The asymptotic behavior of Wilson loops in the large-size limit ($L\rightarrow\infty$) in confining gauge theories with area law is controlled by effective string theory (EST). The $L^{-2}$ term of the large-size expansion for the logarithm of Wilson loop appears within EST as a two-loop correction. Ultraviolet divergences of this two-loop correction for polygonal contours can be renormalized using an analytical regularization constructed in terms of Schwarz-Christoffel mapping. In the case of triangular Wilson loops this method leads to a simple final expression for the $L^{-2}$ term.
In the framework of effective string theory (EST), the asymptotic behavior of a large Wilson loop in confining gauge theories can be expressed via Laplace determinant with Dirichlet boundary condition on the Wilson contour. For a general polygonal region, Laplace determinant can be computed using the conformal anomaly and Schwarz-Christoffel transformation. One can construct ratios of polygonal Wilson loops whose large-size limit can be expressed via computable Laplace determinants and is independent of the (confining) gauge group. These ratios are computed for hexagon polygons both in EST and…
We explicitly show the area law behavior of a circular Wilson loop in confining theories from supergravity. We calculate the correlator of two Wilson loops from supergravity in confining backgrounds. We find that it is dominated by an exchange of a ``scalarball'' that is lighter than the glueballs. We interpret these results in terms of the meson-meson potential in such theories.
We study the correlators of two circular Wilson loops of different radii at strong coupling. In our setup one Wilson loop is located inside the other. We use holography to calculate the connected two-point function. Both an AdS background and a confining background are considered. As the computation for the confining case cannot be carried out analytically we solve the problem numerically. In the AdS case our results agree with similar holographic calculations. In the case of a confining background we find an asymptotic area law, in agreement with the result of the lattice strong coupling expa…
The extraction of Wilson loops of confining gauge systems from their SUGRA (string) duals is reviewed. I start with describing the basic classical setup. A theorem that determines the classical values of the loops associated with a generalized background is derived. In particular sufficient conditions for confining behavior are stated . I then introduce quadratic quantum fluctuations around the classical configurations. I discuss in details the following models of confining behavior: (i) Strings in flat space-time, (ii) $AdS_5$ black hole and its correspondence with pure YM theory in three dim…
In the world line representation of the fermionic effective action for QCD the interaction between Fermions and the gauge field is contained in the fermionic Wilson loop, namely the Wilson loop for a spin-half particle. It is argued that a string representation of the fermionic Wilson loop can provide a link connecting QCD with a dual description of a meson as a quark and an antiquark connected by a string. This is illustrated by obtaining such a representation in compact U(1) gauge theories. The resulting description contains information about the interaction of the spin of the quark with the…
We study solutions of the one-loop beta-functions of the critical bosonic string theory in the framework of the Renormalization Group (RG) approach to string theory, considering explicitly the effects of the 21 extra dimensions. In the RG approach the 26-dimensional manifold is given in terms of a four dimensional Minkowski spacetime times R and a 21-dimensional hyper-plane. In calculating the Wilson loops, as it is wellknown for this kind of confining geometry, two phenomena appear: confinement and over-confinement. There is a critical minimal surface below of which it leads to confinement on…
The deconfining transition in non-Abelian gauge theory is known to occur by a condensation of Wilson lines. By expanding around an appropriate Wilson line background, it is possible at large $N$ to analytically continue the confining phase to arbitrarily high temperatures, reaching a weak coupling confinement regime. This is used to study the high temperature partition function of an $SU(N)$ electric flux tube. It is found that the partition function corresponds to that of a string theory with a number of world-sheet fields that diverges at short distance.
String representation of the Wilson loop in 3D Abelian-projected SU(3)-gluodynamics is constructed in the approximation that Abelian-projected monopoles form a gas. Such an assumption is much weaker than the standard one, demanding the monopole condensation. It is demonstrated that the summation over world sheets, bounded by the contour of the Wilson loop, is realized by the summation over branches of a certain effective multivalued potential of the monopole densities. Finally, by virtue of the so-constructed representation of the Wilson loop in terms of the monopole densities, this quantity i…
We provide formulas for holographically building a bulk metric from given expectation values of rectangular Wilson loops. As a corollary, we prove that any confining quark potential necessarily leads to the existence of a bulk IR bottom.
