Spectrum of Holographic Wilson Loops

Transcription

1 Spectrum of Holographic Wilson Loops Leopoldo Pando Zayas University of Michigan Continuous Advances in QCD 2011 University of Minnesota Based on arxiv: Alberto Faraggi and LPZ Work in Progress, A. Farragi, LPZ and A. Tirziu Continuous Advances in QCD /

2 Introduction Introduction Wilson loops (WL) are an important class of gauge invariant non-local operators in QCD (BFKL). Wilson loops and amplitudes in N = 4 SYM. In N = 4 SYM the half BPS circular WL is captured by a Gaussian Matrix Model (Drukker, Gross; Erickson, Semenoff, Zarembo; Pestun). Summing ladders and other diagrammatic intuition. ( W R (C) = Tr R P exp i ds ( A µ ẋ µ + φ I ẏ I)) C Continuous Advances in QCD /

3 Introduction Localization: Pestun (Topological String)/(String Theory )= (N = 4 SYM)/QCD Want exp(s): exp(s) exp (S + tqv ) d exp (S + tqv ) = {Q, V } exp (S + tqv ) dt {Q, V exp (S + tqv )} = 0 (1) Independence of t, take t Localizes on a Gaussian action. Classical plus one-loop. a is a constant matrix coming from φ I. r is the radius of S 4. S = 4π2 gy 2 r 2 a 2 (2) M Continuous Advances in QCD /

4 Introduction Significant progress in the holographic description of WL (Maldacena; Rey, Yee; Drukker, Gross, Semenoff Ooguri; Drukker, Fiol, Gomis, Passerini; Yamaguchi; Trancanelli; Kruczenski): The branes sum the interaction among strings. Configuration F1 D3 D5 Representation of SU(N) Fundamental Symmetric Antisymmetric We did a semi-classical analysis for the D3-brane. Exact results on the gauge theory side; precision tests of AdS/CFT. Unifying picture in terms of SUSY for all cases. Field theory is ahead (localization): Precision test Continuous Advances in QCD /

5 Holographic Description of Wilson Loops Holographic Description of Wilson Loops For the D3-brane (Drukker, Fiol) ds 2 = L 2 ( cosh 2 (u k )ds 2 AdS 2 + sinh 2 (u k )dω 2 2), F = il 2 cosh(u k )e 0 e 1. AdS 2 S 2 worldvolume with electric flux. sinh(u k ) = k λ 4N κ. Continuous Advances in QCD /

6 Holographic Description of Wilson Loops The bosonic part of the D3 brane action in the is given by S B = T D3 d 4 σ det (g + 2πα F ) T D3 P [C 4 ]. The D-brane provides a 1/N expansion! T D3 = N 2π 2 L 4 [ S circle = 2N κ ] 1 + κ 2 + sinh 1 κ Continuous Advances in QCD /

7 Holographic Description of Wilson Loops How to organize the answer? Has the same bosonic symmetries as the field theory operator SL(2, R) SO(3) SO(5). (P µ, J µν, D, K µ ) of SO(4, 2) The subgroup preserved by the Wilson loop is generated by (P 0, J ij, D, K 0 ). Jij span the SU(2) SO(3) (P0, K 0, D) form SL(2, R) n I breaks the SO(6) R-symmetry down to SO(5) USp(4) Preserves half of the AdS 5 S 5 supersymmetries OSp(4 4) SU(2, 2 4). OSp (4 4) has SL(2, R) SO(3) SO(5) as its even subgroup and 16 fermionic generators Continuous Advances in QCD /

