http://inspirehep.net/record/499969/files/arXiv:hep-th_9905111.pdf

Large N Field Theories, String Theory and Gravity

We prove conjecture due to Erickson-Semenoff-Zarembo and Drukker-Gross which relates supersymmetric circular Wilson loop operators in the ${\mathcal N=4}$ supersymmetric Yang-Mills theory with a Gaussian matrix model. We also compute the partition function and give a new matrix model formula for the expectation value of a supersymmetric circular Wilson loop operator for the pure ${\mathcal N=2}$ and the ${\mathcal N=2^*}$ supersymmetric Yang-Mills theory on a four-sphere. A four-dimensional ${\mathcal N=2}$ superconformal gauge theory is treated similarly.

Localization of Gauge Theory on a Four-Sphere and Supersymmetric Wilson Loops

http://cds.cern.ch/record/546003/files/0204051.pdf

A semi-classical limit of the gauge/string correspondence

We use localization techniques to compute the expectation values of supersymmetric Wilson loops in Chern-Simons theories with matter. We find the path-integral reduces to a non-Gaussian matrix model. The Wilson loops we consider preserve a single complex supersymmetry, and exist in any N = 2 theory, though the localization requires superconformal symmetry. We present explicit results for the cases of pure Chern-Simons theory with gauge group U(N), showing agreement with the known results, and ABJM, showing agreement with perturbative calculations. Our method applies to other theories, such as Gaiotto-Witten theories, BLG, and their variants.

/pdf/exact-results-for-wilson-loops-in-superconformal-chern-1rdc9v6ens.pdf

Exact results for Wilson loops in superconformal Chern-Simons theories with matter

We discuss possible phase factors for the S-matrix of planar gauge theory, leading to modifications at four-loop order as compared to an earlier proposal. While these result in a four-loop breakdown of perturbative BMN scaling, Kotikov?Lipatov transcendentality in the universal scaling function for large-spin twist operators may be preserved. One particularly natural choice, unique up to one constant, modifies the overall contribution of all terms containing odd-zeta functions in the earlier proposed scaling function based on a trivial phase. Excitingly, we present evidence that this choice is non-perturbatively related to a recently conjectured crossing-symmetric phase factor for perturbative string theory on AdS5 ? S5 once the constant is fixed to a particular value. Our proposal, if true, might therefore resolve the long-standing AdS/CFT discrepancies between gauge and string theory.

/pdf/transcendentality-and-crossing-2wc82m8065.pdf

Transcendentality and crossing

The AdS-CFT correspondence suggests that the Wilson loop of the large N gauge theory with $\mathcal{N}=4$ supersymmetry in four dimensions is described by a minimal surface in ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{5}.$ 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 ultraviolet divergence. We formulate the loop equation for such loops. To the extent that we have checked, the minimal surface in ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{5}$ gives a solution of the equation. We also discuss the zigzag symmetry of the loop operator. In the $\mathcal{N}=4$ gauge theory, we expect the zigzag 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.

/pdf/wilson-loops-and-minimal-surfaces-50hfhx6gxs.pdf

Wilson loops and minimal surfaces

We propose that the expectation value of a circular BPS-Wilson loop in N=4 SUSYM can be calculated exactly, to all orders in a 1/N expansion and to all orders in g^2 N. Using the AdS/CFT duality, this result yields a prediction of the value of the string amplitude with a circular boundary to all orders in alpha' and to all orders in g_s. We then compare this result with string theory. We find that the gauge theory calculation, for large g^2 N and to all orders in the 1/N^2 expansion does agree with the leading string theory calculation, to all orders in g_s and to lowest order in alpha'. We also find a relation between the expectation value of any closed smooth Wilson loop and the loop related to it by an inversion that takes a point along the loop to infinity, and compare this result, again successfully, with string theory.

An Exact Prediction of N=4 SUSYM Theory for String Theory

The partition function of $${\mathcal{N}=6}$$
 supersymmetric Chern–Simons-matter theory (known as ABJM theory) on $${\mathbb{S}^3}$$
 , as well as certain Wilson loop observables, are captured by a zero dimensional super-matrix model. This super–matrix model is closely related to a matrix model describing topological Chern–Simons theory on a lens space. We explore further these recent observations and extract more exact results in ABJM theory from the matrix model. In particular we calculate the planar free energy, which matches at strong coupling the classical IIA supergravity action on $${{\rm AdS}_4\times\mathbb{C}\mathbb{P}^3}$$
 and gives the correct N
 3/2 scaling for the number of degrees of freedom of the M2 brane theory. Furthermore we find contributions coming from world-sheet instanton corrections in $${\mathbb{C}\mathbb{P}^3}$$
 . We also calculate non-planar corrections, both to the free energy and to the Wilson loop expectation values. This matrix model appears also in the study of topological strings on a toric Calabi–Yau manifold, and an intriguing connection arises between the space of couplings of the planar ABJM theory and the moduli space of this Calabi–Yau. In particular it suggests that, in addition to the usual perturbative and strong coupling (AdS) expansions, a third natural expansion locus is the line where one of the two ’t Hooft couplings vanishes and the other is finite. This is the conifold locus of the Calabi–Yau, and leads to an expansion around topological Chern–Simons theory. We present some explicit results for the partition function and Wilson loop observables around this locus.

/pdf/from-weak-to-strong-coupling-in-abjm-theory-1kf6pbv8t6.pdf

From weak to strong coupling in ABJM theory

We propose that the expectation value of a circular BPS-Wilson loop in [script N] = 4 supersymmetric Yang–Mills can be calculated exactly, to all orders in a 1/N expansion and to all orders in g2N. Using the AdS/CFT duality, this result yields a prediction of the value of the string amplitude with a circular boundary to all orders in alpha[prime] and to all orders in gs. We then compare this result with string theory. We find that the gauge theory calculation, for large g2N and to all orders in the 1/N2 expansion, does agree with the leading string theory calculation, to all orders in gs and to lowest order in alpha[prime]. We also find a relation between the expectation value of any closed smooth Wilson loop and the loop related to it by an inversion that takes a point along the loop to infinity, and compare this result, again successfully, with string theory.

http://cds.cern.ch/record/473539/files/0010274.pdf

An exact prediction of N=4 supersymmetric Yang–Mills theory for string theory

The standard prescription for calculating a Wilson loop in the AdS/CFT correspondence is by a string world-sheet ending along the loop at the boundary of AdS For a multiply wrapped Wilson loop this leads to many coincident strings, which may interact among themselves In such cases a better description of the system is in terms of a D3-brane carrying electric flux We find such solutions for the single straight line and the circular loop The action agrees with the string calculation at small coupling and in addition captures all the higher genus corrections at leading order in α' The resulting expression is in remarkable agreement with that found from a zero dimensional gaussian matrix model

Nadav Drukker

Papers

Wilson loops and minimal surfaces

An Exact Prediction of N=4 SUSYM Theory for String Theory

From weak to strong coupling in ABJM theory

An exact prediction of N=4 supersymmetric Yang–Mills theory for string theory

All-genus calculation of Wilson loops using D-branes