Supersymmetric Wilson Loops Research Articles

Supersymmetric Wilson loops in [formula omitted] super-Chern–Simons-matter theory

We study supersymmetric Wilson loop operators in ABJM theory from both sides of the AdS 4 / CFT 3 correspondence. We first construct some supersymmetric Wilson loops. The perturbative computations are performed in the field theory side at the first two orders. A fundamental string solution ending on a circular loop is also studied.

Correlators of supersymmetric Wilson-loops, protected operators and matrix models in 𝒩 = 4 SYM

We study the correlators of a recently discovered family of BPS Wilson loops in = 4 supersymmetric U(N) Yang-Mills theory. When the contours lie on a two-sphere in the space-time, we propose a closed expression that is valid for all values of the coupling constant g and for any rank N, by exploiting the suspected relation with two-dimensional gauge theories. We check this formula perturbatively at order (g4) for two latitude Wilson loops and we show that, in the limit where one of the loops shrinks to a point, logarithmic corrections in the shrinking radius are absent at (g6). This last result strongly supports the validity of our general expression and suggests the existence of a peculiar protected local operator arising in the OPE of the Wilson loop. At strong coupling we compare our result to the string dual of the = 4 SYM correlator in the limit of large separation, presenting some preliminary evidence for the agreement.

Boundary condition for a D-brane from the Wilson loop, and gravitational interpretation of an eigenvalue in the matrix model in AdS/CFT correspondence

We study the supersymmetric Wilson loops in the four-dimensional N=4 super Yang-Mills theory in the context of AdS/CFT correspondence. In the gauge theory side, it is known that the expectation value of the Wilson loops of circular shape with winding number k, W{sub k}(C), is calculable by using a Gaussian matrix model. In the gravity side, the expectation value of the loop is conjectured to be given by the classical value of the action S{sub D3} for a probe D3-brane with k electric fluxes as =e{sup -S{sub D3}}. Given such correspondence, we pursue the interpretation of the matrix model eigenvalue density, or more precisely the resolvent, from the viewpoint of the probe D3-brane. We see that the position of an eigenvalue appears as the gauge field plus the scalar field integrated over the boundary of the probe D3-brane. In the course of our analysis, we also clarify the boundary condition on the D3-brane in terms of the Wilson loop.

BPS Wilson loops inN=4supersymmetric Yang-Mills theory: Examples on hyperbolic submanifolds of space-time

In this paper we present a family of supersymmetric Wilson loops of $\mathcal{N}=4$ supersymmetric Yang-Mills theory in Minkowski space. Our examples focus on curves restricted to hyperbolic submanifolds, ${\mathbb{H}}_{3}$ and ${\mathbb{H}}_{2}$, of space-time. Generically they preserve two supercharges, but in special cases more, including a case which has not been discussed before, of the hyperbolic line, conformal to the straight line and circle, which is $1/2$ BPS. We discuss some general properties of these Wilson loops and their string duals and study special examples in more detail. Generically the string duals propagate on a complexification of ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$ and in some specific examples the compact sphere is effectively replaced by a de Sitter space.

Wilson loops in superconformal Chern-Simons theory and fundamental strings in Anti-de Sitter supergravity dual

We study Wilson loop operators in three-dimensional, = 6 superconformal Chern-Simons theory dual to IIA superstring theory on AdS4 × 3. Novelty of Wilson loop operators in this theory is that, for a given contour, there are two linear combinations of Wilson loop transforming oppositely under time-reversal transformation. We show that one combination is holographically dual to IIA fundamental string, while orthogonal combination is set to zero. We gather supporting evidences from detailed comparative study of generalized time-reversal transformations in both D2-brane worldvolume and ABJM theories. We then classify supersymmetric Wilson loops and find at most 1/6 supersymmetry. We next study Wilson loop expectation value in planar perturbation theory. For circular Wilson loop, we find features remarkably parallel to circular Wilson loop in = 4 super Yang-Mills theory in four dimensions. First, all odd loop diagrams vanish identically and even loops contribute nontrivial contributions. Second, quantum corrected gauge and scalar propagators take the same form as those of = 4 super Yang-Mills theory. Combining these results, we propose that expectation value of circular Wilson loop is given by Wilson loop expectation value in pure Chern-Simons theory times zero-dimensional Gaussian matrix model whose variance is specified by an interpolating function of `t Hooft coupling. We suggest the function interpolates smoothly between weak and strong coupling regime, offering new test ground of the AdS/CFT correspondence.

