Electric-magnetic duality in non-abelian gauge theories

The duality transformation of the vacuum expectation value of the operator which creates magnetic vortices (the 't Hooft loop operator in the Higgs phase), is performed in the radial gauge (x_uA_u^a(x) = 0). It is found that in the weak coupling region (small g) of a pure Yang-Mills theory the dual operator creates electric vortices whose strength is $\text{1}\text{g}$. The theory is self-dual in this region, and the effective coupling of the dual Lagrangian is $\text{1}\text{g}$. (It is self-dual also in the extreme strong coupling region.) Thus the above duality transformation reduces to electric-magnetic duality where the electric field in the 't Hooft loop operators transforms into a magnetic field in the dual operator. In a spontaneously broken gauge theory these results are valid only within the region where the vortices (or the monopoles) are concentrated, or in directions of the algebra space of unbroken symmetry, as self-duality holds only for this subset of fields. Noting that the 't Hooft loop operator project into the subspace of these field configurations we find that it is an electric-magnetic duality for the spontaneously broken theory as well. In the strong coupling region a strong coupling expansion in powers $\text{1}\text{g}$ is suggested.     

A study of 't Hooft and Wilson loops

An investigation is undertaken for 't Hooft loop operators in four-dimensional gauge theories. For the first time, a perimeter law is shown to be their behavior in weak coupling Wilson lattice (and continuum) non-abelian SU(N) gauge theories for all N. However, it is also argued that this perimeter law is poor criterion for quark confinement. Rather, it is suggested that non-leading long-distance behavior is what is crucial and relevant in distinguishing non-abelian from abelian (and hence confining from non-confining) theories. A new object, “the 't Hooft line”, is introduced to measure this non-leading behavior and is computed in strong coupling on the lattice. There, one finds magnetic screening characterized by a magnetic screening mass, m_s. It is shown to all orders in strong coupling that m_s is the glueball mass, a result which is expected to persist in weak coupling and in the continuum. Two further consequences of this work are that pure non-abelian gauge theories cannot be in a Higgs phase and that in such models that absence of massless physical particles implies confinement.Finally, non-leading behavior in Wilson loops is examined. The present picture of confinement suggests the absence of van der Waals forces in Yang-Mills theories.     

Comparison of lattice gauge theories with gauge groups Z2 and SU(2)

We study a model of a pure Yang Mills theory with gauge group SU(2) on a lattice in Euclidean space. We compare it with the model obtained by restricting variables to $\text{Z}$₂. An inequality relating expectation values of the Wilson loop integral in the two theories is established. It shows that confinement of static quarks is true in our SU(2) model whenever it holds for the corresponding $\text{Z}$₂-model. The SU(2) model is shown to have high and low temperature phases that are distinguished by a qualitatively different behavior of the 't Hooft disorder parameter.     

Construction of local meson operator from the path-ordered operator in QCD2

A local meson operator is constructed from the path-ordered operator on QCD₂. The meson operator satisfies the Klein-Gordon equation with the mass identical to the eigenvalue of the 't Hooft equation. The interaction of the meson is shown to be nonlocal and to have a coupling proportional to 1/N_c.     

VI. Dynamical and topological considerations of quark confinement

Non-Abelian magnetic superconductivity provides a realisation of the topological algebra of 't Hooft which does confine quarks. Its dynamical and topological aspects are discussed and the related phenomenological theory of quark confinement is sketched.     

Non-planar diagrams in the large N limit of U(N) and SU(N) lattice gauge theories

It is shown that the limit as N → ∞ with g²N fixed of the strong coupling expansion for the vacuum expectation values of a U(N) or SU(N) lattice gauge theory is not given by a sum of planar diagrams. This contradicts a result claimed by De Wit and 't Hooft.     