Starting with the representation of the Wilson average in the Euclidean 4D compact QED as a partition function of the Universal Confining String Theory, we derive for it the corresponding loop equation, alternative to the familiar one. In the functional momentum representation the obtained equation decouples into two independent ones, which describe the dynamics of the transverse and longitudinal components of the area derivative of the Wilson loop. At some critical value of the momentum discontinuity, which can be determined from a certain equation, the transverse component does not propagate…
We study the stability of some classical string worldsheet solutions employed for computing the potential energy between two static fundamental quarks in confining and non-confining gravity duals. We discuss the fixing of the diffeomorphism invariance of the string action, its relation with the fluctuation orientation and the interpretation of the quark mass substraction worldsheet needed for computing the potential energy in smooth (confining) gravity background. We consider various dual gravity backgrounds and show by a numerical analysis the existence of instabilities under linear fluctuati…
We discuss within the weak-field approximation and the derivative expansion how the area law of the Wilson loop follows directly from the vacuum condensate of mass dimension 2, i.e., simultaneous Bose-Einstein condensation of gluon pair and ghost-antighost pair. Such a novel vacuum condensate was recently claimed to exist as the non-vanishing vacuum expectation value of a BRST-invariant composite operator of mass dimension 2. First of all, we use a version of the non-Abelian Stokes theorem to rewrite the Wilson loop line integral to a surface integral. Then we convert the Yang-Mills theory wit…
We use the t'Hoft-Wilson method for the generation of static fermions potential in order to derive a class of confining potentials which can describe the quark confinement. A general pattern for the construction of propagators through the localization of non-local actions is uncovered.
We calculate Wilson loops in lowest order of perturbation theory for triangular contours whose edges are circular arcs. Based on a suitable disentanglement of the relations between metrical and conformal parameters of the contours, the result fits perfectly in the structure predicted by the anomalous conformal Ward identity. The conformal remainder function depends in the generic 4D case on three cusp and on three torsion angles. The restrictions on these angles imposed by the closing of the contour are discussed in detail and also for cases in 3D and 2D.
We study rectangular timelike Wilson loops at long distances in the exact renormalization group flow in the context of the holographic duality. We consider the 5d holographic model with an exponential dilaton potential constructed in [arXiv:1803.06764]. To probe the RG flow backgrounds at zero and finite temperatures we calculate the minimal surfaces of the corresponding string worldsheets. We show that a holographic RG flow at $T=0$, that mimics QCD behaviour of the running coupling, has a confining phase at long distances, while the RG flow with an AdS UV fixed point is non-confining.
The twisted space-time reduced model of large $N$ QCD with various flavours of adjoint Wilson fermions is constructed applying the symmetric twist boundary conditions with flux $k$. The models with one and two flavours show distinctive behaviours. For the two flavor case, the string tension, calculated at $N=289$, approaches zero as we decrease the quark mass in a way consistent with the theory being governed by an infrared fixed point. In contrast, the string tension for the case of a single adjoint Wilson fermion remains finite as the quark mass decreases to zero, supporting that this is a c…
We present a theorem that determines the value of the Wilson loop associated with a Nambu-Goto action which generalizes the action of the $AdS_5\times S_5$ model. In particular we derive sufficient conditions for confining behavior. We then apply this theorem to various string models. We go beyond the classical string picture by incorporating quadratic quantum fluctuations. We show that the bosonic determinant of $D_p$ branes with 16 supersymmetries yields a Luscher term. We confirm that the free energy associated with a BPS configuration of a single quark is free from divergences. We show tha…
This paper discusses some results on semiclassical string configurations lying in the IR sector of supergravity backgrounds dual to confining gauge theories. On the gauge theory side, the string states we analyse correspond to Wilson loops, glueballs and KK-hadrons. The 1-loop correction to the classical energy is never subleading, but can be viewed as a coupling dependent rescaling of the dimensionful parameters of the theory.
