Minimal Surfaces In AdS Research Articles

Amplitudes at Strong Coupling as Hyper-Kähler Scalars.

Alday and Maldacena conjectured an equivalence between string amplitudes in AdS_{5}×S^{5} and null polygonal Wilson loops in planar N=4 super-Yang-Mills (SYM) theory. At strong coupling this identifies SYM amplitudes with areas of minimal surfaces in anti-de Sitter space. For minimal surfaces in AdS_{3}, we find that the nontrivial part of these amplitudes, the remainder function, satisfies an integrable system of nonlinear differential equations, and we give its Lax form. The result follows from a new perspective on "Y systems," which defines a new psuedo-hyper-Kähler structure directly on the space of kinematic data, via a natural twistor space defined by the Y-system equations. The remainder function is the (pseudo-)Kähler scalar for this geometry. This connection to pseudo-hyper-Kähler geometry and its twistor theory provides a new ingredient for extending recent conjectures for nonperturbative amplitudes using structures arising at strong coupling.

TBA equations and resurgent Quantum Mechanics

We derive a system of TBA equations governing the exact WKB periods in one-dimensional Quantum Mechanics with arbitrary polynomial potentials. These equations provide a generalization of the ODE/IM correspondence, and they can be regarded as the solution of a Riemann-Hilbert problem in resurgent Quantum Mechanics formulated by Voros. Our derivation builds upon the solution of similar Riemann-Hilbert problems in the study of BPS spectra in mathcal{N} = 2 gauge theories and of minimal surfaces in AdS. We also show that our TBA equations, combined with exact quantization conditions, provide a powerful method to solve spectral problems in Quantum Mechanics. We illustrate our general analysis with a detailed study of PT-symmetric cubic oscillators and quartic oscillators.

On a new type of divergence for spiky Wilson loops and related entanglement entropies

We study the divergences of Wilson loops for a contour with a cusp of zero opening angle, combined with a nonzero discontinuity of its curvature. The analysis is performed in lowest order, both for weak and strong coupling. Such a spike contributes a leading divergent term proportional to the inverse of the square root of the cutoff times the jump of the curvature. As nextleading term appears a logarithmic one in the supersymmetric case, but it is absent in QCD. The strong coupling result, obtained from minimal surfaces in AdS via holography, can be used also for applications to entanglement entropy in (2+1)-dimensional CFT's.

Wilson loops at strong coupling for curved contours with cusps

We construct the minimal surface in AdS, relevant for the strong coupling behaviour of local supersymmetric Wilson loops in SYM for a closed contour formed out of segments of two intersecting circles. Its regularised area is calculated including all divergent parts and the finite renormalised term. Furthermore we prove that for generic planar curved contours with cusps the cusp anomalous dimensions are functions of the respective cusp angles alone. They do not depend on other local data of the cusps.

Spinning strings and minimal surfaces in AdS 3 with mixed 3-form fluxes

Motivated by the recent proposal for the S-matrix in $AdS_3\times S^3$ with mixed three form fluxes, we study classical folded string spinning in $AdS_3$ with both Ramond and Neveu-Schwarz three form fluxes. We solve the equations of motion of these strings and obtain their dispersion relation to the leading order in the Neveu-Schwarz flux $b$. We show that dispersion relation for the spinning strings with large spin ${\cal S}$ acquires a term given by $-\frac{\sqrt{\lambda}}{2\pi} b^2\log^2 {\cal S}$ in addition to the usual $\frac{\sqrt\lambda}{\pi} \log {\cal S}$ term where $\sqrt{\lambda}$ is proportional to the square of the radius of $AdS_3$. Using SO(2,2) transformations and re-parmetrizations we show that these spinning strings can be related to light like Wilson loops in $AdS_3$ with Neveu-Schwarz flux $b$. We observe that the logarithmic divergence in the area of the light like Wilson loop is also deformed by precisely the same coefficient of the $ b^2 \log^2 {\cal S}$ term in the dispersion relation of the spinning string. This result indicates that the coefficient of $ b^2 \log^2 {\cal S}$ has a property similar to the coefficient of the $\log {\cal S}$ term, known as cusp-anomalous dimension, and can possibly be determined to all orders in the coupling $\lambda$ using the recent proposal for the S-matrix.