8 Holographic Description of Wilson Loops We studied bosonic and fermionic fluctuations explicitly. Bosonic action: S (2) φ = T D3 coth(u k ) d 4 σ ( ) ĝĝ αβ φˆ4 α β φˆ4 2 + αφî β φî, S a (2) = T D3 coth(u k ) d 4 σ ĝĝ αβ ĝ γδ f αγ f βδ. 4 Six massless scalars and a massless gauge field in AdS 2 S 2. Fermionic action: S (2) Θ = T D3 coth(u k ) d 4 σ ĝ Θ ˆ/ Θ. 2 Four massless Weyl fermions in AdS 2 S 2. Deformed AdS 2 S 2 geometry: dŝ 2 = L 2 sinh 2 (u k ) ( ds 2 AdS 2 + dω 2 ) 2 Continuous Advances in QCD /

9 Holographic Description of Wilson Loops We compactify on S 2 : 2d field 4d origin SL(2, R) SO(3) SO(5) embedding in AdS 5 l + 1 l 1 l 0 Bosons embedding in S 5 l + 1 l 5 l 0 gauge field along AdS 2 l + 1 l 1 l 1 gauge field along S 2 l + 1 l 1 l 1 Fermions IIB spinor l + 1 l 4 l 1 2 Lowest lying modes: 6 massless and 6 massive (two triplets) This is quite different from the fundamental string: 8 lowest lying modes (five massless (Sphere) and three massive) Continuous Advances in QCD /

10 Holographic Description of Wilson Loops The KK modes organize into multiplets of OSp(4 4). j = (j + 1, j, 5) (j + 3 2, j + 1 2, 4) (j + 2, j + 1, 1) j 1, (j + 1 2, j 1 2, 4) (j + 1, j, 1) (j, j 1, 1), 0 = (1, 0, 5) ( 3 2, 1 2, 4) (2, 1, 1), j = 0. SL(2, R) SO(3) SO(5) content of supermultiplets. Continuous Advances in QCD /

11 D5 Branes Holographic dual of k-antisymmetric WL: D5 on AdS 5 S 5 D5 brane with worldvolume AdS 2 S 4 in AdS 5 S 5 and flux in its worldvolume. D5 brane wraps the S 4 inside the S 5. ds 2 = L 2 ( cosh 2 (u)ds 2 H + sinh 2 (u)dω du 2 + dθ 2 + sin 2 (θ)dω 2 4), (3) Classical solution: sits at θ k Where k is the fundamental string charge on the brane θ = θ k, k = 2N ( 1 π 2 θ k 1 ) 4 sin 2θ k, 2πα F = il 2 cos(θ k )e 0 e 1, (4) Continuous Advances in QCD /

12 D5 Branes Some fluctuations [Camino, Paredes and Ramallo]: θ = θ k + ξ, F 0r = cos(θ k ) + f. (5) m 2 l = { (l + 3)(l + 4) for l = 0, 1,... l(l 1) for l = 1, 2,... (6) Here l is related to the eigenvalue l(l + 2) of the spherical harmonic on S 4. Continuous Advances in QCD /

13 D5 Branes The full spectrum h triplet = f + 1 m 2 triplet = f(f + 1), h singlet1 = f + 2 m 2 singlet1 = (f + 1)(f + 2), h singlet2 = f m 2 singlet2 = (f 1)f h singlet3 = f + 1 m 2 singlet3 = f(f + 1). (7) Continuous Advances in QCD /

14 One-loop Corrections One-loop Corrections Z (j) B = [ det Taking into account the SO(3) SO(5) quantum numbers, the partition function for one multiplet j 1 is ( + [ ( det + Z (j) F = [ det ( i / + j + 1 R γ )] j(j + 1) 5(2j+1)/2 [ ( R 2 det + j(j + 1) R 2 )] (2j+1)/2 [ ( det + )] 4(j+1) [ ( det i / + j )] 4j R γ. (j + 1)(j + 2) R 2 (j 1)j R 2 )] (2j+3)/2 )] (2j 1)/2, (8) (9) Continuous Advances in QCD /

15 One-loop Corrections The Fundamental String In the case j = 0 we have [Gross, Tseytlin] ( Z (0) B [det = [det ( )] 5/2 + 2 )] 3/2 R 2, ( Z (0) F [det = i / + 1 )] 4 R γ. (12) Continuous Advances in QCD /