We explore the consequences of imposing Polyakov's zig/zag-invariance in the search for a confining string. We first find that the requirement of zig/zag-invariance seems to be incompatible with spacetime supersymmetry. We then try to find zig/zag-invariant string backgrounds on which to implement the minimal-area prescription for the calculation of Wilson loops considering different possibilities.
String representations of the Wilson loop are constructed in the SU(N)-version of compact QED in three and four dimensions. This is done exactly in the case of the fundamental Wilson loop and in the large-N limit in the case of the adjoint Wilson loop. Using for concreteness the three-dimensional fundamental case, it is demonstrated how the resulting SU(N)-generalization of the so-called theory of confining strings can be obtained in various ways. In its weak-field limit (corresponding to the limit of low monopole densities), this theory enables one to fix the value of the string tension, whic…
Starting from the strong-coupling SU(2) Wilson action in D=3 dimensions, we derive an effective, semi-local action on a lattice of spacing L times the spacing of the original lattice. It is shown that beyond the adjoint color-screening distance, i.e. for $L \ge 5$, thin center vortices are stable saddlepoints of the corresponding effective action. Since the entropy of these stable objects exceeds their energy, center vortices percolate throughout the lattice, and confine color charge in half-integer representations of the SU(2) gauge group. This result contradicts the folklore that confinement…
Schwarz-Christoffel (SC) mapping plays a crucial role in the calculation of the large-size expansion for polygonal Wilson loops in confining gauge theories using effective string theory (EST). Recently a new analytical regularization based on SC mapping was suggested and successfully applied to the calculation of the two-loop contribution of EST in the case of triangular Wilson loops. We prove that this analytical renormalization produces finite results for arbitrary polygonal Wilson loops and show that the result of the analytical renormalization for a given polygonal contour is independent o…
We study semiclassical rotating strings in AdS/CFT backgrounds that exhibit both confinement and finite-size effects. The energy versus spin dispersion relation for short strings is the expected Regge trajectory behaviour, with the same string tension as is measured by the Wilson loop. Long strings probe the interplay between confinement and finite-size effects. In particular, the dispersion relation for long strings shows a characteristic dependence on the string tension and the finite-size scale.
In these lectures we describe the attempt to extract the expectation values of Wilson loops from the string/gauge correspondence. We start with the original calculation in $AdS_5$. It is then extended to the non-conformal background of $D_p$ in the near horizon limit. We discuss the computation at finite temperature. Supergravity models that admit confinement in 3d and 4d are described. A theorem that determines the classical values of loops associated with a generalized background is derived.In particular we determine sufficient conditions for confining behavior. We then apply the theorem to …
The spectral density for two dimensional continuum QCD has a non-analytic behavior for a critical area. Apparently this is not reflected in the Wilson loops. However, we show that the existence of a critical area is encoded in the winding Wilson loops: Although there is no non-analyticity or phase transition in these Wilson loops, the dynamics of these loops consists of two smoothly connected domains separated by the critical area, one domain with a confining behavior for large winding Wilson loops, and one (below the critical size) where the string tension disappears. We show that this can be…
We discuss three dimensional compact QED with a theta term due to an axionic field. The variational gauge invariant functional is considered and it is shown that the ground state energy is independent of theta in a leading approximation. The mass gap of the axionic field is found to be dependent upon theta, the mass gap of the photon field and the scalar potential. The vacuum expectation of the Wilson loop is shown to be independent of theta in a leading approximation, to obey the area law and to lead to confinement. We also briefly discuss the properties of axionic confining strings.
