We study Wilson loops in N=6 superconformal Chern-Simons theory with gauge group $U(M) \times \bar{U(N)}$ that is dual to N M2-branes and (M-N) fractional M2-branes, equivalently, discrete 3-form holonomy at C4/Zk orbifold singularity. We give description of these Wilson loops in terms of macroscopic fundamental string and D6-branes in the dual AdS$_4\times$ CP$^3$ geometry with NS-NS B field turned on over CP$^1 \subset$ CP$^3$.
We study solutions for rigidly rotating strings on AdS$_4 \times$ CP$^3$ in the presence of $B_{\rm NS}$ holonomy turned on over $CP^1 \subset CP^3$. We construct general solutions for rotating strings with two and three angular momenta in $CP^3$ and discuss various limits corresponding to giant magnon and spike like solutions.
The classical Wilson loop is the gauge-invariant trace of the parallel transport around a closed path with respect to a connection on a vector bundle over a smooth manifold. We build a precise mathematical model of the super Wilson loop, an extension introduced by Mason-Skinner and Caron-Huot, by endowing the objects occurring with auxiliary Graßmann generators coming from S-points. A key feature of our model is a supergeometric parallel transport, which allows for a natural notion of holonomy on a supermanifold as a Lie group valued functor. Our main results for that theory comprise an Ambros…
We use twist deformation techniques to analyse the behaviour under area-preserving diffeomorphisms of quantum averages of Wilson loops in Yang-Mills theory on the noncommutative plane. We find that while the classical gauge theory is manifestly twist covariant, the holonomy operators break the quantum implementation of the twisted symmetry in the usual formal definition of the twisted quantum field theory. These results are deduced by analysing general criteria which guarantee twist invariance of noncommutative quantum field theories. From this a number of general results are also obtained, su…
We study the holographic dual string configuration of 1/2 BPS circular Wilson loops in N=4 super Yang-Mills theory by using the global coordinate of AdS. The dual string worldsheet is given by the Poincare disk AdS_2 sitting at a constant global time slice of AdS_5. We also analyze the correlator of two concentric circular Wilson loops from the global AdS perspective and study the phase transition associated with the instability of annulus worldsheet connecting the two Wilson loops.
The strong subadditivity is the most important inequality which entanglement entropy satisfies. Based on the AdS/CFT conjecture, entanglement entropy in CFT is equal to the area of the minimal surface in AdS space. It is known that a Wilson loop can also be holographically computed from the minimal surface in AdS space. In this paper, we argue that Wilson loops also satisfy a similar inequality, and find several evidences of it.
We study supersymmetric AdS$_D$ backgrounds of eleven-dimensional or type II supergravity preserving $\mathcal{N}$ supersymmetries using generalised geometry. We show that a large class correspond precisely to spaces admitting a generalised $G_{D,\mathcal{N}}$ structure with a weak integrability condition, which we call "weak generalised holonomy". This class contains all such backgrounds with odd $D$, but for even $D$ we must impose a technical assumption concerning the inner products of Killing spinors. It is known that all compact smooth solutions are included, though our analysis is purely…
The construction of gravitational Wilson lines in the Chern-Simons formulation of $AdS_3$ gravity in terms of composite operators in the dual boundary conformal field theory is reviewed. New evidence is presented that the Wilson line, dimensionally regularized and suitably renormalized, behaves as a bi-local operator of two conformal primaries whose dimension is predicted by SL(2,R) current algebra.
The Wilson loop in N=4 supersymmetric Yang-Mills theory admits a dual description as a macroscopic string configuration in the adS/CFT correspondence. We discuss the correction to the quark anti-quark potential arising from the fluctuations of the superstring.
We study the Wilson loops and defects in the three dimensional QFT from the D-branes in the AdS(4) x CP**3 geometry. We find out explicit D-brane configurations in the bulk which correspond to both straight and circular Wilson lines extended to the boundary of AdS(4). We analyze critically the role of boundary contributions to the D2-branes with various topology and to the fundamental string actions.
