Dual representations are constructed for non-abelian lattice spin models with U(N) and SU(N) symmetry groups, for all N and in any dimension. These models are usually related to the effective models describing the interaction between Polyakov loops in the strong coupled QCD. The original spin degrees of freedom are explicitly integrated out and a dual theory appears to be a local theory for the dual integer-valued variables. The construction is performed for the partition function and for the most general correlation function. The latter include the two-point function corresponding to quark-anti-quark free energy and the N-point function related to the free energy of a baryon. We consider both pure gauge models and models with static fermion determinant for both the staggered and Wilson fermions with an arbitrary number of flavours. While the Boltzmann weights of such models are complex in the presence of non-zero chemical potential the dual Boltzmann weights appear to be strictly positive on admissible configurations. An essential part of this work with respect to previous studies is an extension of the dual representation to the case of 1) an arbitrary value of the temporal coupling constant in the Wilson action and 2) an arbitrary number of flavours of static quark determinants. The applications and extensions of the results are discussed in detail. In particular, we outline a possible approach to Monte-Carlo simulations of the dual theory, to the large N expansion and to the development of a tensor renormalization group.
We use gauge/string duality to analytically evaluate the renormalized Polyakov loop in pure Yang-Mills theories. For SU(3), the result is in a quite good agreement with lattice simulations for a broad temperature range.
We construct a new order parameter for finite temperature QCD by considering the quark condensate for U(1)-valued temporal boundary conditions for the fermions. Fourier transformation with respect to the boundary condition defines the dual condensate. This quantity corresponds to an equivalence class of Polyakov loops, thereby being an order parameter for the center symmetry. We explore the duality relation between the quark condensate and these dressed Polyakov loops numerically, using quenched lattice QCD configurations below and above the QCD phase transition. It is demonstrated that the Di…
The dual quark condensate $Σ^{(n)}$ proposed recently as a new order parameter of the spontaneous breaking of the ${\mathbb Z}_3$ symmetry are evaluated by the Polyakov-loop extended Nambu--Jona-Lasinio model, where $n$ are winding numbers. The Polyakov-loop extended Nambu--Jona-Lasinio model well reproduces lattice QCD data on $Σ^{(1)}$ measured very lately. The dual quark condensate $Σ^{(n)}$ at higher temperatures is sensitive to the strength of the vector-type four-quark interaction in the Polyakov-loop extended Nambu--Jona-Lasinio model and hence a good quantity to determine the strength.
We discuss the effect of the choice of an Abelian projection on the dual superconductor mechanism of confinement in SU(2) gluodynamics. Using qualitative arguments we show that the dual superconductor Lagrangian corresponding to the Abelian Polyakov gauge has a different structure compared to the dual Lagrangian in the Maximal Abelian gauge. A difference between these Lagrangians reflects the fact that in continuum limit the monopoles should be static in the Abelian Polyakov gauge and therefore these monopoles can not give rise to the confinement of static quarks. Using the Abelian Polyakov ga…
We consider a multi-string configuration that provides a new way to compute the expectation value of the Polyakov loop in a five-dimensional framework known as AdS/QCD. The obtained results are in reasonably good agreement with those obtained by lattice simulations for pure SU(3) gauge theory and also by the usual single string configuration.
We consider the expectation value of a Polyakov loop in 3d SU(2) lattice Yang--Mills theory and transform it to the dual representation in terms of sums over spins. The spin dependence of the amplitudes is computed explicitly by a graphical method. We also determine the asymptotic (large spin) limit of the amplitude factors.
We study dual Abrikosov vortices in Abelian projected SU(2) gauge theory in the Polyakov gauge. We show that vortices are present in this gauge but they are suppressed with respect to the maximal Abelian gauge. We interpret this difference in terms of the shielding of the electric charge by the charged coset fields.