On the Next-leading order Heavy-Quark Potential from AdS/CFT

Applying the AdS/CFT correspondence,the expansion of the heavy-quark potential of N = 4 supersymmetric Yang-Mills theory at large N c is carried out to the next-leading term in the large ’t Hooft coupling at zero temperature.The strong coupling corresponds to the semi-classical expansion of the stringsigma model,the gravity dual of the Wilson loop operator,with the next-leading term expressed in terms of functional determinants of fluctuations.The singularities of these determinants are examined and their contributions are evaluated numerically.We find the next-leading order correction is negative and suppressed by minus square root of the ’t Hooft coupling relative to the leading order.     

Functional integrals for order-disorder variables in terms of Bohm-Aharonov strings

We generalize our previously presented framework for the functional integral formalism of kinks and disorder variables to the direct construction of order variables in relativistic two-dimensional spinor theories. The important new objects here are “_γ5 Bohm-Aharonov strings”. We choose as an illustration the construction of the order variable in the Ising field theory. A _γ5 string brings about a C self-conjugate normalizable mode which has a more subtle nature than the _γ5 skew Atiyah-Singer 't Hooft zero modes. As a result there is a transmutation of quantum numbers similar to that observed with the help of the quasiclassical methods of Jackiw, Rebbi, 't Hooft and Hasenfratz.     

Creation and annihilation operators for Nielsen-Olesen vortices in the coherent state approximation

Using the standard techniques of canonical quantization, we construct approximate expressions for the creation and annihilation operators for Nielsen-Olesen vortices. The forms for the creation and annihilation operators are appropriate to situations where large-scale vortex condensation takes place. In particular, a phenomenon such as this is basic in the quark confinement schemes of Mandelstam and 't Hooft.     

Adler-Bell-Jackiw anomaly and fermion-number breaking in the presence of a magnetic monopole

In (V - A) theories, fermion number is broken in the presence of the 't Hooft-Polyakov magnetic monopole through the Adler-Bell-Jackiw anomaly. An exactly solvable zeroth-order approximation for evaluating Green functions of zero-angular-momentum fermions in the presence of a monopole is developed in the case of an SU(2) model with massless left-handed fermions. Within this approximation the density of the fermion-number breaking condensate is calculated. This density is found to be O(1), i.e. to be independent of the coupling constant and of the vacuum expectation value of the Higgs field. The corrections to the approximation are estimated. It is argued that the above effect can give rise to the strong baryon-number breaking in monopole-fermion interactions in SU(5) grand unified theory.     

Evaluation of higher-order effective potentials with dimensional regularization

A new and simple method for evaluating high-order effective potentials is presented. It is based on the renormalization procedure of 't Hooft and Veltman used in conjunction with the scheme of dimensional regularization. Explicit calculations of the effective potential of a scalar field in two-loop approximation are carried out.     

Zig zag symmetry in AdS/CFT duality

The validity of the Bianchi identity, which is intimately connected with the zig zag symmetry, is established, for piecewise continuous contours, in the context of Polakov’s gauge field–string connection in the large ’t Hooft coupling limit, according to which the chromoelectric ‘string’ propagates in five dimensions with its ends attached on a Wilson loop in four dimensions. An explicit check in the wavy line approximation is presented.     

Green functions of vortex operators

We study the euclidean Green functions of the 't Hooft vortex operator, primarily for abelian gauge theories. The operator is written in terms of elementary fields, with emphasis on a form in which it appears as the exponential of a surface integral. We explore the requirement that the Green functions depend only on the boundary of this surface. The Dirac veto problem appears in a new guise. We present a two-dimensional “solvable model” of a Dirac string, which suggests a new solution of the veto problem. The renormalization of the Green functions of the abelian Wilson loop and abelian vortex operator is studied with the aid of the operator product expansion. In each case, an overall multiplication of the operator makes all Green functions finite; a surprising cancellation of divergences occurs with the vortex operator. We present a brief discussion of the relation between the nature of the vacuum and the cluster properties of the Green functions of the Wilson and vortex operators, for a general gauge theory. The surface-like cluster property of the vortex operator in an abelian Higgs theory is explored in more detail.     