Spectral curves, emergent geometry, and bubbling solutions for Wilson loops

We study the supersymmetric circular Wilson loops of N=4 super Yang-Mills in large representations of the gauge group. In particular, we obtain the spectral curves of the matrix model which captures the expectation value of the loops. These spectral curves are then proven to be precisely the hyperelliptic surfaces that characterize the bubbling solutions dual to the Wilson loops, thus yielding an example of a geometry emerging from an eigenvalue distribution. We finally discuss the Wilson loop expectation value from the matrix model and from supergravity.

Matching gluon scattering amplitudes and Wilson loops in off-shell regularization

We construct a regularization for light-like polygonal Wilson loops in N = 4 SYM, which matches them to the off-shell MHV gluon scattering amplitudes. Explicit calculations are performed for the 1-loop four gluon case. The off light cone extrapolation has to be based on the local supersymmetric Wilson loop. The observed matching concerns Feynman gauge. Furthermore, the leading infrared divergent term is shown to be gauge parameter independent on 1-loop level.

Wilson loop correlators at strong coupling: from matrices to bubbling geometries

We compute at strong coupling the large N correlation functions of supersymmetric Wilson loops in large representations of the gauge group with local operators of = 4 super Yang-Mills. The gauge theory computation of these correlators is performed using matrix model techniques. We show that the strong coupling correlator of the Wilson loop with the stress tensor computed using the matrix model exactly matches the semiclassical computation of the correlator of the 't Hooft loop with the stress tensor, providing a non-trivial quantitative test of electric-magnetic duality of = 4 super Yang-Mills. We then perform these calculations using the dual bulk gravitational picture, where the Wilson loop is described by a ``bubbling'' geometry. By applying holographic methods to these backgrounds we calculate the Wilson loop correlation functions, finding perfect agreement with our gauge theory results.

Supersymmetric Wilson loops at two loops

We study the quantum properties of certain BPS Wilson loops in ${\cal N}=4$ supersymmetric Yang-Mills theory. They belong to a general family, introduced recently, in which the addition of particular scalar couplings endows generic loops on $S^3$ with a fraction of supersymmetry. When restricted to $S^2$, their quantum average has been further conjectured to be exactly computed by the matrix model governing the zero-instanton sector of YM$_2$ on the sphere. We perform a complete two-loop analysis on a class of cusped Wilson loops lying on a two-dimensional sphere, finding perfect agreement with the conjecture. The perturbative computation reproduces the matrix-model expectation through a highly non-trivial interplay between ladder diagrams and self-energies/vertex contributions, suggesting the existence of a localization procedure.

BPS Wilson loops onS2at higher loops

We consider supersymmetric Wilson loops of the variety constructed by Drukker, Giombi, Ricci, and Trancanelli, whose spatial contours lie on a two-sphere. Working to second order in the 't Hooft coupling in planar N=4 Supersymmetric Yang-Mills Theory (SYM), we compute the vacuum expectation value of a wavy-latitude and of a loop composed of two longitudes. We evaluate the resulting integrals numerically and find that the results are consistent with the zero-instanton sector calculation of Wilson loops in 2-d Yang-Mills on S^2 performed by Bassetto and Griguolo. We also consider the connected correlator of two distinct latitudes to third order in the 't Hooft coupling in planar N=4 SYM. We compare the result in the limit where the latitudes become coincident to a perturbative calculation in 2-d Yang-Mills on S^2 using a light-cone Wu-Mandelstam-Leibbrandt prescription. We are not able to calculate the SYM result at the required order in the separation between the latitudes necessary for a match with 2-d Yang-Mills; the result, however, does not preclude such a match.

Supersymmetric Wilson loops onS3

This paper studies in great detail a family of supersymmetric Wilson loop operators in N=4 supersymmetric Yang-Mills theory we have recently found. For a generic curve on an S^3 in space-time the loops preserve two supercharges but we will also study special cases which preserve 4, 8 and 16 supercharges. For certain loops we find the string theory dual explicitly and for the general case we show that string solutions satisfy a first order differential equation. This equation expresses the fact that the strings are pseudo-holomorphic with respect to a novel almost complex structure we construct on AdS_4 x S^2. We then discuss loops restricted to S^2 and provide evidence that they can be calculated in terms of similar observables in purely bosonic YM in two dimensions on the sphere.

