Flat coordinates,topological Landau-Ginzburg models and the Seiberg-Witten period integrals

We study the Picard-Fuchs differential equations for the Seiberg-Witten period integrals in N = 2 supersymmetric Yang-Mills theory. For A-D-E gauge groups we derive the Picard-Fuchs equations by using the flat coordinates in the A-D-E singularity theory. We then find that these are equivalent to the Gauss-Manin system for two-dimensional A-D-E topological Landau-Ginzburg models and the scaling relation for the Seiberg-Witten differential. This suggests an interesting relationship between four-dimensional N = 2 gauge theories in the Coulomb branch and two-dimensional topological field theories.     

Hopping conductivity and weak localization in two-dimensional disordered systems

The low temperature dependence of hopping conductivity σ(T) in 2d-DS with weak localization of electron states is investigated. Hopping between neighbouring localized states is suggested and a power law for σ(T) is obtained. Coulomb effects are shown to be negligible for this process. The transition from the logarithmic T dependence and to the exponential Mott's law is discussed. The density distribution of localized electron states is proved to be exponential. The dielectric permeability ?′(ω) is found to be proportional to |ω|^?1 in the field of validity of the logarithmic corrections.     

Quark confinement in 2+1 dimensional pure Yang-Mills theory

We report some progress on the quark confinement problem in 2 + 1 dim. pure Yang-Mills theory, using Euclidean instanton methods. The instantons are regularized Wu-Yang ‘monopoles’, whose long range Coulomb field is screened by collective effects. Such configurations are stable to small perturbations unlike the case of singular, undressed monopoles. Using exact non-perturbative results for the 3-dim. Coulomb gas, where Debye screening holds for arbitrarily low temperatures, we show in a self-consistent way that a mass gap is dynamically generated in the gauge theory. The mass gap also determines the size of the monopoles. We also identify the disorder operator of the model in terms of the Sine-Gordon field of the Coulomb gas and hence obtain a dual representation whose symmetry is the centre ofSU(2).     

Exponentiation of leading infrared divergences in massless Yang-Mills theories

In the Coulomb gauge, the leading infrared divergences of massless theories occur in external line self-energy parts only. This fact, together with the gauge invariance of S-matrix elements, leads to a simple proof of exponentiation.     

Coulomb gauge vacua of SU(3) gauge theory

SU(3) gauge field theory is studied in the Coulomb gauge, and the topologically distinct, but gauge equivalent, vacuum configurations are analysed. Considering the gauge transformations of the form U ε U(2) ?SU(3)/U(2), we have obtained a new class of vacuum fields characterized by the topological quantum number η = ±1.     

Exact analysis of Coulomb gauge vacuum degeneracies in (2 + 1) dimensions

The solutions of the two-dimensional euclidean σ-model provide an infinite number of pure gauge field configurations satisfying the Coulomb gauge condition, in (2 + 1) dimensions. For vacuum gauge fields associated with finite action instanton solutions of the σ-model, we find that the winding number n configuration leads to n negative eigenvalues for the ghost operator, up to a finite calculable degeneracy.     

Meron configurations in the two-dimensional O(3) σ-model

Singular solutions of the Euclidean field equations for the O(3) σ-model in two dimensions are found. These are analogues of the meron field configurations that exist in four-dimensional gauge theories. Explicit multiple meron and anti-meron fields are constructed. It is shown that the long range meron-meron-anti-meron interactions are identical with those of a two-dimensional Coulomb gas. The geometrical meaning and the dynamics of the merons are discussed.     

Elliptic Hypergeometry of Supersymmetric Dualities II. Orthogonal Groups, Knots, and Vortices

We consider Seiberg electric-magnetic dualities for 4d ${\mathcal{N} = 1}$ SYM theories with SO(N) gauge group. For all such known theories we construct superconformal indices (SCIs) in terms of elliptic hypergeometric integrals. Equalities of these indices for dual theories lead both to proven earlier special function identities and new conjectural relations for integrals. In particular, we describe a number of new elliptic beta integrals associated with the s-confining theories with the spinor matter fields. Reductions of some dualities from SP(2N) to SO(2N) or SO(2N + 1) gauge groups are described. Interrelation of SCIs and the Witten anomaly is briefly discussed. Possible applications of the elliptic hypergeometric integrals to a two-parameter deformation of 2d conformal field theory and related matrix models are indicated. Connections of the reduced SCIs with the state integrals of knot theory, generalized AGT duality for (3 + 3)d theories, and a 2d vortex partition function are described.     

On quantum mechanics with extended supersymmetry and nonabelian gauge constraints

We analyze a weakly restricted general class of quantum mechanical models with at least four real supercharges and nonabelian gauge constraints. The innocent-looking restrictions lead automatically and exclusively to the quantum mechanics which are the dimensionally reduced counterparts of supersymmetric Yang-Mills field theories. This result provides in turn an independent proof that N = 1, N = 2 and N = 4 Yang-Mills fields are the only possible supersymmetric gauge field theories (without central charges) in four dimensions.     