In gauge systems coupled to matter, the static potential flattens out at a scale where the confining string breaks by formation of a dynamical pair of particles. Surprisingly, such a breaking is invisible in Wilson loops even when the inter-charge separation is much larger than the flattening scale. Observing string breaking also requires using different operators. A known string mechanism provides us with a simple explanation, leading to the area law for large Wilson loops, as observed in most gauge models coupled to whatever kind of matter. It is also pointed out that in a simple 3D Z_2 gaug…
The open Wilson lines are gauge-invariant operators made with a gauge transporter along an open path saturated at the end-points with matter fields. Here it is shown that numerical experiments on 3D Z2 Higgs model provide useful guidance in addressing the problem of the functional form of their vacuum expectation values. It turns out they satisfy, as long as their size does not exceed the string breaking scale, a remarkable factorization, related to finite-size scaling of the Fredenhagen-Marcu order parameter. This leads to conjecture a decay area law, like ordinary Wilson loops, with the diff…
In this letter we study the dilatonic corrections to the static gauge potential between heavy sources. These corrections come from the solutions to the the lowest order beta equations. In the energetically favoured branch, the potential obtained is characterised by having a linear confining term, an $L$ independent term and another 1/L piece. This is indicative of a Lüscher-type behaviour in the strong-coupling regime of the dual gauge theory. On the other hand, we also explore the singularity as a point where the theory becomes free.
We study the boundary contribution to the low energy effective action of the open string describing the confining flux tube in gauge theories. The form of the boundary terms is strongly constrained by the requirement of Lorentz symmetry, which is spontaneously broken by the formation of a long confining flux tube in the vacuum. Writing the boundary action as an expansion in the derivatives of the Nambu-Goldstone modes describing the transverse fluctuations of the string, we single out and put in a closed form the first few Lorentz invariant boundary terms. We also evaluate the leading deviatio…
The renormalization group approach towards the string representation of non abelian gauge theories translates, in terms of the string sigma model beta function equations, the renormalization group evolution of the gauge coupling constant and Zamolodchikov`s $c$ function. Tachyon stability, glueball mass gap, renormalization group evolution of the $c$ function and the area law for the Wilson loop are studied for a critical bosonic string vacuum corresponding to a non abelian gauge theory in four dimensional space-time. We prove that the same intrinsic geometry for the string vacuum is universal…
SU(N) gauge theories, extended with adjoint fermions having periodic boundary conditions, are confining at high temperature for sufficiently light fermion mass $m$. In the high temperature confining region, the one-loop effective potential for Polyakov loops has a Z(N)-symmetric confining minimum. String tensions associated with Polyakov loops are calculable in perturbation theory, and display a novel scaling behavior in which higher representations have smaller string tensions than the fundamental representation. In the magnetic sector, the Polyakov loop plays a role similar to a Higgs field,…
We point out that the world sheet swept by the confining string in presence of dynamical quarks can belong to two different phases, depending on the number of charge species and the quark masses. When it lies in the normal phase (as opposed to the tearing one) the string breaking is invisible in the Wilson loop, while is manifest in operators composed of disjoint sources, as observed in many numerical experiments. We work out an explicit formula for the correlator of Polyakov loops at finite temperature, which is then compared with recent lattice data, both in the quenched case and in presence…
We discuss the effective action for Polyakov-Wilson loops winding around compact Euclidean time, which serve as order parameters for the finite temperature deconfinement transition in $SU(N)$ Yang-Mills gauge theory. We then apply our results to the study of the high temperature continuation of the confining phase, and to the analysis of certain $Z_N$ domain walls that have been argued to play a role in cosmology. We argue that the free energy of these walls is much larger than previously thought.
