y = ln[4πµy(1 y] where J(x = 2 π +J(µy + J(µ(1 y J(µ, 1 ln(1 e 2πxz z z 2 dz. 1 Since J(x is of order exp[ 2πx] for large x, we can now derive asymptotic results that include all inverse powers of µ. For example, F (µ, y ln[4πµy(1 y]. This means that we can now make O(g 2 predictions for correlators of BMN operators to all orders in λ 1/µ 2. 18

19 A GAUGE THEORY PREDICTION As a specific example, let us consider the three-string vertex for the three string states 1 = 0 ã i mã j m, 2 = 0, 3 = 0 ã i nã j n, We find to all finite orders in 1/µ that H g 2 sin 2 (πny 2π 2 y(1 y ω 1m ω 3n. where i and j are distinct SO(4 indices. This gauge theory prediction agrees with the known result at leading order (one loop in the λ expansion. However, the two-loop result of Beisert et al. in hep-th/ appears to disagree with it by a factor of two. 19