Dynamical supersymmetry breaking in a finite field theoretical model

We present a supersymmetric field theory in two or three space-time dimensions with an internal symmetry of the O(N) type. In the large-N limit the model is finite and supersymmetry is spontaneously broken. The fields representing the order parameters of the broken supersymmetry phase acquire dynamics through quantum corrections. In particular the Goldstone fermion is a zero-mass fermionic bound state.     

Lifting a conformal field theory from d-dimensional flat space to (d+1)-dimensional AdS space

A quantum field theory on anti-de Sitter space can be constructed from a conformal field theory on its boundary Minkowski space by an inversion of the holographic mapping. The resulting theory is defined by its Green functions and is conformally covariant. The structure of operator product expansions is carried over to AdS space. We show that this method yields a higher spin field theory HS(4) from the minimal conformal O(N) sigma model in three dimensions.     

BPHZL renormalization of 1/N expansion and critical behaviour of the three-dimensional chiral field

A modified soft mass renormalization scheme for 1/N expansion of the three dimensionalO(N)-invariant chiral field in the high- and low-temperature phases, as well as at the critical point, is constructed, free of infrared divergencies in each separate diagram. Generalized quantum chirality identities for composite operators are derived, from which the renormalizability of the model follows. The approach formulated here is applied to a rigorous analysis of the universal critical behaviour of theN-component chiral field in three dimensions.     

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.     

Non-trivial saddle points and band structure of bound states of the two-dimensional O(N) vector model

We discuss O(N) invariant scalar field theories in 0 + 1 space-time dimensions (quantum mechanics) and in 1 + 1 space-time dimensions (field theory). Combining ordinary “Large N” saddle point techniques and simple properties of the diagonal resolvent of one-dimensional Schrödinger operators we find non-trivial (non-constant) solutions to the saddle point equations of these models in addition to the saddle point describing the ground state of the theory. In the “Large N” limit these saddle points are exact for the quantum mechanical case, but only approximate in the two-dimensional theory. In the latter case they are the leading contributions to the time evolution kernel at short times, or equivalently, the leading contribution to the high temperature expansion of partition function stemming from space dependent static configurations in case of the Euclidean theory. We interpret these novel saddle points as collective O(N) singlet excitations of the field theory, each embracing a host of finer quantum states arranged in O(N) multiplets, in an analogous manner to the band structure of molecular spectra.     

The effective potential,vacuum tunneling and the 1/N expansion

The problem of vacuum decay in quantum field theory, when the instability is the result of radiative corrections, is discussed. The large-N expansion of the O(N) quartic model in four (Euclideanized) dimensions is analysed in detail and it is shown that, although the effective potential has no lower bound, tunnelling solutions of the usual type (instantons) do not exist. This is shown to happen because that expansion of the action which begins with the effective potential is inappropriate for the kind of field configurations in question. The relevance of this result to the related problem in the Salam-Weinberg model is also discussed.     

Kosterlitz-thouless transition and self-duality in three dimensions

A class of self-dual globally symmetric Z_N models in three dimensions is presented. The limit N → ∞ is a type of anisotropic U(1) model (XY model) dual to a gas of integer point charges, interacting via a logarithmic potential in three dimensions. The latter is, at low temperature, nothing but a sine-Gordon theory with an anisotropic, logarithmic propagator. It therefore has a low-temperature Kosterlitz-Thouless phase and KT phase transition to a massive phase.The relation of the U(1) model to the thermodynamics of a helical magnet along the ferromagnetic-helical phase boundary in zero applied field (or to the smectic A to amectic C phase boundary in a liquid crystal) is indicated.     