We examine in non-Abelian gauge theory the heavy quark limit in the presence of the (anti-)self-dual homogeneous background field and see that a confining potential emerges, consistent with the Wilson criterion, although the potential is quadratic and not linear in the quark separation. This builds upon the well-known feature that propagators in such a background field are entire functions. The way in which deconfinement can occur at finite temperature is then studied in the static temporal gauge by calculation of the effective potential at high temperature. Finally we discuss the problems to …
The derivation of the $ q \bar q $ and the $ 3q $ potential for two dynamical quarks in a Wilson--loop context is reviewed. Some improvements are introduced. Only the usual assumptions in the evaluation of the Wilson loop integrals and expansions in the quark velocities are required for the result. It is shown that under the same assumptions it is possible to obtain the relativistic flux--tube lagrangian and a $ q \bar{q}$ Bethe--Salpeter equation with a confining kernel for spinless quarks. IFUM 482/FT
We discuss the confining features of the Schwinger model on the Poincare half plane. We show that despite the fact that the expectation value of the large Wilson loop of massless Schwinger model displays the perimeter behavior, the system can be in confining phase due to the singularity of the metric at horizontal axis. It is also shown that in the quenched Schwinger model, the area dependence of the Wilson loop, in contrast to the flat case, is a not a sign of confinement and the model has a finite energy even for large external charges separation. The presence of dynamical fermions can not m…
We discuss the impact of quadratic quantum fluctuations on the Wilson loop extracted from classical string theory. We show that a large class of models, which includes the near horizon limit of D_p branes with 16 supersymmetries, admits a Lüscher type correction to the classical potential. We confirm that the quantum determinant associated with a BPS configuration of a single quark in the AdS_5 \times S^5 model is free from divergences. We find that for the Wilson loop in that model, unlike the situation in flat space-time, the fermionic determinant does not cancel the bosonic one. For string …
We review certain properties of confinement with added focus on the ones we study with holography. Then we discuss observables whose unique behavior can indicate the presence of confinement. Using mainly the Wilson loop in the gauge/gravity formalism, we study two main features of the QCD string: the string tension dependence on the temperature while in the confining phase, and the logarithmic broadening of the flux tube between the heavy static charges that turns out to be a generic property of all confining theories. Finally, we review the k-string bound state and we show that for a wide cla…
We address the problem of computing scattering amplitudes related to the correlation function of two Wilson lines and/or loops elongated along light-cone directions in strongly coupled gauge theories. Using the AdS/CFT correspondence in the classical approximation, the amplitudes are shown to be related to minimal surfaces generalizing the {\em helicoid} in various $AdS_5$ backgrounds. Infra-red divergences appearing for Wilson lines can be factorized out or can be cured by considering the IR finite case of correlation functions of two Wilson loops. In non-conformal cases related to confining …
In this work the solution generated by performing a U-duality on a deformation of the Maldacena-Nastase solution is studied. This is a solution of type-IIA with a metric that is asymptotically AdS4 and supports a G2 structure. It is believed to be dual to a 2+1d, N=1 gauge theory similar to the baryonic branch of Klebanov-Strassler with an additional intermediate scale. An improved radial coordinate is used allowing the derivation of UV series solutions to the BPS equation that persist to all orders. A study of the properties of the dual field theory is performed which includes an operator ana…
Three-dimensional bond or site percolation theory on a lattice can be interpreted as a gauge theory in which the Wilson loops are viewed as counters of topological linking with random clusters. Beyond the percolation threshold large Wilson loops decay with an area law and show the universal shape effects due to flux tube quantum fluctuations like in ordinary confining gauge theories. Wilson loop correlators define a non-trivial spectrum of physical states of increasing mass and spin, like the glueballs of ordinary gauge theory. The crumbling of the percolating cluster when the length of one pe…
We study the spectrum of the minimal supersymmetric extension of the Carroll-Field-Jackiw model for Electrodynamics with a topological Chern-Simons-like Lorentz-symmetry violating term. We identify a number of independent background fermion condensates, work out the gaugino dispersion relation and propose a photonic effective action to consider aspects of confinement induced by the SUSY background fermion condensates, which also appear to signal Lorentz-symmetry violation in the photino sector of the action. Our calculations of the static potential are carried out within the framework of the g…
We calculate configuration energies of string-like membranes in M2- and M5-brane spaces. In the near horizon approximation these backgrounds reduce to $AdS_4\times S^7$ and $AdS_7\times S^4$ spaces and the dual theories are supersymmetric $SU(N)$ gauge theories, in accordance with the AdS/CFT correspondence. In this case the string-like configuration energy is identified with the quark and anti-quark energy interaction in the dual theories. Far from the horizon the dual theory is more involved but any how we were able to calculate the string-like configuration energy. For limiting regimes we f…
We show that the QCD vacuum (without dynamical quarks) is a dual superconductor at least in the low-energy region in the sense that monopole condensation does really occur. In fact, we derive the dual Ginzburg-Landau theory (i.e., dual Abelian Higgs model) directly from the SU(2) Yang-Mills theory by adopting the maximal Abelian gauge. The dual superconductor can be on the border between type II and type I, excluding the London limit. The masses of the dual Abelian gauge field is expressed by the Yang-Mills gauge coupling constant and the mass of the off-diagonal gluon of the original Yang-Mil…
We derive a solution of type IIB supergravity which is asymptotic to AdS_5 x S^5, has SO(6) symmetry, and exhibits some of the features expected of geometries dual to confining gauge theories. At the linearized level, the solution differs from pure AdS_5 x S^5 only by a dilaton profile. It has a naked singularity in the interior. Wilson loops follow area law behavior, and there is a mass gap. We suggest a field theory interpretation in which all matter fields of N=4 gauge theory acquire a mass and the infrared theory is confining.