Wilson loops: From 4D supersymmetric Yang-Mills theory to 2D Yang-Mills theory

In this paper we study supersymmetric Wilson loops restricted to an ${S}^{2}$ submanifold of four-dimensional space in $\mathcal{N}=4$ super Yang-Mills theory. We provide evidence from both perturbation theory and the AdS dual that those loops are equal to the analogous observables in two-dimensional Yang-Mills on ${S}^{2}$ (excluding nonperturbative contributions). This relates a subsector of $\mathcal{N}=4$ SYM to a low-dimensional soluble model and also suggests that this subsector of $\mathcal{N}=4$ SYM is invariant under area preserving diffeomorphisms.

A prediction for bubbling geometries

We study the supersymmetric circular Wilson loops in N=4 Yang-Mills theory. Their vacuum expectation values are computed in the parameter region that admits smooth bubbling geometry duals. The results are a prediction for the supergravity action evaluated on the bubbling geometries for Wilson loops.

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.

Calibrated surfaces and supersymmetric Wilson loops

We study the dual gravity description of supersymmetric Wilson loops whose expectation value is unity. They are described by calibrated surfaces that end on the boundary of anti de-Sitter space and are pseudo-holomorphic with respect to an almost complex structure on an eight-dimensional slice of AdS_5 x S^5. The regularized area of these surfaces vanishes, in agreement with field theory non-renormalization theorems for the corresponding operators.

Small deformations of supersymmetric Wilson loops and open spin-chains

We study insertions of composite operators into Wilson loops in = 4 supersymmetric Yang-Mills theory in four dimensions. The loops follow a circular or straight path and the composite insertions transform in the adjoint representation of the gauge group. This provides a gauge invariant way to define the correlator of non-singlet operators. Since the basic loop preserves an SL(2,) subgroup of the conformal group, we can assign a conformal dimension to those insertions and calculate the corrections to the classical dimension in perturbation theory. The calculation turns out to be very similar to that of single-trace local operators and may also be expressed in terms of a spin-chain. In this case the spin-chain is open and at one-loop order has Neumann boundary conditions on the type of scalar insertions that we consider. This system is integrable and we write the Bethe ansatz describing it. We compare the spectrum in the limit of large angular momentum both in the dilute gas approximation and the thermodynamic limit to the relevant string solution in the BMN limit and in the full AdS5 × S5 metric and find agreement.

Wilson loops and vertex operators in a matrix model

We systematically construct wave functions and vertex operators in the type IIB (IKKT) matrix model by expanding a supersymmetric Wilson loop operator. They form a massless multiplet of the $\mathcal{N}=2$ type IIB supergravity and automatically satisfy conservation laws.

Supersymmetric Wilson loops

I construct 1/16,1/8 and 1/4 BPS Wilson loops in N=4 supersymmetric Yang–Mills theory and argue that expectation values of 1/4 BPS loops do not receive quantum corrections. At strong coupling, non-renormalization of supersymmetric Wilson loops implies subtle cancellations in the partition function of the AdS string with special boundary conditions. The cancellations are shown to occur in the semiclassical approximation.

Supersymmetric Wilson lines and loops, and super non-Abelian Stokes theorem

We generalize the standard product integral formalism to incorporate Grassmann valued matrices and show that the resulting supersymmetric product integrals provide a natural framework for describing supersymmetric Wilson lines and Wilson loops. We use this formalism to establish the supersymmetric version of the non-Abelian Stokes' theorem.

Supersymmetric Wilson loops in a type-IIB matrix model

We show that the supersymmetric Wilson loops in a type-IIB matrix model give a transition operator from reduced supersymmetric Yang-Mills theory to supersymmetric space-time theory. In comparison with Green-Schwarz superstring we identify the supersymmetric Wilson loops with the asymptotic states of a type-IIB superstring. It is pointed out that the supersymmetry transformation law of the Wilson loops is the inverse of that for the vertex operators of massless modes in the $U(N)$ open superstring with a Dirichlet boundary condition.