Gauge field configuration for a magnetic monopole and its dual configuration are studied in SU(2) gauge theory. We present a relation between the monopole field and its dual field. Since these fields can become massive, their massive Lagrangians are derived. In the dual case, an additional term appears. We show this term is necessary to produce a linear potential between a monopole charge and an antimonopole charge.
SU(2) gauge theory in the nonlinear gauge of the Curci-Ferrari type is studied in low-energy region. We give a classical solution that connects color electric charges. Its dual solution, which has a configuration of monopole, is also presented. Due to the gluon condensation subsequent to the ghost condensation, these classical fields become massive. The massive Lagrangian with the classical solution and that with the dual solution are derived. We show that these Lagrangians produce a linear potential between a quark and an antiquark. This is the mechanism of quark confinement that is different…
We determine the gluonic spectral function $SU(3)$ Yang-Mills theory as well as we found fermionic spectral functions in the strong quenched QED where we found new solutions. Our novel technique provides solutions with the usual branch cut for propagators while not showing any pole structure at the first Riemann sheet (identical with entire complex plane) of complex square of momentum. Implications and further utilizations are briefly adressed for QCD and Standard model calculations.
The dual conformal box integral in Minkowski space is not fully determined by the conformal invariants $z$ and $\bar{z}$. Depending on the kinematic region its value is on a 'branch' of the Bloch-Wigner function which occurs in the Euclidean case. Dual special conformal transformations in Minkowski space can change the kinematic region in such a way that the value of the integral jumps to another branch of this function, encoding a subtle breaking of dual conformal invariance for the integral. We classify conformally equivalent configurations of four points in compactified Minkowski space. We …
We prove that a scalar quantum field theory defined on noncommutative Minkowski spacetime with noncommuting momentum coordinates is covariant with respect to the UV/IR duality which exchanges coordinates and momenta. The proof is based on suitable resonance expansions of charged noncommutative scalar fields in a background electric field, which yields an effective description of the field theory in terms of a coupled complex two-matrix model. The two independent matrix degrees of freedom ensure unitarity and manifest CT-invariance of the field theory. The formalism describes an analytic contin…
We extend the duality between massive and topologically massive antisymmetric tensor gauge theories in arbitrary space-time dimensions to include topological defects. We show explicitly that the condensation of these defects leads, in 4 dimensions, to confinement of electric strings in the two dual models. The dual phase, in which magnetic strings are confined is absent. The presence of the confinement phase explicitly found in the 4-dimensional case, is generalized, using duality arguments, to arbitrary space-time dimensions.
Interquark confinement potential is calculated in the dual Monopole Nambu-Jona-Lasinio model with dual Dirac strings as a functional of a dual Dirac string length. The calculation is carried out by the explicit integration over quantum fluctuations of a dual-vector field (monopole-antimonopole collective excitation) around the Abrikosov flux line and string shape fluctuations. The contribution of the scalar field (monopole-antimonopole collective excitation) exchange is taken into account in the tree approximation due to the London limit regime.
Recent development on non-Abeliann vortices and monopoles is reviewed with an emphasis on their relevance on confinement and duality. A very recent construction of non-Abelian vortices which do not dynamically Abelianize is crucial in this context.
Electric self-dual vortices arising as BPS states in the strong coupling limit of N=2 supersymmetric Yang-Mills theory, softly broken to N=1, are reported.
We derive the Maxwell's equations on the $κ$-deformed spacetime, valid up to first order in the deformation parameter, using the Feynman's approach. We show that the electric-magnetic duality is a symmetry of these equations. It is also shown that the laws of electrodynamics are {\it different} for particles of equal charges, but with different masses. We show that the Poincare angular momentum, required to maintain the usual Lorentz algebra structure, do not get any $κ$-dependent corrections.
Evidence is discussed, from lattice simulations, that QCD vacuum is a dual superconductor in the confining phase, and undergoes a phase transition to normal at the deconfining temperature.