Monopoles and topological field theory

We present a topological quantum field theory for magnetic monopoles in an SU(N) Yang-Mills-Higgs model. This field theory is obtained by gauge fixing the topological action defining the monopole charge. This work extends to the three-dimensional case the quantization of invariant polynomials in four dimensions. We choose the Bogomolny self-duality equations as gauge conditions for the magnetic monopole topological field theory. In this way the geometrical equation discussed e.g. in Atiyah and Hitchin's work are recovered as ghost equations of motion. We give the cocycles of the corresponding topological symmetry. In the N→∞ limit interesting phenomena occur. The functional integration is forced to cover only the moduli space and the role of the ghosts stemming from the gauge fixing process is to provide a smooth semiclassical approximation.     

New principles for string/membrane unification

The target space theory of the N = (2,1) heterotic string may be interpreted as a theory of gravity coupled to matter in either 1 + 1 or 2 + 1 dimensions. Among the target space theories in 1 + 1 dimensions are the bosonic, type II, and heterotic string world-sheet field theories in a physical gauge. The (2 + 1)-dimensional version describes a consistent quantum theory of supermembranes in 10 + 1 dimensions. The unifying framework for all of these vacua is a theory of (2 + 2)-dimensional self-dual geometries embedded in 10 + 2 dimensions. There are also indications that the N = (2,1) string describes the strong-coupling dynamics of compactifications of critical string theories to two dimensions, and may lead to insights about the fundamental degrees of freedom of the theory.     

Non-perturbative effects in supersymmetric sigma models

A scheme for taking account of crucial non-perturbative effects in a quantum field theory ahead of developing perturbation series for it is extended here from bosonic to supersymmetric sigma models in two dimensions. The scheme writes field products in the lagrangian in terms of suitably defined normal ordered products and VEVs of field products. The exact values of the latter can be inferred directly from the symmetry and supersymmetry Ward identities of the theory, so that a lagrangian with scale breaking effects explicitly treated, is available for use in perturbation theory. The supersymmetric sigma model on the manifold S^N is used to illustrate many aspects of the scheme.     

Non-abelian traveling wave solutions for massless QCD2

The field equations for quantum chromodynamics in 1 + 1 dimensions (QCD₂) with massless fermions are shown to admit classical non-abelian traveling wave solutions. In this case, the field equations reduce to the linear Frenet-Serret equations for a curve in the three-space corresponding to an SU(2) subalgebra of the SU(N) gauge group.     

Large N classical equations and their quantum significance

We show that the large N limits of a wide variety of vector models may be obtained by studying the classical equations of motion. In particular, we derive a constraint which allows us to choose solutions of the classical field equations which directly give the correlation functions of N → ∞ quantum system. Models studied here include quantum mechanics on a sphere, two-dimensional linear and nonlinear O(N) field theories and the CP^N model.     

Quantum free Yang-Mills on the plane

We construct a free-probability quantum Yang-Mills theory on the two dimensional plane, determine the Wilson loop expectation values, and show that this theory is the N=∞ limit of U(N) quantum Yang-Mills theory on the plane. Our model provides an example of a stochastic geometry, motivated by quantum field theory, based on free probability theory.     

Integrability of a family of quantum field theories related to sigma models

A method is introduced for constructing lattice discretizations of large classes of integrable quantum field theories. The method proceeds in two steps: The quantum algebraic structure underlying the integrability of the model is determined from the algebra of the interaction terms in the light-cone representation. The representation theory of the relevant quantum algebra is then used to construct the basic ingredients of the quantum inverse scattering method, the lattice Lax matrices and R-matrices. This method is illustrated with four examples: The sinh-Gordon model, the affine sl(3) Toda model, a model called the fermionic sl(2|1) Toda theory, and the N=2 supersymmetric sine-Gordon model. These models are all related to sigma models in various ways. The N=2 supersymmetric sine-Gordon model, in particular, describes the Pohlmeyer reduction of string theory on AdS₂×S², and is dual to a supersymmetric non-linear sigma model with a sausage-shaped target space.     

