Approximate Bogomol'nyi-Prasad-Sommerfield states

We consider dimensionally reduced three-dimensional supersymmetric Yang-Mills-Chern-Simons theory. Although the N=1 supersymmetry of this theory does not allow local massive Bogomol'nyi-Prasad-Sommerfield (BPS) states, we find approximate BPS states which have nonzero masses that are almost independent of the Yang-Mills coupling constant and which are a reflection of the massless BPS states of the underlying N=1 super-Yang-Mills theory. The masses of these states at large Yang-Mills coupling are exactly at the n-particle continuum thresholds. This leads to a relation between their masses at zero and large Yang-Mills coupling.     

Cosmological models with Yang-Mills fields

Cosmological models with an SU(2) Yang-Mills field are studied. For a specific model with a minimally coupled Yang-Mills Lagrangian, which includes an arbitrary function of the second-order term and a fourth-order term, a corresponding reconstruction program is proposed. It is shown that the model with minimal coupling has no de Sitter solutions, for any nontrivial function of the second-order term. To get de Sitter solutions, a gravitational model with nonminimally coupled Yang-Mills fields is then investigated. It is shown that the model with non-minimal coupling has in fact a de Sitter solution, even in absence of the cosmological constant term.     

The classical Yang-Mills field of a quark source

The Coulomb solution of the classical Yang-Mills equations with external source is unstable for sufficiently strong coupling constant. We show that the Yang-Mills system has no other physically static, spherically symmetric solution. Some consequences for the quark confinement problem in non-abelian gauge theories are pointed out.     

Gauge fixing and renormalization in topological quantum field theory

We show how Witten's topological Yang-Mills and gravitational quantum field theories may be obtained by a straightforward BRST gauge fixing procedure. We investigate some aspects of the renormalization of the topological Yang-Mills theory. It is found that the beta function for the Yang-Mills coupling constant is not zero.     

On YM2 measures and area-preserving diffeomorphisms

For a given gauge group and compact Riemannian two-manifold, it is known that the associated Yang-Mills measure can be defined directly as a finitely additive measure on the space of connections, and this finitely additive measure is invariant with respect to SDiff, the group of all area-preserving diffeomorphisms of the surface. The first question we address is whether this symmetry essentially characterizes the projection of the Yang-Mills measure to the space of gauge equivalence classes. The proper formulation of this question entails the construction of an SDiff-equivariant equivariant completion of the space of continuous connections, such that the projection of the Yang-Mills measure to the space of gauge equivalence classes has a countably additive extension. We also consider the coupling of the Yang-Mills measure to determinants of Dirac operators. The basic problems are to prove that the coupled measure is absolutely continuous with respect to the background Yang-Mills measure, to find a reasonable formula for the Radon-Nikodym derivative, and to analyze the action of SDiff.     

Lattice Yang-Mills theory at nonzero temperature and the confinement problem

We discuss finite temperature lattice Yang-Mills theory with special attention to the confinement problem. The relationship between the confinement criteria of Wilson, Polyakov, and 't Hooft is clarified by establishing a string of inequalities between the corresponding string tensions. The close connection between finite temperature Yang-Mills models and spin models is exploited to obtain new and rather sharp upper bounds for the critical coupling constant above which there is confinement. This same analogy also allows us to establish infrared bounds for the gauge models that yield a lower bound for this critical coupling and thereby show the existence of a weak coupling regime without confinement at nonzero temperature in three or more space dimensions. Finally we discuss extension of our results to other forms of the lattice action, the Hamiltonian lattice models of Kogut and Susskind and 't Hooft'sN → ∞ limit.     

Operators with Large R-Charge in N=4 Yang-Mills Theory

It has been recently proposed that string theory in the background of a plane wave corresponds to a certain subsector of the N=4 supersymmetric Yang-Mills theory. This correspondence follows as a limit of the AdS/CFT duality. As a particular case of the AdS/CFT correspondence, it is a priori a strong-weak coupling duality. However, the predictions for the anomalous dimensions which follow from this particular limit are analytic functions of the 't Hooft coupling constant λ and have a well-defined expansion in the weak coupling regime. This allows one to conjecture that the correspondence between the strings on the plane wave background and the Yang-Mills theory works at the level of perturbative expansions. In our paper we perform perturbative computations in the Yang-Mills theory that confirm this conjecture. We calculate the anomalous dimension of the operator corresponding to the elementary string excitation. We verify at the two-loop level that the anomalous dimension has a finite limit when the R-charge J→∞ keeps λ/J² finite. We conjecture that this is true at higher orders of perturbation theory. We show, by summing an infinite subset of Feynman diagrams, under the above assumption, that the anomalous dimensions arising from the Yang-Mills perturbation theory are in agreement with the anomalous dimensions following from the string worldsheet sigma-model.     

A model regularizing gauge fields in the strong coupling region

We propose a model using an effective lagrangian for bilocal gauge extended fields, looking somehow as an average over all open strings with fixed end points. The locality of the action is retrieved as a range parameter, which acts as a regulator, shrinks, and the Yang-Mills theory is recovered in this limit. The system is invariant by rotation and translation, generalizes lattice gauge theories and allows regularized strong coupling expansions for Yang-Mills field theory.     