We solve Klein-Gordon equation for massless scalars on d+1 dimensional Minkowski (Euclidean) space in terms of the Cauchy data on the hypersurface t=0. By inserting the solution into the action of massless scalars in Minkowski (Euclidean) space we obtain the action of dual theory on the boundary t=0 which is exactly the holographic dual of conformally coupled scalars on d+1 dimensional (Euclidean anti) de Sitter space obtained in (A)dS/CFT correspondence. The observed equivalence of dual theories is explained using the one-to-one map between conformally coupled scalar fields on Minkowski (Eucl…
We study low-energy dynamics of three-dimensional $\mathcal{N}=2$ $SU(N)$ "chiral" gauge theories with $F$ fundamental and $\bar{F}$ anti-fundamental matters without a Chern-Simons term. Compared to a naive semi-classical analysis of the Coulomb branch, its quantum structure is highly richer than expected due to so-called "dressed" Coulomb branch (monopole) operators. We propose dualities and confinement phases for the "chiral" $SU(N)$ theories. The theories with $N>F > \bar{F}$ exhibit spontaneous supersymmetry breaking. The very many Coulomb branch operators generally remain exactly massless…
Quantum operators of coordinates and momentum components of a particle in the Minkowski spacetime can belong to the generalized Snyder-Yang algebra and produce a quantum phase space with three new constants in the general case. With account for the O(2,6) invariance in the quantum phase space of a color particle the equation of motion is obtained, which contains a oscillator rising potential. The presence of the oscillator potential can simulate a confinement of a color particle. A parameter of the oscillator potential is estimated and a relationship between current and constituent quark masse…
Recent attempts to find a ``holographic dual'' to QCD-like theories included a suggestion by Karsh et al (below referred to as KKSS) to incorporate confinement via a potential quadratically increasing into the 5-th direction of the $AdS_5$ space. We show that the same conclusion follows from completely different line of arguments. If instantons are promoted into the 5d space by identifying the instanton size $ρ$ at the 5-th coordinate, the background geometry necessarily should be the AdS$_5$. As I argued already in 1999, confinement described via ``dual superconductivity'' leads to a factor $…
The presence of magnetic monopole like excitations of nonabelian varieties is one of the subtlest consequences of spontaneously broken gauge symmetries. Important hints about their quantum mechanical properties, which remained long mysterious, are coming from a detailed knowledge of the dynamics of supersymmetric gauge theories which has become available recently. These developments might shed light on the problem of confinement and dynamical symmetry breaking in QCD. We discuss here some beautiful features of vortex-monopole systems, in which dual nonabelian transformations among monopoles ar…
In this thesis we review the Seiberg-Witten solution of four dimensional N=2 gauge theories, the six-dimensional construction of these models recently proposed by Gaiotto and the BPS quiver technique for the determination of the BPS spectrum of N=2 theories. We then study how these recent developments allow to better understand nonperturbative aspects such as confinement and dynamical symmetry breaking in the N=1 theories obtained starting from N=2 SQCD and adding a mass term for the chiral multiplet in the adjoint of the gauge group. We find an unusual realization of the 't Hooft-Mandelstam m…
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…
We study an exactly marginal deformation of N=4 SUSY Yang-Mills with gauge group U(N) using field theory and string theory methods. The classical theory has a Higgs branch for rational values of the deformation parameter. We argue that the quantum theory also has an S-dual confining branch which cannot be seen classically. The low-energy effective theory on these branches is a six-dimensional non-commutative gauge theory with sixteen supercharges. Confinement of magnetic and electric charges, on the Higgs and confining branches respectively, occurs due to the formation of BPS-saturated strings…
We consider four dimensional $\mathcal{N}=1$ gauge theories that are S-confining, that is they are dual to a Wess-Zumino model. S-confining theories with a simple gauge group have been classified. We prove all the S-confining dualities in the list, when the matter fields transform in rank-$1$ and/or rank-$2$ representations. Our only assumptions are the S-confining dualities for $SU(N)$ with $N+1$ flavors and for $Usp(2N)$ with $2N+4$ fundamentals. The strategy consists in a sequence of deconfinements and re-confinements. We pay special attention to the explicit superpotential at each step.