Instantons and transverse pure gauge fields

The phenomenon of the ambiguity of the Coulomb gauge in Yang-Mills theories, first discovered by Gribov, is studied for the instanton solutions.It is shown that in the Coulomb gauge the instanton connects a non-vanishing transverse pure gauge field in the remote (Euclidean) past to an analogous one in the far future.     

Radial and Regge excitations in unified,grand unified and subconstituent models

Necessary group theoretic conditions for all elementary gauge bosons and fermions of an arbitrary renormalizable gauge theory to lie on Regge trajectories are reviewed. It is then argued that in properly unified gauge theories all particles of a given spin lie on Regge trajectories. This then implied that a properly unified gauge theory has no local U(1) factor groups, and no massive fermion singlets. A consideration of the general pattern of Regge and radial recurrences to be expected in quantum field theories suggests that the presence or absence of spin $\text{3}\text{2}$ quarks and/or leptons in the TeV region will provide crucial clues to enable one to distinguish between various classes of unified, grand unified, and subconstituent models. The correct interpretation of such excited fermions will require correlation with the Higgs boson mass and possible radial and Regge excitations of the weak vector bosons.     

Mean-field approach to Z(N) gauge systems with generalized action

The phase diagram of Z(N) lattice gauge theories with generalized action is examined in the mean-field approach. The phase diagram is well reproduced, with the exception of the Coulomb phase, which is absent. A previously identified mechanism that dynamically generates the Coulomb phase from quantum fluctuations is shown to give agreement with Monte Carlo data in four dimensions.     

Chemical potentials in gauge theories

One-loop calculations of the thermodynamic potential Ω are presented for temperature gauge and non-gauge theories. Prototypical formulae are derived which give Ω as a function of both (i) boson and/or fermion chemical potential, and in the case of gauge theories (ii) the thermal vacuum parameter A₀=const (A_μ is the euclidean gauge potential). From these basic abelian gauge theory formulae, the one-loop contribution to Ω can readily be constructed for Yang-Mills theories, and also for non-gauge theories.     

The propagator in the A0 = 0 gauge

We compute the Wilson loop in the A₀ = 0 gauge for abelian and non-abelian theories. We find to fourth order that only two choices for the longitudinal propagator are consistent with the results obtained in the Feynman and Coulomb gauges. In particular the principal value presciption does not work.     

States of one-dimensional Coulomb systems as simple examples of θ vacua and confinement

The one-dimensional Coulomb system is known to have equilibrium states with nonvanishing electric field. These states are shown here to be analogous, and related, to the vacua which have been discussed for gauge theories in two or more space-time dimensions. The system exhibits confinement of fractional charges, which we dicuss with the purpose of offering a simple example of the-vacua phenomenology. Precise relations and connections between one-dimensional Coulomb gases and two-dimensional Abelian gauge theories, and quantum-mechanical matter systems, are discussed.Supported in part by National Science Foundation Grant PHY-2825390 A01.     

Spontaneous symmetry breaking in massless field theories,finite temperature and criticality

We study the behaviour at finite temperature of massless field theories exhibiting spontaneously broken solutions. We establish the occurence of a phase transition of the first kind at some critical point T_c which can be calculated to any finite order in perturbation theory. Similarly, perturbative methods can be used for thermodynamic functions in all regions, including the critical region. For the case of a gauge theory, we demonstrate the gauge independence of the critical point, the thermodynamic potentials and the order parameter to all orders of perturbation theory.     

Structure constants for the osp(1|2) current algebra

We study the free field realization of the two-dimensional osp(1|2) current algebra. We consider the case in which the level of the affine osp(1|2) symmetry is a positive integer. Using the Coulomb gas technique we obtain integral representations for the conformal blocks of the model. In particular, from the behaviour of the four-point function, we extract the structure constants for the product of two arbitrary primary operators of the theory. From this result we derive the fusion rules of the osp(1|2) conformal field theory and we explore the connections between the osp(1|2) affine symmetry and the N = 1 superconformal field theories.     

Dynamics of N = 2 supersymmetric gauge theories in three dimensions

We study the structure of the moduli spaces of vacua and superpotentials of N = 2 supersymmetric gauge theories in three dimensions. By analyzing the instanton corrections, we compute the exact superpotentials and determine the quantum Coulomb and Higgs branches of the theories in the weak coupling regions. We find candidates for non-trivial N = 2 superconformal field theories at the singularities of the moduli spaces. The analysis is carried out explicitly for gauge groups U(N_c) and SU(N_c) with N _f flavors. We show that the field theory results are in complete agreement with the intersecting branes picture. We also compute the exact superpotentials for arbitrary gauge groups and arbitrary matter content.