On the Yang-Mills mass gap problem

Theorem 4.1 of the author’s paper “Quantum Yang-Mills-Weyl dynamics in the Schroedinger paradigm”, RJMP 21 (2), 169–188 (2014) claims the relative ellipticity of cutoff Yang-Mills quantum energy-mass operators in von Neumann algebras with regular traces. This implies that the spectra of cutoff self-adjoint Yang-Mills energy-mass operators in a nonperturbative quantum Yang-Mills theory (with an arbitrary compact simple gauge group) are nonnegative sequences of the eigenvalues converging to +∞. The spectra are self-similar in the inverse proportion to the running coupling constant. In particular, they have self-similar positive spectral mass gaps. Presumably, this is a solution of the Yang-Mills Millennium problem. The present note shows that the fundamental spectral value of a cutoff quantum Yang-Mills energy-mass operator is the simple zero eigenvalue with the vacuum eigenvector. The direct proof (without von Neumann algebras) is based on the domination over the number operator (with simple fundamental eigenvalue) and the standard spectral variational principle.     

Optimizedδ — expansion and triviality or non-triviality of field theories

We use a very simple version of the optimized (linear)δ-expansion by scaling the free part of the Lagrangian with a variational parameter. This method is well suited to calculate the renormalized coupling constant in terms of the free one and the cutoff. One never has to calculate any new Feynman graphs but simply can modify existing results from the literature. We find that Φ ₄ ⁴ -theory as well as QED are free in the limit where the cutoff goes to infinity. In contrast to this, the structure of Yang-Mills theories enforces a special choice of the Lagrangian of theδ-expansion. Together with the change in the sign of theβ-function, this leads to a different behavior and allows Yang-Mills theory to become non trivial.     

Optimized�� �� expansion and triviality or non-triviality of field theories

We use a very simple version of the optimized (linear)??-expansion by scaling the free part of the Lagrangian with a variational parameter. This method is well suited to calculate the renormalized coupling constant in terms of the free one and the cutoff. One never has to calculate any new Feynman graphs but simply can modify existing results from the literature. We find that ?? ₄ ⁴ -theory as well as QED are free in the limit where the cutoff goes to infinity. In contrast to this, the structure of Yang-Mills theories enforces a special choice of the Lagrangian of the??-expansion. Together with the change in the sign of the??-function, this leads to a different behavior and allows Yang-Mills theory to become non trivial.     

Chiral U(1) anomaly in D=4 super Yang-Mills theory coupled to supergravity

We evaluate the chiral U(1) anomaly in D=4 supersymmetric Yang-Mills theory coupled to supergravity. We consider not only the minimal coupling between the gauge fields and fermions but also the interaction term which mixes the gravitino and the gaugino. We show that the mixing interaction gives new contributions to the anomaly.     

Chaotic behavior of confining lattice Gauge field configurations

We analyze the leading Lyapunov exponents of SU(2) Yang-Mills field configurations on the lattice which are initialized by quantum Monte-Carlo simulations. We find that configurations in the strong coupling phase at finite temperature are substantially more chaotic than in deconfinement.     

Gravitational Gauge Theory Developed Based on the Stephenson-Kilmister-Yang Equation

A Yang-Mills formulation of Einstein gravity with spin-affine connection as the dynamical variable of gravitational field is suggested based on the Stephenson-Kilmister-Yang (SKY) equation. A physically interesting property of the present formalism is that the Einstein field equation appears as a first-integral solution to the Yang-Mills type gravitational gauge field equation. The gravitational current density, the law of conservation and the gravitational gauge field strength in vierbein formulation are discussed. The present scheme could provide us with new insight into a possible way to include both Yang-Mills field and gravitational gauge field into one framework of generalized vierbein fields.     

Color screening by a Yang-Mills instability

The Coloumb field set up by a classical Yang-Mills particle is shown to be unstable above a critical coupling strength. The instability is related to color screening and represents a classical phase transition. Possible connections with the confinement problem are discussed.     

Multi-Regge form of gluon-exchange amplitudes in supersymmetric Yang-Mills theories

Within supersymmetric Yang-Mills theories, all effective vertices for the interaction of Reggeized gluons with particles were found both in the leading-logarithm approximation and in the next-to-leading-order approximation. The contributions of scalars to the eigenfunction of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) kernel for the adjoint representation of the gauge group and to impact factors similar to those in QCD were calculated. The impact factors that arise in supersymmetric Yang-Mills theories, but which are not present in QCD, are also found. Fulfillment of all bootstrap conditions, which guarantees the multi-Regge form of multiparticle amplitudes in supersymmetric Yang-Mills theories in the next-to-leading-order approximation, was proven. A method for testing bootstrap relations without calculating explicit expressions for vertices and impact factors was developed. This method was used to prove fulfillment of bootstrap conditions in theories featuring any number of fermions and scalar particles transformed according to any representations of the color group and involved in interactions with one another via Yukawa coupling.     

Stability in the Yang-Mills system with an external source

We investigate stability of the Coulomb solution in the Yang-Mills system under finite perturbations below the critical coupling strength. The role of recently discovered low-energy solutions is discussed.     

New anomaly-free theories beyond superstrings?

Many new anomaly-free configurations are found group-theoretically, assuming that they contain N = 1 supergravity and Yang-Mills matter with a simple group and allowing the existence of many Yang-Mills singlets (shadow matter).     

Fermion condensation in the field strength approach to non-abelian yang-mills theories

Within the field strength approach to Yang-Mills theories the fermionic sectors of gauge theories are bosonized for the SU(2) and SU(3) gauge group. The emerging effective meson theories are studied in the tree approximation. In this approximation the original minimal gauge coupling of the quarks to gluons is rendered into an effective local four-fermion interaction with non-trivial Lorentz and gauge structure. The Schwinger-Dyson equation is solved in the strong coupling limit and the quark condensates and constituent masses are evaluated.