Many Polyakov loop models can be written in a dual formulation which is free of sign problem even when a non-vanishing baryon chemical potential is introduced in the action. Here, results of numerical simulations of a dual representation of one such effective Polyakov loop model at finite baryon density are presented. We compute various local observables such as energy density, baryon density, quark condensate and describe in details the phase diagram of the model. The regions of the first order phase transition and the crossover, as well as the line of the second order phase transition, are e…
We show that in four-dimensional gauge theories dual to five-dimensional Einstein gravity coupled to a single scalar field in the bulk the derivative of the single heavy quark free energy in the deconfined phase is $dF_{Q}(T)/dT \sim -1/c_s^2(T)$, where $c_s(T)$ is the speed of sound. This general result provides a direct link between the softest point in the equation of state of strongly-coupled plasmas and the deconfinement phase transition described by the expectation value of the Polyakov loop. We give an explicit example of a gravity dual with black hole solutions that can reproduce the l…
We consider a dual representation of an effective three-dimensional Polyakov loop model for the SU(3) theory at nonzero real chemical potential. This representation is free of the sign problem and can be used for numeric Monte-Carlo simulations. These simulations allow us to locate the line of second order phase transitions, that separates the region of first order phase transition from the crossover one. The behavior of local observables in different phases of the model is studied numerically and compared with predictions of the mean-field analysis. Our dual formulation allows us to study als…
We construct a novel observable for finite temperature QCD that relates confinement and chiral symmetry. It uses phases as boundary conditions for the fermions. We discuss numerical and analytical aspects of this observable, like its spectral behavior below and above the critical temperature, as well as the connection to chiral condensate, center symmetry and the canonical ensemble.
Confinement in SU($N$) gauge theory is due to the linear potential between colored objects. At short distances, the linear contribution could be considered as the quadratic correction to the leading Coulomb term. Recent lattice data show that such quadratic corrections also appear in the deconfined phase, in both the thermal quantities and the Polyakov loop. These contributions are studied systematically employing the gauge/string duality. "Confinement" in ${\mathcal N}=4$ SU($N$) Super Yang-Mills (SYM) theory could be achieved kinematically when the theory is defined on a compact space manifo…
We prove the absence of a mass gap and confinement in the Polyakov model with massless complex fermions in any representation of the gauge group. A $U(1)_{*}$ topological shift symmetry protects the masslessness of one dual photon. This symmetry emerges in the IR as a consequence of the Callias index theorem and abelian duality. For matter in the fundamental representation, the infrared limits of this class of theories interpolate between weakly and strongly coupled conformal field theory (CFT) depending on the number of flavors, and provide an infinite class of CFTs in $d=3$ dimensions. The l…
We explore vortex formation for Abelian projected SU(2) in the Polyakov gauge and compare the results with those calculated in the maximal Abelian gauge. In both gauges, a non-zero vacuum expectation value of a monopole field operator signals confinement. We find vortices in the Polyakov projection, confirming the connection between the dual superconductor order parameter and the existence of vortices. However we find that the Polyakov Abelian projection is problematic, leaving the maximal Abelian projection as the best candidate to define an effective theory of confinement in this scenario.
We show that the Polyakov loop approaches its asymptotic value at high temperatures {\it from below} in any strongly-coupled gauge theory at large $N_c$ with a dual supergravity description. In these theories the Polyakov loop can be used to define a quantity that behaves as a free energy of a heavy quark in the medium. This should be compared to the recent lattice results in pure glue SU(3) theory at high temperatures where the Polyakov loop approaches its non-interacting limit {\it from above}. We show that the "heavy quark free energy" obtained from the renormalized loop computed on the lat…
To obtain deeper understanding of QCD properties at finite temperature, we consider the Fourier decomposition of the grand-canonical partition function based on the canonical ensemble method via the imaginary chemical potential. Expectation values are, then, represented by summation over each canonical sector. We point out that the modified Polyakov-loop can play an important role in the canonical ensemble; for example, the Polyakov-loop paradox which is known in the canonical ensemble method can be evaded by considering the quantity. In addition, based on the periodicity issue of the modified…