We study supersymmetric $SO(N_c)$ gauge theories with $N_f$ flavors of quarks in the vector representation. Among the phenomena we find are dynamically generated superpotentials with physically inequivalent branches, smooth moduli spaces of vacua, confinement and oblique confinement, confinement without chiral symmetry breaking, massless composites (glueballs, exotics, monopoles and dyons), non-trivial fixed points of the renormalization group and massless magnetic quarks and gluons. Our analysis sheds new light on a recently found duality in $N=1$ supersymmetric theories. The dual forms of so…
It is argued that the dual transformation of non-Abelian monopoles occurring in a system with gauge symmetry breaking G -> H is to be defined by setting the low-energy H system in Higgs phase, so that the dual system is in confinement phase. The transformation law of the monopoles follows from that of monopole-vortex mixed configurations in the system (with a large hierarchy of energy scales, v_1 >> v_2) G -> H -> 0, under an unbroken, exact color-flavor diagonal symmetry H_{C+F} \sim {\tilde H}. The transformation property among the regular monopoles characterized by π_2(G/H), follows from th…
A generalization of Born-Infeld non-linear vacuum electrodynamics involving axion and dilaton fields is constructed with couplings dictated by electromagnetic duality and SL(2,R) symmetries in the weak field limit. Besides the Newtonian gravitational constant the model contains a single fundamental coupling parameter b0. In the absence of axion and dilaton interactions it reduces, in the limit as b0 tends to infinity, to Maxwell's linear vacuum theory while for finite b0 it reduces to the original Born-Infeld model. The spherically symmetric static sector of the theory is explored in a backgro…
Quantum theory of the free Maxwell field in Minkowski space is constructed using a representation in which the self dual connection is diagonal. Quantum states are now holomorphic functionals of self dual connections and a decomposition of fields into positive and negative frequency parts is unnecessary. The construction requires the introduction of new mathematical techniques involving ``holomorphic distributions''. The method extends also to linear gravitons in Minkowski space. The fact that one can recover the entire Fock space --with particles of both helicities-- from self dual connection…
We discuss an attempt to understand confinement and dynamical symmetry breaking in a 4-dimensional non-supersymmetric gauge theory using S-duality in type IIB string theory. The electric theory is a USp(2n) gauge theory and its magnetic dual is argued to be an SO(2n) or SO(2n-1) gauge theory. These theories are obtained as the low-energy effective theory of O3-D3bar systems in type IIB string theory, which are related by S-duality. Confinement and dynamical symmetry breaking in the USp(2n) gauge theory are caused by the condensation of scalar fields in the magnetic dual description, which is c…
The nonabelian generalization of a recently proposed abelian axial gauge model for tensor matter fields is obtained. In both cases the model can be derived from a $\vf^{4}-$type theory for antisymmetric fields obeying a complex self-dual condition.
We study dual superconductivity of the ground state of SU(2) gauge theory, in connection with confinement. We do that measuring on the lattice a disorder parameter describing condensation of monopoles. Confinement appears as a transition to dual superconductor, independent of the abelian projection defining monopoles. Some speculations are made on the existence of a more appropriate disorder parameter. A similar study for SU(3) is presented in a companion paper.