We provide a refined interpretation of a gravitational Wilson line in AdS$_3$ in terms of Ishibashi states in the dual CFT$_2$. Our strategy is to give a method to evaluate the Wilson line that accounts for all the information contained in the representation, and clarify the role of boundary conditions at the endpoints of the line operator. This gives a novel way to explore and reconstruct the local bulk dynamics which we discuss. We also compare our findings with other interpretations of Ishibashi states in AdS$_3$/CFT$_2$.
We compute the one-loop world-sheet correction to partition function of $AdS_5 \times S^5$ superstring that should be representing $k$-fundamental circular Wilson loop in planar limit. The 2d metric of the minimal surface ending on $k$-wound circle at the boundary is that of a cone of $AdS_2$ with deficit $2π(1-k)$. We compute determinants of 2d fluctuation operators by first constructing heat kernels of scalar and spinor Laplacians on the cone using the Sommerfeld formula. The final expression for the k-dependent part of the one-loop correction has simple integral representation but is differ…
We calculate the expectation value of the Wilson line in the pure AdS$_3$ Einstein gravity theory and also the entanglement entropy in the boundary theory. Our one-loop calculation of entanglement entropy shows a shift of the central charge 26. Finally, we show that the Wilson line provides the equivalent description to the boundary entanglement entropy. This equivalence leads to a concrete example of the building of "minimum surface=entanglement entropy".
The AdS/CFT correspondence predicts a phase transition in Wilson loop correlators in the strong coupling N=4, D=4 SYM theory which arises due to instability of the classical string stretched between the loops. We study this transition in detail by solving equations of motion for the string in the particular case of two circular Wilson loops. The transition is argued to be smoothened at finite `t Hooft coupling by fluctuations of the string world sheet and to be promoted to a sharp crossover. Some general comments about Wilson loop correlators in gauge theories are made.
We consider the expectation value of Wilson lines in two defect versions of N = 4 SYM, both with supersymmetry completely broken, where one is described in terms of an integrable boundary state, the other one not. For both cases, imposing a certain double scaling limit, we find agreement to two leading orders between the expectation values calculated from respectively the field theory and the string theory side of the AdS/dCFT correspondence.
Guided by the paper hep-th/0002106 by Polyakov and Rychkov, we compute the second variational derivative of a wavy plane Wilson surface observable, to find that a necessary condition for a proposed surface equation to be satisfied in the large $N$ limit is that we are in the critical dimension D=6.
For the asymptotic string solution in AdS_3 which is represented by the AdS_3 Poincare coordinates and yields the planar multi-gluon scattering amplitude at strong coupling in arXiv:0904.0663, we express it by the AdS_4 Poincare coordinates and demonstrate that the hexagonal and octagonal Wilson loops surrounding the string surfaces take closed contours consisting of null vectors in R^{1,2} owing to the relations of Stokes matrices. For the tetragonal Wilson loop we construct a string solution characterized by two parameters by solving the auxiliary linear problems and demanding a reality cond…
In hep-th/9803002, Maldacena argued that in the light of the AdS/CFT correspondence as formulated by Witten and Gubser, Klebanov and Polyakov as a relation between partition functions, the expectation value of the Wilson loop in $\cal N$=4 SU(N) SYM is given by the worldsheet partition function with the action formulated on an $AdS_5\times S^5$ background and the world sheet ending on the loop on the boundary of $AdS_5$. What we propose to do in this paper is to provide some alternative arguments as to why it should be so.