The vortex free energy in the screening phase of the Z(2) Higgs model

The vortex free energy was proposed to distinguish between the confinement and the Higgs phase (in the sense of 't Hooft) in lattice gauge theory, when matter fields are present that transform according to an arbitrary representation of the gauge group. In this paper I consider the Z(2) Higgs model and calculate the vortex free energy in the screening part of the confining/screening phase of Fradkin and Shenker. The result does not agree with the expected behavior that corresponds to the structure of the phase diagram. Therefore the vortex free energy is no longer a good indicator for confinement when matter fields transform non-trivially under the center of the gauge group (such as Z(2) Higgs scalars).     

Some aspects of a unified approach to gauge,dual and Gribov theories

We consider quantum chromodynamics (QCD) with N_c colours and N_f flavours. Large N expansions for this theory are discussed and their advantages are pointed out, especially in relation to the possibility of unifying gauge, dual and Gribov theories of strong interactions. We first recall how the 1/N_c expansion of 't Hooft can be related to a dual loop expansion with a fixed coupling constant. We point out the necessity for quarkless (purely gluonic) bound states to appear and their importance in maintaining confinement at higher orders in 1/N_c. We show how non-orientable dual loops are reinterpreted in QCD and how a paradox appears when N_f is such that asymptotic freedom is lost. Some recent results of Cornwall and Tiktopoulos are analyzed in leading order in 1/N_c. We then introduce a 1/N expansion at ? ≡ N_f/N_c fixed and show how it is related to the hadronic topological expansion (TE). This allows an unambiguous definition of reggeon field theory concepts such as the bare pomeron and diffractive dissociation in QDC. We are able to relate the parameter ? to the clustering of hadronic final states into resonances. Decreasing ? corresponds to increasing cluster over gap size. Renormalization of the dual coupling constant as a function of ? is discussed and an apparent paradox is resolved. We are also able to shed some new light on the problem of f extinction in the TE.Finally, we compare our approach to other schemes trying to relate different aspects of hadron physics.     

Kinematics of periodic structures in SU(N) gauge theories

We study periodicity by use of the holonomy group. For a general field configuration, which spans the whole SU(N) group, we show that periodicity of gauge-invariant quantities implies that either the flux is quantized or that there is no flux at all. Finally we show that in a special gauge, the quantized 't Hooft flux can be expressed as a line integral over one of the components of the vector potential. The other components then have “Higgs-like” zeros, with winding numbers related to the 't Hooft flux.     

Scalar quantum chromodynamics in two dimensions and the parton model

We study SU(N) scalar quantum chromodynamics in two space-time dimensions in the large-N limit. This is the model of color gauge fields interacting with scalar quarks. We find that the consensual properties of four-dimensional QCD, i.e. infrared slavery, quark confinement, the charmonium picture, etc. are all realized. Moreover, the current in this model mimics nicely the behavior of the current in four-dimensional QCD, in contrast to the original model of 't Hooft.     

ZQCD and universal matching to domain walls

We discuss the one loop matching procedure in ZQCD. Universality and Casimir scaling leave – in terms of the 't Hooft coupling – just two combinations of parameters to be fixed numerically. These numbers are then the same for any number of colours.     

Explicit non-'t Hooft rational solutions of the Atiyah-Drinfeld-Hitchin-Manin instanton matrix equations for SU(2r)

The Atiyah-Drinfeld-Hitchin-Manin instanton matrix equations may be written as a pair, one quadratic and one linear. The explicit general solution of the quadratic equation is given in terms of rational functions of free parameters, thus reducing the ADHM equations to a single equation. For SU(2r), large families of explicit solutions, rational in free parameters, are constructed by this method. These new families are generally about twice as large as the previously known 't Hooft families, and contain non-'t Hooft solutions for every r ? 1.     

Instantons on a line

Witten's multi-instanton solutions are shown explicitly to be gauge equivalent to fields obtained using the ansatz developed by Corrigan, Fairlie, 't Hooft and Wilczek.