Slight Violation of the Alday-Maldacena Duality for a Wavy Circle
with A. Mironov and A. Morozov
IMM2
see also IMMIM8I)

arXiv : 0803.1547
arXiv : 0712.0159
arXiv : 0712.2316

Introduction

・ Gauge / gravity correspondence (& matrix models) continue tobe the central themes of string theory
・ ① BDS’s (Bern, Dixon, Smirnov) conjectured exponentiationa la Sudarkov of the all order planar n- gluon amplitudes
for perturbative N=4 SYM,
which is now known to be slightly violated at n=6, L=2loop level.
BDKKRSVV 08031465
DHKS 08031466
1

② A new version of gauge-string duality by Alday-Maldacena :computation at strong coupling by the minimal surface ofan AdS5 string・ ① and ② ⇒ A fruitful assessment of the issue
and its reformulation are called for today.
Hints in the strong coupling limit would be supplied by IMM2
・ Note that, if it were true, would give a resolution
to the AdS-minimal surface problem.

2ContentsI) IntroductionII) The BDS conjecture, which now calls for a modification
III) Representing
as an abelian Wilson loop in momentumspaceIV) T-dualized AdS5 space and string amplitudes
V) Nambu-Goto equation and the linearized form
VI) The computation at a wavy circle3II) symbolicallyfactorizes & exponentiates
To be more precise,

Define, with the help of MHV & N=4 SUSY,

4BDS conjecture

known
BES ↓
planar cusp anomalous dimension
planar collinear anomalous dimension

5III)
・is expressed as a sum of the so-called ``two mass easy
box functions’’ F2me : (Bern, Dunbar, Dixon, Kosower)

masslessmassivewith・ dilogarithmic rep. of F2me6・is known to reproduce, after computation,
the rep. ofin momentum space

Drummond, Korchemsky, Sokatchev; Brandhuber, Heslop, Travaglini

twoidentical adjacent nonadjacent7

・log of abelian Wilson loop
free field・ situation ; T-dualities operating
rep. on D instantons

rep. on D3 branes
weak coupling

our cornerstrong coupling8IV)・ AdS / CFT duality ;assumed・ compute the same gluon amplitude at strong coupling using
tree level semiclassical string theory
・ The original AdS5 geometry
embedding into R2,4・ place D3 brane at(IR regulator)
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・ take T-duality indirections

・ T-dualized geometry, which is again AdS5

10

・ semiclassical string amplitudes

・ Equivalence of Neuman rep. and Dirichlet rep. (AdS5 σ-model)
N:

11

D:

The 2nd term, after T-duality relation, and `` the D-instantoncondition ‘’12V)・ work on the Euclidean worldsheet
・ choose・ The 1st ansatz ;・・・ ①recognize this as
・ The 2nd ansatz ;・・・ ②

IMM1, IM8・ ① and ② form AdS3 ansatz, which contains the AldayMaldacena rhombus solution.
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・ Eq. of motion

・ linear approximation to NG
eliminatethroughand ②, linearize w.r.t.
, wheresolution

14VI)・ how to deal with the issue
coupling side・ explicit examples containing

fruitfully in the strong
ly many parameters needed⇒ an infinitesimal deformation of the unit circle into anarbitrary curve on the plane・ circle solution (formalAdS3 ansatz

limit of lightlike n-gon){now15

・The only candidate to the solution which lie in these ansatze is, which in fact solves

・ formulation

IMM1unit circle→・ consider the conformal map
・ find the shape of the minimal surface
by solving the NG eq. forsubject to the b.c.
16action・ some simplification due to・ write

・ need to compute a=a(h) (at least) to the lowest order inh
・ regularization needed

Eq. of motion
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has appeared again in a-linearized problem

・ The solution which satisfies the boundary condition is・ remaining procedure・ substitute the solution into the regularized area
・ and evaluate it18

・ Evaluating the regularized area

Waysi) μ – reg. : replacein the denominate byii) c – reg. : cut the integral overatiii) dimensional reg. : we did not accomplish・・・i) ・ direct evaluation without partial integration・ manage to find the local boundary counterterm

19ii) ・ partial integration to isolate the singularities
・ hard to find local boundary counterterms
・ shift the boundary condition to a(h) to the regularizedboundary omitting the surface terms

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・ Double contour integral
・ For example, in `` r’/r regularization ‘’ : two radii are s.t.

・ The alternative ; λ regularization
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・ Results

diagonaloff-diagonalThe red denote discrepancies
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・ comments on our resultin front of the bracket

・

inside the bracket
5% discrepancy
・ part inwhich is linear in
is consistent withthe expression which is made of the Schwarzian derivative :unit circle

・ The planar nature of our wavy circle may have obscured
some of the symmetry properties that this problem possesses.
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