We find 1/6 BPS string configurations in AdS_4 x CP^3, which we identify as the duals of certain 1/6 BPS circular Wilson loops in N=6 super Chern-Simons-matter gauge theory. We use our results to verify -in the strong coupling limit- a proposal made in arXiv:1402.4128 for a relation between the expectation value of these Wilson loops and the Bremsstrahlung function from deforming 1/2 BPS Wilson lines with a cusp. We also derive an analogous relation between the expectation value of some particular 1/12 BPS Wilson loops and the Bremsstrahlung function from deforming 1/6 BPS Wilson lines with an…
We use the string sigma model action in AdS_5 \times S^5 to reconstruct the open string solution ending on the Wilson loop in S^3\times R parametrized by a geometric angle in S^3 and an angle in flavor space. Under the interchange of the world sheet space and time coordinates and the T-duality transformation with the radial inversion, the static open string configuration associated with the BPS Wilson loop with two equal angle parameters becomes a long open spinning string configuration which is produced by taking the equal limit of two frequencies for the folded spinning closed string with tw…
Wilson loops are calculated within the AdS/CFT correspondence by finding a classical solution to the string equations of motion in AdS_5 x S^5 and evaluating its action. An important fact is that this sigma-model used to evaluate the Wilson loops is integrable, a feature that has gained relevance through the study of spinning strings carrying large quantum numbers and spin-chains. We apply the same techniques used to solve the equations for spinning strings to find the minimal surfaces describing a wide class of Wilson loops. We focus on different cases with periodic boundary conditions on the…
We study AdS$_{7}$/CFT$_{6}$ correspondence between M-theory on AdS$_{7} \times S^{4}$ and the 6D $\mathcal{N} = (2,0)$ superconformal field theory. In particular we focus on Wilson surfaces. We use the conjecture that the (2,0) theory compactified on $S^{1}$ is equivalent to the 5D maximal super Yang-Mills (MSYM) and Wilson surfaces wrapping this $S^1$ correspond to Wilson loops in 5D MSYM. The Wilson loops in 5D MSYM obtained by the localization technique result in the Chern-Simons matrix model. We calculate the expectation values of Wilson surfaces in large rank symmetric representations an…
Based on an extension of the holographic principle to superspace, we provide a strong-coupling description of smooth super Wilson loops in terms of minimal surfaces of the $AdS_5 \times S^5$ superstring. We employ the classical integrability of the Green-Schwarz superstring on $AdS_5 \times S^5$ to derive the superconformal and Yangian $Y[\mathfrak{psu}(2,2|4)]$ Ward identities for the super Wilson loop, thus extending the strong coupling results obtained for the Maldacena-Wilson loop. In the course of the derivation, we determine the minimal surface solution up to third order in an expansion …
Recent progresses in the computation of quantum string corrections to holographic Wilson loops are extended to the case of strings in $AdS_{4}\times CP^{3}$. For this, the ratio of 1/2-BPS circular and 1/6-BPS latitude fermionic Wilson loops in ABJM is considered at strong coupling by studying the corresponding semiclassical string partition functions. Explicit evaluation of fluctuation determinants using phaseshifts and diffeomorphism invariant regulators leads to exact matching with the recent field theory proposal. Key to this computation is the choice of boundary conditions for massless fe…
We give a short introduction to and a partial review of the work on the calculation of Wilson loops and $Q\bar Q$-potentials via the conjectured AdS/CFT duality. Included is a discussion of the relative weight of the stringy correction to the target space background versus the correction by the quantum fluctuations of the string world sheet.
We study the two circular Wilson loop correlator in which one is of anti-symmetric representation, while the other is of fundamental representation in 4-dimensional ${\cal N}=4$ super Yang-Mills theory. This correlator has a good AdS dual, which is a system of a D5-brane and a fundamental string. We calculated the on-shell action of the string, and clarified the Gross-Ooguri transition in this correlator. Some limiting cases are also examined.