We show that a hairy black hole solution can provide a holographically dual description of quark confinement. There exists a one-parameter sensible metric which receives the backreaction of matter contents in the holographic action, where the scalar and gauge field are responsible for the condensation of chromomagnetic monopoles. This model features a preconfining phase triggered by second-order monopole condensation and a first-order confinement/deconfinement phase transition. To confirm the confinement, the quark-antiquark potential is calculated by probing a QCD string in both phases. At la…
We study whether broken dual gauge symmetry, as detected by a monopole order parameter introduced by the Pisa group, is necessarily associated with the confinement phase of a lattice gauge theory. We find a number of examples, including SU(2) gauge-Higgs theory, mixed fundamental-adjoint SU(2) gauge theory, and pure SU(5) gauge theory, which appear to indicate a dual gauge symmetry transition in the absence of a transition to or from a confined phase. While these results are not necessarily fatal to the dual superconductor hypothesis, they may pose some problems of interpretation for the prese…
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…
Arguments for the confinement of transverse gauge field excitations, which are based upon superconvergence relations of the propagator, and upon the BRST algebra, are reviewed and applied to supersymmetric models. They are shown to be in agreement with recent results obtained as a consequence of holomorphy and duality in certain $N=1$ SUSY models. The significance of the one loop anomalous dimension of the gauge field in the Landau gauge is emphasized. For the models considered, it is shown to be proportional, with negative relative sign, to the one loop coefficient of the renormalization grou…
We give a theoretical framework for defining and extracting non-Abelian magnetic monopoles in a gauge-invariant way in SU(N) Yang-Mills theory to study quark confinement. Then we give numerical evidences that the non-Abelian magnetic monopole defined in this way gives a dominant contribution to confinement of fundamental quarks in SU(3) Yang-Mills theory, which is in sharp contrast to the SU(2) case in which Abelian magnetic monopoles play the dominant role for quark confinement.
The dual superconductivity of the vacuum in SU(3) gauge theory is investigated by constructing a disorder parameter which signals monopole condensation in various abelian projections and by studying numerically on the lattice its behaviour at finite temperature. We find that the vacuum is a dual superconductor with respect to each U(1) of the residual gauge group after abelian projection independently of the abelian projection chosen. Like in the SU(2) case (discussed in a companion paper) a finite size scaling analysis enables us to extract the indices of the phase transition and our analysis…
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…
We investigate the non-perturbative structure of two planar $Z_p \times Z_p$ lattice gauge models and discuss their relevance to two-dimensional condensed matter systems and Josephson junction arrays. Both models involve two compact U(1) gauge fields with Chern-Simons interactions, which break the symmetry down to $Z_p \times Z_p$. By identifying the relevant topological excitations (instantons) and their interactions we determine the phase structure of the models. Our results match observed quantum phase transitions in Josephson junction arrays and suggest also the possibility of {\it oblique…
Traditionally the physics of the Unruh effect, i.e. the q.f.t. in the wedges $W_R$ or $W_L$ in Minkowski space is related to the physics in the Rindler Fock space, which is a proplematical strategy. In a careful analysis we show that the correct dual q.f.t. lives rather in the thermal Rindler Hilbert space and turns out to be unitarily equivalent to the corresponding Minkowski space theory in contrast to the Rindler Fock space theory. We show in particular that in thermal Rindler Hilbert space a new sort of objects occurs, viz., quasi-particle/hole creation/annihilation operators of thermal Ri…
We study the gravity dual of four dimensional pure Yang-Mills theory through D4 branes, as proposed by Witten (holographic QCD). In this holographic QCD, it has been widely believed that the confinement phase in the pure Yang-Mills theory corresponds to the AdS D4 soliton in gravity and the deconfinement phase corresponds to the black D4 brane. We inspect this conjecture carefully and show that the correspondence between the black D4 brane and the deconfinement phase is not correct. Instead, by using a slightly different set up, we find an alternative gravity solution called "localized soliton…
Suggested holographic duals of QCD, based on AdS/CFT duality, predict that one should be able to vary the scales of colour confinement and chiral-symmetry breaking independently. Furthermore they suggest that such independent variation of scales can be achieved by the inclusion of extra 4-fermion interactions in QCD. We simulate lattice QCD with such extra 4-fermion terms at finite temperatures and show that for strong enough 4-fermion couplings the deconfinement transition occurs at a lower temperature than the chiral-symmetry restoration transition. Moreover the separation of these transitio…
The concept of Euclidean time is proposed which is dual to the usual Minkowski time. The De Sitter solution is shown to be dual to the anti-De Sitter solution under the dual transformation in which Euclidean time and Minkowski time are interchanged. This observation enables us to make a proposal in which supersymmetry is broken, but the four dimensional cosmological constant remains zero. An explicit model is proposed in which we calculate the cosmological constant. The phenomenology based on this model is presented, including the issue of the quark-lepton-Higgs mass matrix.