More about one-loop effective action of open superstring in AdS5 × S5

We reconsider the calculation of the one-loop effective action for an open Green-Schwarz superstring in the $AdS_5\times S^5$ background for a circular boundary loop. By an explicit computation of the ratio of relevant determinants, describing semi-classical fluctuations about the minimal surface in AdS and flat spaces, we show that it does not depend upon the AdS regularizing parameter $\epsilon$. The only dependence upon $\epsilon$ resides in the reparametrization path integral of the exponential of the classical boundary action. We analyze how the result depends on the choice of the boundary condition imposed on fluctuating fields and show that, despite the fact that the contribution of individual angular modes changes, the product over the modes remains unchanged.

G-functions and gluon scattering amplitudes at strong coupling

We study gluon scattering amplitudes/Wilson loops in N=4 super Yang-Mills theory at strong coupling by calculating the area of the minimal surfaces in AdS_3 based on the associated thermodynamic Bethe ansatz system. The remainder function of the amplitudes is computed by evaluating the free energy, the T- and Y-functions of the homogeneous sine-Gordon model. Using conformal field theory (CFT) perturbation, we examine the mass corrections to the free energy around the CFT point corresponding to the regular polygonal Wilson loop. Based on the equivalence between the T-functions and the g-functions, which measure the boundary entropy, we calculate corrections to the T- and Y-functions as well as express them at the CFT point by the modular S-matrix. We evaluate the remainder function around the CFT point for 8 and 10-point amplitudes explicitly and compare these analytic expressions with the 2-loop formulas. The two rescaled remainder functions show very similar power series structures.

Space-like minimal surfaces in AdS × S

We present a four parameter family of classical string solutions in AdS 3 ×S 3, which end along a light-like tetragon at the boundary of AdS 3 and carry angular momentum along two cycles on the sphere. The string surfaces are space-like and their projections on AdS 3 and on S 3 have constant mean curvature. The construction is based on the Pohlmeyer reduction of the related sigma model. After embedding in AdS 5 × S 5, we calculate the regularized area and analyze conserved charges. Comments on possible relations to scattering amplitudes are presented. We also sketch time-like versions of our solutions.

Series solution and minimal surfaces in AdS

According to the Alday-Maldacena program the strong coupling limit of Super Yang-Mills scattering amplitudes is given by minimal area surfaces in AdS spacetime with a boundary consisting of a momentum space polygon. The string equations in AdS systematically reduce to coupled Toda type equations whose Euclidean classical solutions are then of direct relevance. While in the simplest case of AdS_3 exact solutions were known from earlier studies of the sinh-Gordon equation, there exist at present no similar exact forms for the generalized Toda equations related to AdS_d with d>=4. In this paper we develop a series method for the solution to those equations and evaluate their contribution to the finite piece of the worldsheet area. For the known sinh-Gordon case the method is seen to give results in excellent agreement with the exact answer.

Some comments on spacelike minimal surfaces with null polygonal boundaries in AdS m

We discuss some geometrical issues related to spacelike minimal surfaces in AdS m with null polygonal boundaries at conformal infinity. In particular for AdS 4, two holomorphic input functions for the Pohlmeyer reduced system are identified. This system contains two coupled differential equations for two functions $ \alpha \left( {z,\bar z} \right) $ and $ \beta \left( {z,\bar z} \right) $ , related to curvature and torsion of the surface. Furthermore, we conjecture that, for a polynomial choice of the two holomorphic functions, the relative positions of their zeros encode the conformal invariant data of the boundary null 2n-gon.

Scattering Amplitudes, the AdS/CFT Correspondence, Minimal Surfaces, and Integrability

We focus on the computation of scattering amplitudes of planar maximally supersymmetric Yang‐Mill in four dimensions at strong coupling by means of the AdS/CFT correspondence and explain how the problem boils down to the computation of minimal surfaces in AdS in the first part of this paper. In the second part of this review we explain how integrability allows to give a solution to the problem in terms of a set of integral equations. The intention of the review is to give a pedagogical, rather than very detailed, exposition.