The minimization of the Nambu-Goto action for a surface whose contour defines a circular Wilson loop of radius a placed at a finite value of the coordinate orthogonal to the boundary is considered. This is done for asymptotically AdS spaces. The condensates of even dimension $n=2$ through $10$ are calculated in terms of the coefficient of $a^{n}$ in the expansion of the on-shell subtracted Nambu-Goto action for small $a$ The subtraction employed is such that it presents no conflict with conformal invariance in the AdS case and need not introduce an additional infrared scale for the case of con…
We compute the one-loop string corrections to the Wilson loop, glueball Regge trajectory and stringy hadron masses in the Witten model of non supersymmetric, large-N Yang-Mills theory. The classical string configurations corresponding to the above field theory objects are respectively: open straight strings, folded closed spinning strings, and strings orbiting in the internal part of the supergravity background. For the rectangular Wilson loop we show that besides the standard Luescher term, string corrections provide a rescaling of the field theory string tension. The one-loop corrections to …
We discuss the critical behavior of several three-dimensional magnetic systems, such as pure and randomly dilute (anti)ferromagnets and stacked triangular antiferromagnets. We also discuss the nature of the multicritical points that arise in the presence of two distinct O(n)-symmetric order parameters and, in particular, the nature of the multicritical point in the phase diagram of high-T_c superconductors that has been predicted by the SO(5) theory. For each system, we consider the corresponding Landau-Ginzburg-Wilson field theory and review the field-theoretical results obtained from the ana…
We offer a physicists' proof that center-vortex theory requires the area in the Wilson-loop area law to involve an extremal area. Area-law dynamics is determined by integrating over Wilson loops only, not over surface fluctuations for a fixed loop. Fluctuations leading to to perimeter-law corrections come from loop fluctuations as well as integration over finite -thickness center-vortex collective coordinates. In d=3 (or d=2+1) we exploit a contour form of the extremal area in isothermal which is similar to d=2 (or d=1+1) QCD in many respects, except that there are both quartic and quadratic t…
We revisit the dyon-anti-dyon liquid model for the Yang-Mills confining vacuum discussed by Diakonov and Petrov, by retaining the effects of the classical interactions mediated by the streamline between the dyons and anti-dyons. In the SU(2) case the model describes a 4-component strongly interacting Coulomb liquid in the center symmetric phase. We show that in the linearized screening approximation the streamline interactions yield Debye-Huckel type corrections to the bulk parameters such as the pressure and densities, but do not alter significantly the large distance behavior of the correlat…
We discuss a solution of the equations of motion of five-dimensional gauged type IIB supergravity that describes confining SU(N) gauge theories at large N and large 't Hooft parameter. We prove confinement by computing the Wilson loop, and we show that our solution is generic, independent of most of the details of the theory. In particular, the Einstein-frame metric near its singularity, and the condensates of scalar, composite operators are universal. Also universal is the discreteness of the glueball mass spectrum and the existence of a mass gap. The metric is also identical to a generically…
