Mean field and Monte Carlo studies of SU(N) chiral models in three dimensions

SU(N) chiral models defined on three-dimensional cubic lattices arestudied using mean field and Monte Carlo techniques. Mean field theory predicts first-order transitions for all finite N greater than two. The mean field estimates of the transition temperature and discontinuity of the order parameter are in good agreement with computer simulations for N = 3 and 4. The N → ∞ limit of mean field theory has a first-order phase transition.     

Thermodynamic properties of the φ4 lattice field theory near the Ising limit

For a d-dimensional φ⁴ lattice field theory consisting of N spins with nearest-neighbor interactions, the partition function is transformed for large bare coupling constant λ into an Ising-like system with additional neighbor interactions. For d = 2 a mean field approximation is then used to estimate the difference in critical temperature between the lattice φ⁴ field theory and its Ising limit (λ = ∞). Expansions are obtained for the susceptibility and specific heat. The critical exponents are shown to be identical to the Ising exponents.     

CPN−1 coupled to gauge fields: Mean field theory and monte carlo results

We define a two parameter lattice field theory which interpolates between the O (2N) Heisenberg model, pure U(1) gauge theory, and a lattice version of the CP^N?1 model. The phase diagram in space-time dimension d=4 is obtained by Monte Carlo simulation on a 4⁴ lattice, and the nature of the phases is discussed in mean field approximation.     

On the functional integration measure of supersymmetric Yang-Mills theories

For the simplest N = 1 supersymmetric Yang-Mills theory, we construct a bosonic field transformation which linearizes the Yang-Mills action in each topological sector and whose jacobian equals the product of Fadeev-Popov and Matthews-Salam determinants up to cubic order at least. Some possible implications of this result for the maximally extended N = 4 theory are discussed.     

The energy tensor of matter in the projectve theory of relativity

In the projective theory of relativity the 5-dimensional field equation \(_{\mu \nu } \) and the resulting equation of motion Tμυ_;μ = 0 are investigated. There Tμυ stands for the 5-dimensional tensor of macroscopic matter. The 4-dimensional field equations and equation of motion obtained by projection are a generalization of Einstein's theory of general relativity and Maxwell's electrodynamics, involving a scalar field φ.They contain a single constant φ₀.The weak field approximation is investigated for the case of an ideal fluid and leads to Newton's mechanics, including Newton's gravitational law, and to Maxwell's electrodynamics. For the constant φ₀ one obtains the approximate value φ₀c⁴/γ_N with Newton's gravitational constant γ_N.For homogeneous and isotropic cosmological models consisting of matter only the general solution for the radius K of curvature is given. This solution is independent of the equation of state of matter For a pure dust universe the general solution for the scalar field φ is given. For a closed universe a power law φ ?K^?1 is valid which leads to Mach's principle. The calculation of the age of a closed universe yields over 7×10⁹y,if one uses mean values of the present cosmological data.     

Higher order corrections to the Lipatov asymptotics in the ϕ<Superscript>4</Superscript> theory

Higher orders in perturbation theory can be calculated by the Lipatov method [1]. For most field theories, the Lipatov asymptotics has the functional form ca ^N Γ(N+b (N is the perturbation theory order); relative corrections to this asymptotics have the form of a power series in 1/N. The coefficients of higher order terms of this series can be calculated using a procedure analogous to the Lipatov approach and are determined by the second instanton in the field theory in question. These coefficients are calculated quantitatively for the n-component ?⁴ theory under the assumption that the second instanton is (i) a combination of elementary instantons and (ii) a spherically asymmetric localized function. A technique of two-instanton computations, as well as the method for integrating over rotations of an asymmetric instanton in the coordinate state, is developed.     

Observations on the moduli space of superconformal field theories

Some aspects of the moduli space of superconformal field theories are discussed. It is helpful to consider the conformal field theory as a background for propagation of strings and to exploit the space-time interpretation. Using this point of view we show that the metric on the moduli space of N = 4 superconformal field theory with c = 6 is locally that of O(20,4)/O(20) × O(4). We also discover some properties of the moduli space of N = 2 superconformal field theories with c = 9. Particular examples of these conformal field theories are sigma models on four- and six-dimensional Calabi-Yau spaces. Therefore, we can use this technique to learn about the moduli space of these spaces. For c = 6 we recover the known moduli space of K₃. Our analysis of the c = 9 system leads to a new coupling in four dimensional supergravity. As a by-product, we prove that gauge couplings cannot depend on the moduli of N = 1 space-time supersymmetric compactifications.     