Wilson loops which are small deviations from straight, infinite lines, called wavy lines, are considered in the context of the AdS/CFT correspondence. A single wavy line and the connected correlation function of a straight and wavy line are considered. It is argued that, to leading order in ``waviness'', the functional form of the loop is universal and the coefficient, which is a function of the 't Hooft coupling, is found in weak coupling perturbation theory and the strong coupling limit using the AdS/CFT correspondence. Supersymmetric arguments are used to simplify the computation and to sho…
We directly compute Wilson loop/Wilson loop correlators on ${\mathbb R}\times $S$^3$ in AdS/CFT by constructing space-like minimal surfaces that connect two space-like circular contours on the boundary of global AdS that are separated by a space-like interval. We compare these minimal surfaces to the disconnected "double cap" solutions both to regulate the area, and show when the connected/disconnected solution is preferred. We find that for sufficiently large Wilson loops no transition occurs because the Wilson loops cannot be sufficiently separated on the sphere. This may be considered an ef…
The evaluation of BPS Wilson loops in N=6, D=3 Chern-Simons matter theory is reduced to ordinary matrix integrals via localization technique. It is easy to check that the vacuum expectation value of 1/2 BPS Wilson loops at leading order in planar limit agrees with the regularized classical string action, via AdS/CFT. Then the subleading terms in principle can be calculated by treating the string theory semi-classically. In this article we calculate the one-loop determinant for fluctuation modes of holographic Wilson loop in the dual geometry AdS4xCP3. The fermionic normal mode frequencies are …
We revisit the computation of the 1-loop string correction to the "latitude" minimal surface in $AdS_5 \times S^5$ representing 1/4 BPS Wilson loop in planar $\cal N$=4 SYM theory previously addressed in arXiv:1512.00841 and arXiv:1601.04708. We resolve the problem of matching with the subleading term in the strong coupling expansion of the exact gauge theory result (derived previously from localization) using a different method to compute determinants of 2d string fluctuation operators. We apply perturbation theory in a small parameter (angle of the latitude) corresponding to an expansion nea…
A previous calculation of the phase transition in the Wilson loop correlator in the zero temperature AdS/CFT correspondence is extended to the case where the loops are concentric circles of unequal radii. This phase transition occurs due to the instability of the classical string stretched between the loops. We compute the string action and its expansion in the distance h between the loops for small h. We also find that the connected minimal surface is subleading or does not even exist when h=0 and the radii are considerably different. This feature has no analogue in flat space.
We study correlation functions of local operator insertions on the 1/2-BPS Wilson line in ${\cal N}=4$ super Yang-Mills theory. These correlation functions are constrained by the 1d superconformal symmetry preserved by the 1/2-BPS Wilson line and define a defect CFT$_1$ living on the line. At strong coupling, a set of elementary operator insertions with protected scaling dimensions correspond to fluctuations of the dual fundamental string in AdS$_5 \times S^5$ ending on the line at the boundary and can be thought of as light fields propagating on the AdS$_2$ worldsheet. We use AdS/CFT techniqu…
Previous attempts to match the exact N=4 super Yang-Mills expression for the expectation value of the 1/2-BPS circular Wilson loop with the semiclassical AdS(5)xS(5) string theory prediction were not successful at the first subleading order. There was a missing prefactor ~ lambda^(-3/4) which could be attributed to the unknown normalization of the string path integral measure. Here we resolve this problem by computing the ratio of the string partition functions corresponding to the circular Wilson loop and the special 1/4-supersymmetric latitude Wilson loop. The fact that the latter has a triv…
We discuss AdS$_2\times S^3\times{\text{CY}}_2\times I_ρ$ solutions to massive Type IIA supergravity with 4 Poincaré supersymmetries. We propose explicit dual quiver quantum mechanics built out of D0 and D4 colour branes coupled to D4' and D8 flavour branes. We propose that these quivers describe the interactions of instantons and Wilson lines in 5d gauge theories with 8 Poincaré supersymmetries. Using the RR Maxwell fluxes of the solutions, conveniently put off-shell, we construct a functional from which the holographic central charge can be derived through a geometrical extremisation princip…
We study correlation functions of Wilson loops and local operators in a subsector of N=4 SYM which preserves two supercharges. Localization arguments allow to map the problem to a calculation in bosonic two-dimensional Yang-Mills theory. In turn, this can be reduced to computing correlators in certain Gaussian multi-matrix models. We focus on the correlation function of a Wilson loop and two local operators, and solve the corresponding three-matrix model exactly in the planar limit. We compare the strong coupling behavior to string theory in AdS_5xS^5, finding precise agreement. We pay particu…
The AdS/CFT correspondence relates Wilson loops in N=4 SYM theory to minimal area surfaces in AdS5xS5 space. If the Wilson loop is Euclidean and confined to a plane (t,x) then the dual surface is Euclidean and lives in Minkowski AdS3. In this paper we study such minimal area surfaces generalizing previous results obtained in the Euclidean case. Since the surfaces we consider have the topology of a disk, the holonomy of the flat current vanishes which is equivalent to the condition that a certain boundary Schroedinger equation has all its solutions anti-periodic. If the potential for that Schro…
The $p$-adic AdS/CFT is a holographic duality based on the $p$-adic number field $\mathbb{Q}_p$. For a $p$-adic CFT living on $\mathbb{Q}_p$ and with complex-valued fields, the bulk theory is defined on the Bruhat-Tits tree, which can be viewed as the bulk dual of $\mathbb{Q}_p$. We propose that bulk theory can be formulated as a lattice gauge theory of PGL$(2,\mathbb{Q}_p)$ on the Bruhat-Tits tree, and show that the Wilson line networks in this lattice gauge theory can reproduce all the correlation functions of the boundary $p$-adic CFT.
