Wilson loops and Riemann theta functions II

In this paper we extend and simplify previous results regarding the computation of Euclidean Wilson loops in the context of the AdS/CFT correspondence, or, equivalently, the problem of finding minimal area surfaces in hyperbolic space (Euclidean AdS3). If the Wilson loop is given by a boundary curve X(s) we define, using the integrable properties of the system, a family of curves X(lambda,s) depending on a complex parameter lambda known as the spectral parameter. This family has remarkable properties. As a function of lambda, X(lambda,s) has cuts and therefore is appropriately defined on a hyperelliptic Riemann surface, namely it determines the spectral curve of the problem. Moreover, X(lambda,s) has an essential singularity at the origin lambda=0. The coefficients of the expansion of X(lambda,s) around lambda=0, when appropriately integrated along the curve give the area of the corresponding minimal area surface. Furthermore we show that the same construction allows the computation of certain surfaces with one or more boundaries corresponding to Wilson loop correlators. We extend the area formula for that case and give some concrete examples. As the main example we consider a surface ending on two concentric circles and show how the boundary circles can be deformed by introducing extra cuts in the spectral curve.Comment: LaTeX, 45 pages, 10 figures. v2: typos corrected, references adde

Notes on Euclidean Wilson loops and Riemann Theta functions

The AdS/CFT correspondence relates Wilson loops in N=4 SYM theory to minimal area surfaces in AdS5 space. In this paper we consider the case of Euclidean flat Wilson loops which are related to minimal area surfaces in Euclidean AdS3 space. Using known mathematical results for such minimal area surfaces we describe an infinite parameter family of analytic solutions for closed Wilson loops. The solutions are given in terms of Riemann theta functions and the validity of the equations of motion is proven based on the trisecant identity. The world-sheet has the topology of a disk and the renormalized area is written as a finite, one-dimensional contour integral over the world-sheet boundary. An example is discussed in detail with plots of the corresponding surfaces. Further, for each Wilson loops we explicitly construct a one parameter family of deformations that preserve the area. The parameter is the so called spectral parameter. Finally, for genus three we find a map between these Wilson loops and closed curves inside the Riemann surface.Comment: 35 pages, 7 figures, pdflatex. V2: References added. Typos corrected. Some points clarifie

Measurement of the separation dependence of resonant energy transfer between CdSe/ZnS core/shell nanocrystallite quantum dots

The separation dependence of the interaction between two resonant groups of CdSe/ZnS nanocrystallite quantum dots is studied at room temperature. A near-field scanning optical microscope is used to bring a group of mono-disperse ~6.5 nm diameter nanocrystallite quantum dots which are attached to the microscope probe, into close proximity of `~8.5 nm diameter group of nanocrystallite quantum dots which are deposited on a solid immersion lens. Information extracted from photoluminescence, photoluminescence excitation and absorption curves as well as numerical calculations of the energy levels, show that the third excited excitonic energy level of the large quantum dots nearly matches the ground excitonic energy level for the small quantum dots. Quenching of the small quantum dots photoluminescence signal has been observed as they approach the large quantum dots. On average, the separation between microscope probe and solid immersion lens changed in the 15-50 nm range. The transition probability between these two groups of quantum dots is calculated to be (2.60 x 10-47 m6)/R6, within the (0.70 x 10-47 m6)/R6 - (11.0 x 10-47 m6)/R6 experimentally obtained range of transition probabilities. The F\"orster radius, as a signature of energy transfer efficiency, is experimentally found to be in the 14-22 nm range.Comment: 8 pages-8 figures Accepted Physical Review B 201

Holographic calculations of Euclidean Wilson loop correlator in Euclidean anti-de Sitter space

The correlation functions of two or more Euclidean Wilson loops of various shapes in Euclidean anti-de Sitter space are computed by considering the minimal area surfaces connecting the loops. The surfaces are parametrized by Riemann theta functions associated with genus three hyperelliptic Riemann surfaces. In the case of two loops, the distance $L$ by which they are separated can be adjusted by continuously varying a specific branch point of the auxiliary Riemann surface. When $L$ is much larger than the characteristic size of the loops, then the loops are approximately regarded as local operators and their correlator as the correlator of two local operators. Similarly, when a loop is very small compared to the size of another loop, the small loop is considered as a local operator corresponding to a light supergravity mode.Comment: 30 pages, 10 figure

Wilson loops and riemann theta functions in the gauge/gravity duality

One important implication of the AdS/CFT conjecture is that the expectation value of a Wilson loop operator in a conformally invariant field theory may be computed in the dual string theory by calculating the regularized area of the minimal area surface that ends on the Wilson loop in the boundary of AdS space. As a consequence, Euclidean Wilson loops correspond to minimal area surfaces in Euclidean AdS space. Many examples of Euclidean Wilson loops have been computed including the parallel lines which give the quark-antiquark energy. We approach the study of Wilson loops from the point of view of finding Riemann theta function solution to the cosh-gordon equation. We compute an infinite set of equivalent classes of simple Wilson loops. Each equivalent class consists of Wilson loops that, though having different shapes and lengths, have the same regularized area of their dual minimal area surfaces. An analytic formula for the area of their dual surfaces is derived. Furthermore new examples of Wilson loops which consist of multiple curves are calculated. For instance we compute cases of concentric Wilson loops which may be viewed as perturbed concentric circular Wilson loops. The trace of their monodromy matrix which gives information about the conserved charges is determined to be a simple function of the spectral parameter