The Wick rotation provides the standard technique of computing Feynman diagrams by means of Euclidean propagators. Let us suppose that quantum fields in an interaction zone are really Euclidean. In contrast with the well-known Euclidean field theory dealing with the Wightman and Schwinger functions of free quantum fields, we address complete Green's functions of interacting fields, i.e., causal forms on the Borchers algebra of quantum fields. They are the Laplace transform of the Euclidean states obeying a certain condition. If Euclidean states of a quantum field system, e.g., quarks do not sa…
The purpose of this paper is to review the recent progress in understanding quark confinement. The emphasis of this review is placed on how to obtain a manifestly gauge-independent picture for quark confinement supporting the dual superconductivity in the Yang-Mills theory, which should be compared with the Abelian projection proposed by 't Hooft. The basic tools are novel reformulations of the Yang-Mills theory based on change of variables extending the decomposition of the $SU(N)$ Yang-Mills field due to Cho, Duan-Ge and Faddeev-Niemi, together with the combined use of extended versions of t…
It is argued that a dual symmetry is needed to naturally explain experimental limits on color confinement. Since color is an exact symmetry the only possibility is that this symmetry be a dual symmetry, related to non trivial spatial homotopy. The sphere at infinity of 3-dimensional space being 2-dimensional, the relevant homotopy is $Π_2$, the corresponding configurations monopoles, and the mechanism dual superconductivity. The consistency of the order-disorder nature of the deconfining transition is compared with lattice data . It is also shown that the only dual quantum number is magnetic c…
Starting from SU(2) Yang-Mills theory in 3+1 dimensions, we prove that the abelian-projected effective gauge theories are written in terms of the maximal abelian gauge field and the dual abelian gauge field interacting with monopole current. This is performed by integrating out all the remaining non-Abelian gauge field belonging to SU(2)/U(1). We show that the resulting abelian gauge theory recovers exactly the same one-loop beta function as the original Yang-Mills theory. Moreover, the dual abelian gauge field becomes massive if the monopole condensation occurs. This result supports the dual …
Odd Toric Code in a tilted field: Higgs-confinement multicriticality, spontaneous self-duality symmetry breaking, and valence bond solids
Racah Institute of Physics, The Hebrew University of Jerusalem（耶路撒冷希伯来大学拉卡物理研究所）；Max Planck Institute of Quantum Optics（马克斯·普朗克量子光学研究所）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University of Jerusalem（耶路撒冷希伯来大学弗里茨·哈伯分子动力学研究中心）
We study the confinement between vortex strings in the dual lattice gauge theory of the abelian Higgs model. The dual lattice gauge theory is described by a 2-form gauge field. We calculate the string-antistring potential from the surface operator of the 2-form gauge field. The linear confining potential appears in a confinement phase and it disappears in a deconfinement phase. The phase diagram of the theory is also obtained.
The type of the vacuum is studied numerically in the maximally Abelian (MA) gauge and in the Landau (LA) gauge of SU(2) gluodynamics. The type of the vacuum is determined by a ratio between the dual coherence and the dual penetration lengths. The dual penetration length is determined from correlations between Wilson loops and electric fields in both gauges. The dual coherence length is found from correlations between Wilson loops and dimension-2 operators both in the MA and the LA gauges. This determination of the coherence length is supported by theoretical and numerical observation that the …
We study Yang-Baxter deformations of 4D Minkowski spacetime. The Yang-Baxter sigma model description was originally developed for principal chiral models based on a modified classical Yang-Baxter equation. It has been extended to coset curved spaces and models based on the usual classical Yang-Baxter equation. On the other hand, for flat space, there is the obvious problem that the standard bilinear form degenerates if we employ the familiar coset Poincaré group/Lorentz group. Instead we consider a slice of AdS$_5$ by embedding the 4D Poincaré group into the 4D conformal group $SO(2,4)$. With …