The algebraic curve (finite-gap) classification of rotating string solutions was very important in the development of integrability through comparison with analogous structures at weak coupling. The classification was based on the analysis of monodromy around the closed string cylinder. In this paper we show that certain classical Wilson loop minimal surfaces corresponding to the null cusp and qqbar potential with trivial monodromy can, nevertheless, be described by appropriate algebraic curves. We also show how a correlation function of a circular Wilson loop with a local operator fits into t…
Holonomy invariants in strict higher gauge theory have been studied in depth, aiming to applications to higher Chern-Simons theory. For a flat 2-connection, the holonomy of surface knots of arbitrary genus has been defined and its covariance properties under 1-gauge transformation and change of base data have been determined. Using quandle theory, a definition of trace over a crossed module has been given that yields surface knot invariants upon application to 2-holonomies.
We study at quantum level correlators of supersymmetric Wilson loops with contours lying on Hopf fibers of $S^3$. In $\mathcal{N}=4$ SYM theory the strong coupling analysis can be performed using the AdS/CFT correspondence and a connected classical string surface, linking two different fibers, is presented. More precisely, the string solution describes oppositely oriented fibers with the same scalar coupling and depends on an angular parameter, interpolating between a non-BPS configuration and a BPS one. The system can be thought as an alternative deformation of the ordinary antiparallel lines…
We study Wilson loops in string theory realizations of AdS/BCFT and wedge holography. The field theories are based on 3d $\mathcal N=4$ long quiver gauge theories engineered by D3, D5 and NS5 branes, and on BCFTs involving 4d $\mathcal N=4$ SYM coupled to such 3d theories. The holographic duals have geometry $AdS_4\times S^2\times S^2$ warped over a strip. We identify the holographic representation of antisymmetric Wilson loops associated with individual 3d gauge nodes in terms of probe D5-branes. The expectation values obtained holographically are matched to supersymmetric localization comput…
We describe a simple method of extending AdS_5 black string solutions of 5d gauged supergravity in a supersymmetric way by addition of Wilson lines along a circular direction in space. When this direction is chosen along the string, and due to the specific form of 5d supergravity that features Chern-Simons terms, the existence of magnetic charges automatically generates conserved electric charges in a 5d analogue of the Witten effect. Therefore we find a rather generic, model-independent way of adding electric charges to already existing solutions with no backreaction from the geometry or brea…
We initiate the calculation of quantum corrections to Wilson loops in a class of four-dimensional defect conformal field theories with vacuum expectation values based on N=4 super Yang-Mills theory. Concretely, we consider an infinite straight Wilson line, obtaining explicit results for the one-loop correction to its expectation value in the large-N limit. This allows us to extract the particle-interface potential of the theory. In a further double-scaling limit, we compare our results to those of a previous calculation in the dual string-theory set-up consisting of a D5-D3 probe-brane system …
The string theory calculation of the 1/2-BPS circular Wilson loop of N=4 SYM in the planar limit at next to leading order at strong coupling is revisited in the ratio of its semiclassical string partition function and the one dual to a latitude Wilson loop with trivial expectation value. After applying a conformal transformation from the disk to the cylinder, this problem can be approached by means of the Gel'fand-Yaglom formalism. Using results from the literature and the exclusion of zero modes from a modified Gel'fand-Yaglom formula, we obtain matching with the known field theory result. As…
We revisit the calculation of one-loop corrections to the partition functions of a string ending on a circular or straight line in type IIB string theory on $AdS_3\times S^3 \times T^4$ supported by a mixture of 3-form fluxes depending on a parameter $q$ (with $q=0$ being R-R and $q=1$ pure NS-NS cases). Correcting earlier results in the literature, we find that the partition function depends on $q$ only through rescaled tension $T= T_0/(1-q^2)$.