We compare the grand canonical partition function at fixed chemical potential mu with the canonical partition function at fixed baryon number B, formally and by numerical simulations at mu=0 and B=0 with four flavours of staggered quarks. We verify that the free energy densities are equal in the thermodynamic limit, and show that they can be well described by the hadron resonance gas at T < T_c and by the free fermion gas at T>T_c. Small differences between the two ensembles, for thermodynamic observables characterising the deconfinement phase transition, vanish with increasing lattice size. T…
The quantum mechanical mass of 't Hooft-Polyakov monopoles in the four-dimensional Georgi-Glashow is calculated non-perturbatively using lattice Monte Carlo simulations. This is done by imposing twisted boundary conditions that ensure there is one unit of magnetic charge on the lattice, and measuring the free energy difference between this ensemble and the vacuum. In the weak-coupling limit, the results can be used to determine the quantum correction to the classical mass, once renormalisation of couplings is taken properly into account. The methods can also be used to study the masses at stro…
In this paper we re-derive the effective Nambu-Goto theory result for the Polyakov loop correlator, starting from the free bosonic string and using a covariant quantization. The boundary conditions are those of an open string attached to two D0-branes at spatial distance R, in a target space with compact euclidean time. The one-loop free energy contains topologically distinct sectors corresponding to multiple covers of the cylinder in target space bordered by the Polyakov loops. The sector that winds once reproduces exactly the Nambu-Goto partition function. In our approach, the world-sheet du…
The dual superconductivity is a promising mechanism for quark confinement. We have proposed the non-Abelian dual superconductivity picture for SU(3) Yang-Mills theory, and showed the restricted field dominance (called conventionally Abelian dominance), and non-Abelian magnetic monopole dominance in the string tension. We have further demonstrated by measuring the chromoelectric flux that the non-Abelian dual Meissner effect exists and determined that the dual superconductivity for SU(3) case is of type I, which is in sharp contrast to the SU(2) case: the border of type I and type II. In this t…
Lattice Yang-Mills theories at finite temperature can be mapped onto effective 3d spin systems, thus facilitating their numerical investigation. Using strong-coupling expansions we derive effective actions for Polyakov loops in the $SU(2)$ and $SU(3)$ cases and investigate the effect of higher order corrections. Once a formulation is obtained which allows for Monte Carlo analysis, the nature of the phase transition in both classes of models is investigated numerically, and the results are then used to predict -- with an accuracy within a few percent -- the deconfinement point in the original 4…
The phase structure of Polyakov-loop extended Nambu-Jona-Lasinio (PNJL) model is explored at imaginary chemical potential, with particular emphasis on the deconfinement transition. We point out that the statistical confinement nature of the model naturally leads to characteristic dependence of the chiral condensate $<\bar{q}q>$ on $θ=μ_I/T$. We introduce a dual parameter for the deconfinement transition by making use of this dependence. By changing a four-fermion coupling constant, we tune the location of the critical endpoint of the deconfinement transition.
We compute, in SU(3) pure gauge theory, the vacuum expectation value (vev) of the operator which creates a $Z_3$ vortex wrapping the lattice through periodic boundary conditions (dual Polyakov line). The technique used is the same already tested in the SU(2) case. The dual Polyakov line proves to be a good disorder parameter for confinement, and has a similar behaviour to the monopole condensate. The new features which characterise the construction of the disorder operator in SU(3) are emphasised.
The dual superconductivity is a promising mechanism for quark confinement. We have presented a new formulation of the Yang-Mills theory on the lattice that enables us to change the original non-Abelian gauge field into the new field variables such that one of them called the restricted field gives the dominant contribution to quark confinement in the gauge independent way. We have pointed out that the SU(3) Yang-Mills theory has another reformulation using new field variables (minimal option), in addition to the way adopted by Cho, Faddeev and Niemi (maximal option). In the past lattice confer…
We study a simple lattice model with local symmetry, whose construction is based on a crossed module of finite groups. Its dynamical degrees of freedom are associated both to links and faces of a four-dimensional lattice. In special limits the discussed model reduces to certain known topological quantum field theories. In this work we focus on its dynamics, which we study both analytically and using Monte Carlo simulations. We prove a factorization theorem which reduces computation of correlation functions of local observables to known, simpler models. This, combined with standard Krammers-Wan…
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,…