16 One-loop Corrections The method (Straight String) ( det + ) j(j + 1) R 2 = ) ) det ( σ ( p σ 2 + j(j + 1). p Z (13) Solving the associated equation [ σ 2 ( p 2 + σ 2 ) ± pσ + j 2 1 ] θ ±p j (σ) = 0 (14) 4 In the interval [ɛ, R] with initial conditions θ ±p j (ɛ) = 0 σ θ ±p j (ɛ) = 1 (15) Continuous Advances in QCD /

17 One-loop Corrections The (preliminary) Result ln Z (j) = 1 2 ( [ 5 (2j + 1) ln Kj+1/2 (pɛ) ] (2j + 3) ln [ K j+3/2 (pɛ) ] p (2j + 1) ln [ K j+1/2 (pɛ) ] (2j 1) ln [ K j 1/2 (pɛ) ] [ K 2 j+3/2 (pɛ) Kj+1/ (j + 1) ln (pɛ) ] 2(j + 1) [ K 2 j+1/2 (pɛ) Kj 1/2 2 +4j ln (pɛ) ] ) + 4 (2j + 1) ln [pɛ] 2j (16) Continuous Advances in QCD /

18 One-loop Corrections Divergences ln Z (j) = T ɛ ( ( ) ( Λɛ (2j + 1) ln + g j, Λɛ )) T T (17) Where g(j, x) is finite as x. -T/ɛ is the Euler characteristic of AdS 2 with a cutoff ɛ The Euler characteristic is not Weyl invariant [Alvarez]. The D-brane is a string? Not really. Continuous Advances in QCD /

19 One-loop Corrections The Matrix Model F (X) = 1 Z N dx i 2 (x)f (x) exp ( 2N λ N x 2 i ) (18) i=1 i=1 where ( (x) = det x j 1 i ) (19) The Vandermonde determinant. P j is any polynomial of order j. F (X) = 1 N dx i det [P j 1 (x i )] 2 F Z i=1 ( ) ( λ 2N x exp N i=1 x 2 i ) (20) Continuous Advances in QCD /

20 One-loop Corrections The Fundamental WL: Exact Result T re M = 1 N L1 N 1( 4Nλ?) exp[2nλ?] (21) L 1 N 1 Associated Laguerre polynomial. You probably know this in the large N limit: lim N 1 N L1 N 1 (2/ λ)i 1 ( λ) (22) Continuous Advances in QCD /

21 One-loop Corrections k-wound string versus symmetric representation The k-wound string is computed by one integral: F (X) = m (k) (X) Ne kx 1 () For the k-symmetric representation: F (X) = m (k) (X) + m (k 1,1) (X) + Ne kx 1 + N(N 1)e (k 1)x 1+x 2 + (24) (25) Master integral: ( N ) F (X) = exp w i x i i=1 N w i = k (26) i=1 Continuous Advances in QCD /

22 One-loop Corrections Formal Answer Defining I ij (y) = dx e (x y)2 P i 1 (x)p j 1 (x) (27) ( N ) ( exp w i x i = 1 N Z exp i=1 ɛ i 1,...,i N j 1,...,j N i=1 y 2 i ) i 1 i N ɛ j 1 j N I i1 j 1 (y 1 ) I in j N (y N ) (28) (29) where y i = w i λ 8N (30) Continuous Advances in QCD /

23 Conclusions Comments We get a nice picture for F1 and D3 in terms of OSp(4 4) representations. The D5 seems to be consistent with this. Finding the 1-loop correction to the WL expectation value amounts to computing functional determinants. Similar calculation for the D5. Study other less symmetric WL. The cusp anomalous dimension in the fundamental from spinning string(tseytlin, Roiban, Tirziu). Anti-symmetric representation of the cusp anomalous dimension [classical solution is known] (Armoni). Continuous Advances in QCD 2011 /