Deformations of Fermionic Quantum Field Theories and Integrable Models

Considering the model of a scalar massive Fermion, it is shown that by means of deformation techniques it is possible to obtain all integrable quantum field theoretic models on two-dimensional Minkowski space which have factorizing S-matrices corresponding to two-particle scattering functions S ₂ satisfying S ₂(0) = ?1. Among these models there is for example the Sinh-Gordon model. Our analysis provides a complement to recent developments regarding deformations of quantum field theories. The deformed model is investigated also in higher dimensions. In particular, locality and covariance properties are analyzed.     

Progress in lattice gauge theory

Two topics of lattice gauge theory are reviewed. They include string tension and β-function calculations by strong coupling Hamiltonian methods for SU(3) gauge fields in 3 + 1 dimensions, and a 1/N-expansion for discrete gauge and spin systems in all dimensions. The SU(3) calculations give solid evidence for the coexistence of quark confinement and asymptotic freedom in the renormalized continuum limit of the lattice theory. The crossover between weak and strong coupling behavior in the theory is seen to be a weak coupling but non-perturbative effect. Quantitative relationships between perturbative and non-perturbative renormalization schemes are obtained for the O(N) nonlinear sigma models in 1 + 1 dimensions as well as the range theory in 3 + 1 dimensions. Analysis of the strong coupling expansion of the β-function for gauge fields suggests that it has cuts in the complex 1/g²-plane. A toy model of such a cut structure which naturally explains the abruptness of the theory's crossover from weak to strong coupling is presented. The relation of these cuts to other approaches to gauge field dynamics is discussed briefly.The dynamics underlying first order phase transitions in a wide class of lattice gauge theories is exposed by considering a class of models-P(N) gauge theories - which are soluble in the N → ∞ limit and have non-trivial phase diagrams. The first order character of the phase transitions in Potts spin systems for N #62; 4 in 1 + 1 dimensions is explained in simple terms which generalizes to P(N) gauge systems in higher dimensions. The phase diagram of Ising lattice gauge theory coupled to matter fields is obtained in a $\text{1}\text{N}$ expansion. A one-plaquette model (1 time-0 space dimensions) with a first-order phase transitions in the N → ∞ limit is discussed.     

Supersymmetric mini-superspace

A mini-superspace model for quantum cosmology which possesses local supersymmetry is found by reducing N = 1 supergravity in a k = + 1 Friedmann model, and its classical and quantum solutions are discussed. In a more general class of supersymmetric models containing a scalar field, it may still be possible to compute the Hartle-Hawking state.     

Topological matter in four dimensions

Topological models involving matter couplings to Donaldson-Witten theory are presented. The construction is carried out using both the topological algebra and its central extension, which arise from the twisting of N = 2 supersymmetry in four dimensions. The framework on which the construction is based is constituted by the superspace associated to these algebras. The models show new features of topological quantum field theories which could provide either a mechanism for topological symmetry breaking, or the analog of two-dimensional mirror symmetry in four dimensions.     

Interfacial contribution to cluster free energy

The Monte Carlo technique of “overlapping distributions” is used for computing directly the free energy difference F_{N + 1} ?F_N between two clusters containing respectively N + 1 and N solute atoms, in the square and the simple cubic Ising model with nearest neighbour interactions. High accuracy results are obtained within reasonable computer times. The capillarity approximation gives a good fit to the data, provided the following is taken into account: (i) the specific bulk and surface energies are given their macroscopic equilibrium values; (ii) a curvature correction to the surface specific energy has to be introduced: it is positive for two dimensions and negative for three dimensions; (iii) a size independent term is to be added in three dimensions; it may be viewed as a corner contribution; (iv) the coefficient of the logarithmic term is given its more recent value. When introduced in the master equation which describes the kinetics of the cluster population, within the simplifying assumptions of the classical nucleation theory, good agreement is found, for the shapes of the cluster size distributions, with the numerical experiments on the kinetic Ising model in two dimensions. However, the time scales of both computations do not match linearly. Possible reasons for this discrepancy are discussed.