Spectral sums of the Dirac-Wilson operator and their relation to the Polyakov loop are thoroughly investigated. The approach by Gattringer is generalized to mode sums which reconstruct the Polyakov loop locally. This opens the possibility to study the mode sum approximation to the Polyakov loop correlator. The approach is re-derived for the ab initio continuum formulation of Yang-Mills theories, and the convergence of the mode sum is studied in detail. The mode sums are then explicitly calculated for the Schwinger model and SU(2) gauge theory in a homogeneous background field. Using SU(2) latt…
We review the physics of confinement based on non-Abelian dual superconductor picture, relying on exact solutions in N=2 supersymmetric QCD and based on the recent developments in our understanding of non-Abelian vortices and monopoles. The non-Abelian monopoles, though they are basically just the 't Hooft-Polyakov SU(2) monopoles embedded in various corners of the larger gauge group, require flavor symmetry in an essential way for their very existence. The phenomenon of flavor-color-flavor separation characterizes the multiple roles flavor symmetry plays in producing quantum-mechanical non-Ab…
Lattice gauge theories are fundamental to our understanding of high-energy physics. Nevertheless, the search for suitable platforms for their quantum simulation has proven difficult. We show that the Abelian Higgs model in 1+1 dimensions is a prime candidate for an experimental quantum simulation of a lattice gauge theory. To this end, we use a discrete tensor reformulation to smoothly connect the space-time isotropic version used in most numerical lattice simulations to the continuous-time limit corresponding to the Hamiltonian formulation. The eigenstates of the Hamiltonian are neutral for p…
It is proved that for a system of spins $σ_i = \pm 1$ having an interaction energy $-\sum K_{ij} σ_i σ_j $ with all the $K_{ij}$ strictly positive,one can construct a dual formulation by associating a dual spin $S_{ijk} = \pm 1$ to each triplet of distinct sites $i,j$ and $k$. The dual interaction energy reads $-\sum _{(ij)} D_{ij} \prod _{k \neq i,j} S_{ijk}$ with $tanh(K_{ij})\ = \ exp(-2D_{ij})$, and it is invariant under local symmetries. We discuss the gauge-fixing procedure, identities relating averages of order and disorder variables and representations of various quantities as integral…
In this series of three papers, we generalize the derivation of dual photons and monopoles by Polyakov, and Banks, Myerson and Kogut, to obtain approximative models of SU(2) lattice gauge theory. Our approach is based on stationary phase approximations. In this third article, we start from the plaquette representation of 3-dimensional SU(2) lattice gauge theory. By extending a work of Borisenko, Voloshin and Faber, we transform the expectation value of a Wilson loop into a path integral over a dual gluon field and monopole variables. The action contains the tree-level Coulomb interaction and a…
In this series of three papers, we generalize the derivation of dual photons and monopoles by Polyakov, and Banks, Myerson and Kogut, to obtain approximative models of SU(2) lattice gauge theory. Our approach is based on stationary phase approximations. In this second article, we start from the spin foam representation of 3-dimensional SU(2) lattice gauge theory. By extending an earlier work of Diakonov and Petrov, we approximate the expectation value of a Wilson loop by a path integral over a dual gluon field and monopole-like degrees of freedom. The action contains the tree-level Coulomb int…
By dressing Polyakov lines with appropriate functionals of the gauge fields, we construct observables describing a fundamental representation static quark-antiquark pair in the singlet, adjoint and average channels of SU(N) pure gauge theory. Each of the potentials represents a gauge invariant eigenvalue of the Hamiltonian. Numerical simulations are performed for SU(2) in 2+1 dimensions. The adjoint channel is found to be repulsive at small and confining at large separations, suggesting the existence of a metastable (N^2-1)-plet bound state. For small distances and temperatures above the decon…
We discuss the non-zero baryon number formulation of QCD in the quenched limit at finite temperature. This describes the thermodynamics of gluons in the background of static quark sources. Although a sign problem remains in this theory, our simulation results show that it can be handled quite well numerically. The transition region gets shifted to smaller temperatures and the transition region broadens with increasing baryon number. Although the action is in our formulation explicitly Z(3) symmetric the Polyakov loop expectation value becomes non-zero already in the low temperature phase and t…
To find a perfect lattice action in terms of monopole action on the dual lattice, we performed simulations of a monopole effective action obtained numerically from vacuum configurations in SU(2) QCD. Although the Polyakov loop behavior near $T_c$ is well reproduced by the action, a small but repulsive term is needed in addition to get the string tension correctly. It is reported also a monopole effective action in $SU(3)$ QCD which is expressed by one kind of monopole currents.
