4 research outputs found
Some properties of the Alday-Maldacena minimum
The Alday-Maldacena solution, relevant to the n=4 gluon amplitude in N=4 SYM
at strong coupling, was recently identified as a minimum of the regularized
action in the moduli space of solutions of the AdS_5 sigma-model equations of
motion. Analogous solutions of the Nambu-Goto equations for the n=4 case are
presented and shown to form (modulo the reparametrization group) an equally
large but different moduli space, with the Alday-Maldacena solution at the
intersection of the sigma-model and Nambu-Goto moduli spaces. We comment upon
the possible form of the regularized action for n=5. A function of moduli
parameters z_a is written, whose minimum reproduces the BDDK one-loop
five-gluon amplitude. This function may thus be considered as some kind of
Legendre transform of the BDDK formula and has its own value independently of
the Alday-Maldacena approach.Comment: 10 page
Boundary Ring: a way to construct approximate NG solutions with polygon boundary conditions: I. Z_n-symmetric configurations
We describe an algebro-geometric construction of polygon-bounded minimal
surfaces in ADS_5, based on consideration of what we call the "boundary ring"
of polynomials. The first non-trivial example of the Nambu-Goto (NG) solutions
for Z_6-symmetric hexagon is considered in some detail. Solutions are
represented as power series, of which only the first terms are evaluated. The
NG equations leave a number of free parameters (a free function). Boundary
conditions, which fix the free parameters, are imposed on truncated series. It
is still unclear if explicit analytic formulas can be found in this way, but
even approximate solutions, obtained by truncation of power series, can be
sufficient to investigate the Alday-Maldacena -- BDS/BHT version of the
string/gauge duality.Comment: 42 pages, 5 figure
Boundary Ring or a Way to Construct Approximate NG Solutions with Polygon Boundary Conditions. II. Polygons which admit an inscribed circle
We further develop the formalism of arXiv:0712.0159 for approximate solution
of Nambu-Goto (NG) equations with polygon conditions in AdS backgrounds, needed
in modern studies of the string/gauge duality. Inscribed circle condition is
preserved, which leaves only one unknown function y_0(y_1,y_2) to solve for,
what considerably simplifies our presentation. The problem is to find a
delicate balance -- if not exact match -- between two different structures: NG
equation -- a non-linear deformation of Laplace equation with solutions
non-linearly deviating from holomorphic functions, -- and the boundary ring,
associated with polygons made from null segments in Minkovski space. We provide
more details about the theory of these structures and suggest an extended class
of functions to be used at the next stage of Alday-Maldacena program:
evaluation of regularized NG actions.Comment: 45 page
Deviation from Alday-Maldacena Duality For Wavy Circle
Alday-Maldacena conjecture is that the area A_P of the minimal surface in
AdS_5 space with a boundary P, located in Euclidean space at infinity of AdS_5,
coincides with a double integral D_P along P, the Abelian Wilson average in an
auxiliary dual model. The boundary P is a polygon formed by momenta of n
external light-like particles in N=4 SYM theory, and in a certain n=infty limit
it can be substituted by an arbitrary smooth curve (wavy circle). The
Alday-Maldacena conjecture is known to be violated for n>5, when it fails to be
supported by the peculiar global conformal invariance, however, the structure
of deviations remains obscure. The case of wavy lines can appear more
convenient for analysis of these deviations due to the systematic method
developed in arXiv:0803.1547 for (perturbative) evaluation of minimal areas,
which is not yet available in the presence of angles at finite n. We correct a
mistake in that paper and explicitly evaluate the h^2\bar h^2 terms, where the
first deviation from the Alday-Maldacena duality arises for the wavy circle.Comment: 23 page