Deviation from Alday–Maldacena duality for wavy circle

Alday–Maldacena conjecture is stated in this paper that the area A Π of the minimal surface in AdS 5 space with a boundary Π, located in Euclidean space at infinity of AdS 5 , coincides with a double integral D Π along Π, the Abelian Wilson average in an auxiliary dual model. This comes from Alday and Maldacena's original proposal and the BDS conjecture on the extrapolation of the MHV amplitudes. The boundary Π is a polygon formed by momenta of n external light-like particles in N = 4 SYM theory, and in a certain n = ∞ 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 dual 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 [H. Itoyama, A. Mironov, A. Morozov, Anomaly in n = ∞ Alday–Maldacena duality for wavy circle, JHEP 0807 (2008) 024, 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 h ¯ 2 terms, where the first deviation from the Alday–Maldacena duality arises for the wavy circle.

Boundary ring: A way to construct approximate NG solutions with polygon boundary conditions I. [formula omitted]-symmetric configurations

We describe an algebro-geometric construction of polygon-bounded minimal surfaces in AdS 5 , based on the consideration of what we call the “boundary ring” of polynomials. The first non-trivial example of solutions to the Nambu–Goto (NG) equations 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. A better use, albeit being exotic to theory of PDE, of the boundary ring is suggested as well. 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.

ALDAY-MALDACENA DUALITY AND AdS PLATEAU PROBLEM

A short summary of approximate approach to the study of minimal surfaces in AdS, based on solving Nambu-Goto equations iteratively. Today, after partial denunciation of the BDS conjecture, this looks like the only constructive approach to understanding the ways of its possible modification and thus to saving the Alday-Maldacena duality. Numerous open technical problems are explicitly formulated throughout the text.

AdS/CFT correspondence as a consequence of scale invariance

We study an anisotropic scale transformation in the worldsheet description of D3-branes in type IIB theory, and show that the transformation is really a symmetry in a region near D3-branes. AdS/CFT correspondence follows from this symmetry. We will explicitly show that Wilson loops in N = 4 supersymmetric Yang–Mills theory and minimal surfaces in AdS 5 are related by the symmetry. The functional form of a supersymmetric Wilson loop operator is naturally derived from our worldsheet point of view.

Holographic Derivation of Entanglement Entropy from the anti–de Sitter Space/Conformal Field Theory Correspondence

A holographic derivation of the entanglement entropy in quantum (conformal) field theories is proposed from anti-de Sitter/conformal field theory (AdS/CFT) correspondence. We argue that the entanglement entropy in d + 1 dimensional conformal field theories can be obtained from the area of d dimensional minimal surfaces in AdS(d+2), analogous to the Bekenstein-Hawking formula for black hole entropy. We show that our proposal agrees perfectly with the entanglement entropy in 2D CFT when applied to AdS(3). We also compare the entropy computed in AdS(5)XS(5) with that of the free N=4 super Yang-Mills theory.

Wilson loops and minimal surfaces

The AdS/CFT correspondence suggests that the Wilson loop of the large N gauge theory with N=4 supersymmetry in 4 dimensions is described by a minimal surface in AdS_5 x S^5. We examine various aspects of this proposal, comparing gauge theory expectations with computations of minimal surfaces. There is a distinguished class of loops, which we call BPS loops, whose expectation values are free from ultra-violet divergence. We formulate the loop equation for such loops. To the extent that we have checked, the minimal surface in AdS_5 x S^5 gives a solution of the equation. We also discuss the zig-zag symmetry of the loop operator. In the N=4 gauge theory, we expect the zig-zag symmetry to hold when the loop does not couple the scalar fields in the supermultiplet. We will show how this is realized for the minimal surface.

Wilson Loops and Minimum Surfaces

The AdS/CFT correspondence suggests that the Wilson loop of the large N gauge theory with N=4 supersymmetry in 4 dimensions is described by a minimal surface in AdS_5 x S^5. We examine various aspects of this proposal, comparing gauge theory expectations with computations of minimal surfaces. There is a distinguished class of loops, which we call BPS loops, whose expectation values are free from ultra-violet divergence. We formulate the loop equation for such loops. To the extent that we have checked, the minimal surface in AdS_5 x S^5 gives a solution of the equation. We also discuss the zig-zag symmetry of the loop operator. In the N=4 gauge theory, we expect the zig-zag symmetry to hold when the loop does not couple the scalar fields in the supermultiplet. We will show how this is realized for the minimal surface.