Recently, it was argued that the thermal deconfinement transition in pure Yang-Mills theory is continuously connected to a quantum phase transition in softly-broken N=1 SYM theory on R^3 x S^1. The transition is semiclassically calculable at small S^1 size L, occurs as the soft mass m_soft and L vary, and is driven by a competition between perturbative effects and nonperturbative topological molecules. These are correlated instanton--anti-instanton tunneling events, whose constituents are monopole-instantons "bound" by attractive long-range forces. The mechanism driving the transition is unive…
In this series of three papers, we generalize the derivation of dual photons and monopoles by Polyakov, and Banks, Myerson and Kogut, to obtain approximative models of SU(2) lattice gauge theory. The papers take three different representations as their starting points: the representation as a BF Yang-Mills theory, the spin foam representation and the plaquette representation. The derivations are based on stationary phase approximations. In this first article, we cast 3- and 4-dimensional SU(2) lattice gauge theory in the form of a lattice BF Yang-Mills theory. In several steps, the expectation…
We have proposed the non-Abelian dual superconductivity in SU(3) Yang-Mills theory for the mechanism of quark confinement,and we presented the numerical evidences in preceding lattice conferences by using the proposed gauge link decomposition to extract magnetic monopole in the gauge invariant way. In this talk, we focus on the dual Meissner effects in view of the magnetic monopole in SU(3) Yang-Mills theory. We measure the chromoelectric and chromomagnetic flux due to a pair of quark and antiquark source at finite temperature. Then, we measure the correlation function of Polyakov loops and Po…
We determine the quark condensate and the dressed Polyakov loop from the finite temperature Landau gauge quark propagator evaluated with U(1)-valued boundary conditions in an approximation to quenched QCD. These gauge invariant quantities allow for an investigation of the chiral and deconfinement transition. We compare results from Dyson-Schwinger equations on a lattice with infinite volume continuum results and study the temperature and quark mass dependence of both quantities. In particular we investigate the chiral condensate and the dressed Polyakov loop in the chiral limit. We also consid…
The U(1) lattice gauge theory in three dimensions is a perfect laboratory to study the properties of the confining string. On the one hand, thanks to the mapping to a Coulomb gas of monopoles, the confining properties of the model can be studied semi-classically. On the other hand, high-precision numerical estimates of Polyakov loop correlators can be obtained via a duality map to a spin model. This allowed us to perform high-precision tests of the universal behavior of the effective string and to find macroscopic deviations with respect to the expected Nambu-Goto predictions. These correction…
We report on the results of numerical simulations of 1+1 dimensional SU(N) Yang-Mills theory with maximal supersymmetry at finite temperature and compactified on a circle. For large N this system is thought to provide a dual description of the decoupling limit of N coincident D1-branes on a circle. It has been proposed that at large N there is a phase transition at strong coupling related to the Gregory-Laflamme (GL) phase transition in the holographic gravity dual. In a high temperature limit there was argued to be a deconfinement transition associated to the spatial Polyakov loop, and it has…
Recently lattice simulation in pure Yang-Mills theory exposes significant quadratic corrections for both the thermodynamic quantities and the renormalized Polyakov loop in the deconfined phase. These terms are previously found to appear naturally for ${\mathcal N}=4$ Super Yang-Mills~(SYM) on a sphere at strong coupling, through the gauge/gravity duality. Here we extend the investigation to the weak coupling regime, and for general large-$N$ gauge theories. Employing the matrix model description, we find some novel behavior in the deconfined phase, which is not noticed in the literature. Due t…
In any Abelian gauge theory with an action periodic in the link variables one can perform a duality transformation not only in the partition function, but also in correlation functions including Polyakov loops. The calculation of expectation values in the confinement phase, like electric field strength or monopole currents in the presence of external charges, becomes significantly more efficient simulating the dual theory. We demonstrate this using the ordinary Wilson action. This approach also allows a quantitative analysis of the dual superconductor model, because the dual transformed U(1) t…
We have continued our systematic investigations of the numerical simulations of lattice gauge theories in the dual formulation. These include: i) a more practical implementation of the quasi-local updating technique, ii) a thorough investigation of the sign problem, iii) issues related to the ergodicity of the various update algorithms, iv) a novel way of measuring conventional observables like plaquette in the dual formalism and v) investigations of thermalisation. While the dual formulation holds out a lot of promises in principle, there are still some ways to go before it can be made into a…
We have developed the techniques necessary for a numerical simulation of d=3 SU(2) Lattice Gauge theories in the dual formulation as originally developed by Anishetty et al . These include updating techniques that preserve the constrained configuration space, efficient evaluation of 6-j symbols and a certain problem associated with the positive indefiniteness of the weight factors.
The dual superconductivity is a promising mechanism of quark confinement. In the preceding works, we have given a non-Abelian dual superconductivity picture for quark confinement, and demonstrated the numerical evidences on the lattice. In this talk, we focus on the the confinement and deconfinement phase transition at finite temperature in view of the dual superconductivity. By using our new formulation of lattice Yang-Mills theory and numerical simulations on the lattice, we extract the dominant mode for confinement by decomposing the Yang-Mills field, and we investigate the Polyakov loop av…