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 is calculable by using a Gaussian matrix model. On the other hand, in the gravity side, it has been conjectured that the expectation value of the Wilson loop is given by the classical value of the action for a probe D3-brane with k electric fluxes. Given such correspondence, we pursue the interpretation of the matrix model eigenvalue dens…
We describe new solutions for open string moving in AdS_5 and ending in the boundary, namely dual to Wilson loops in N=4 SYM theory. First we introduce an ansatz for Euclidean curves whose shape contains an arbitrary function. They are BPS and the dual surfaces can be found exactly. After an inversion they become closed Wilson loops whose expectation value is $W=\exp(-\sqrtλ)$. After that we consider several Wilson loops for N=4 SYM in a pp-wave metric and find the dual surfaces in an AdS_5 pp-wave background. Using the fact that the pp-wave is conformally flat, we apply a conformal transforma…
We calculate the expectation value of one circular Wilson loop and the correlator of two concentric circular Wilson loops in AdS/QCD using the modified AdS_5-metric given in Ref.[1]. The confinement properties of this metric in AdS/QCD are analyzed and compared with QCD and Nambu-Goto theory in four dimensions.
We use AdS/QCD to analyze the quark and gluon scalar and pseudo-scalar condensates around static color sources described by a circular Wilson loop. We also derive the static dipole-dipole interactions between rectangular Wilson loops in AdS/QCD and discuss their relevance for static string interactions in QCD at strong coupling.
We study the half-BPS circular Wilson loop in ${\cal N}=4$ super Yang-Mills with orthogonal gauge group. By supersymmetric localization, its expectation value can be computed exactly from a matrix integral over the Lie algebra of $SO(N)$. We focus on the large $N$ limit and present some simple quantitative tests of the duality with type IIB string theory in $AdS_5\times \mathbb{RP}^5$. In particular, we show that the strong coupling limit of the expectation value of the Wilson loop in the spinor representation of the gauge group precisely matches the classical action of the dual string theory …
We apply a new description of thermal fundamental string probes to the study of finite temperature Wilson loops in the context of the AdS/CFT correspondence. Previously this problem has been considered using extremal probes even though the background is at finite temperature. The new description of fundamental string probes is based on the blackfold approach. As a result of our analysis we find a new term in the potential between static quarks in the symmetric representation which for sufficiently small temperatures is the leading correction to the Coulomb force potential. We also find an orde…
The 1/2-BPS Wilson loop in $\mathcal{N}=4$ supersymmetric Yang-Mills theory is an important and well-studied example of conformal defect. In particular, much work has been done for the correlation functions of operator insertions on the Wilson loop in the fundamental representation. In this paper, we extend such analyses to Wilson loops in the large-rank symmetric and antisymmetric representations, which correspond to probe D3 and D5 branes with $AdS_2 \times S^2$ and $AdS_2 \times S^4$ worldvolume geometries, ending at the $AdS_5$ boundary along a one-dimensional contour. We first compute the…
The half-supersymmetric Wilson loop in $\mathcal N=4$ SYM is arguably the central non-local operator in the AdS/CFT correspondence. On the field theory side, the vacuum expectation values of Wilson loops in arbitrary representations of $SU(N)$ are captured to all orders in perturbation theory by a Gaussian matrix model. Of prominent interest are the $k$-symmetric and $k$-antisymmetric representations, whose gravitational description is given in terms of D3- and D5-branes, respectively, with fluxes in their world volumes. At leading order in $N$ and $λ$ the agreement in both cases is exact. In …
In this paper, minimal surface in $q$-deformed $AdS_5\times S^5$ with boundary a cusp is studied in detail. This minimal surface is dual to cusped Wilson loop in the dual field theory. We found that the area of the minimal surface has both logarithmic squared divergence and logarithmic divergence. The logarithmic squared divergence can not be removed by either Legendre transformation or the usual geometric substraction. We further make analytic continuation to the Minkowski signature %for the case without the jump and take the limit such that the two edges of the cusp become light-like and ext…