Time evolution of a quantum mechanical maser model

The time evolution of the Dicke maser model describing N spins ($\text{s =}\text{1}\text{2}$) interacting by a dipole coupling with one mode of an electromagnetic field, is studied for finite N. The mean photon number and its mean square deviation can be calculated as functions of time for various initial states. For not too large times, these quantities show a periodic behavior given by elliptic functions.     

On the unitary representations of N = 2 superconformal theory

The Kac formula for superconformal dimensions (generalized to N = 2) is further developed (compared to a previous article). A list of discrete values of the central charge for which unitary representations are expected to exist is proposed. For several of these, unitarity is checked by computer. For two values, unitarity is proven by providing explicit fermionic representations. For one of those values, the N = 2 theory coincides with a sub theory of one of the known unitary N = 1 theories, thus extending a similar situation between N = 0 and N = 1.     

Symmetry conserving generalisation of Hartree Fock Bogoliubov theory

We derive a generalisation of the HFB equations which conserve particle number. This is achieved in using the equation of motion method or alternatively the Green's function technique. The price we have to pay is that there is not only one mean field for the particle numberN but a set of coupled mean field equations for the whole bandN, N±2,N±4... Nevertheless we think that our theory is a quite interesting variant in comparison with the conventional projection technique. We apply our theory to simple models and find that the results are excellent.     

Structure of higher order corrections to the Lipatov asymptotic form

High orders of perturbation theory can be calculated by the Lipatov method, whereby they are determined by saddle-point configurations, or instantons, of the corresponding functional integrals. For most field theories, the Lipatov asymptotic form has the functional form ca ^NΓ(N+b) (N is the order of perturbation theory) and the relative corrections to it are series in powers of 1/N. It is shown that this series diverges factorially and its high-order coefficients can be calculated using a procedure similar to the Lipatov one: the Kth expansion coefficient has the form const[ln(S ₁/S ₀)]^?K Γ(K+(r ₁? r ₀)/2), where S ₀ and S ₁ are the values of the action for the first and second instantons of this particular field theory, and r ₀ and r ₁ are the corresponding number of zeroth-order modes; the instantons satisfy the same equation as in the Lipatov method and are assumed to be renumbered in order of their increasing action. This result is universal and is valid in any field theory for which the Lipatov asymptotic form is as specified above.     

The weak-coupling phase of lattice spin and gauge models

We analyze the lattice weak-coupling (w.c.) expansion of O(N), CP^N?1 and chiral spin models, and of large-N reduced chiral and gauge models.We find that the w.c. expansion always agrees with mean field results, whenever comparable, for arbitrary space-time dimensions, and that the expansion of the reduced models agrees with that of the original ones. However, w.c. results disagree with one-dimensional large-N and (old and new) exact results. We explain this phenomenon as a failure of the analytic continuation from higher dimensions that defines lattice w.c. perturbation theory for massless models (even if infrared singularities always cancel).We use an improved version of the mean field (m.f.) technique suitable for reduced models. We compute the m.f. approximation of chiral models and use this result to determine the large-d (m.f.) behaviour of reduced gauge models, finding agreement with standard Wilson theory results.We give a new characterization of large-N chiral models in terms of the single-link integral for the adjoint representation of SU(N).     

A new approach to the relaxation process in gravitational systems

A reformulation of the kinetic theory of N-body gravitational systems, retaining periodic trajectories in the mean field, leads to an intrinsically non-markovian evolution equation, and a √N dependence of the relaxation time. Explicit calculations are carried out for one-dimensional systems.     

Effective Field Theory for Fermi Systems in a Large N Expansion

A system of fermions with short-range interactions at finite density is studied using the framework of effective field theory. The effective action formalism for fermions with auxiliary fields leads to a loop expansion in which particle-hole bubbles are resummed to all orders. For spin-independent interactions, the loop expansion is equivalent to a systematic expansion in 1/N, where N is the spin-isospin degeneracy g. Numerical results at next-to-leading order are presented and the connection to the Bose limit of this system is elucidated.     

An improved mean field approach to the phase structure in ZN gauge theory in four dimensions

We extend the mean field approximation scheme to include the effect of fluctuation of the gauge field. As a consequence, we successfully obtain for the Z_N theory (N > 5) the phase transition which separates the Coulomb phase from the ordered phase, as well as that separating the Coulomb and disordered phases. The former transition shows characteristics of higher-order phase transitions.     

Equivalent lagrangians in large N field theories

The auxiliary field method is generalized to any O (N)-invariant theory with non-polynomial interactions. In non-supersymmetric theories, two lagrangians with and without an auxiliary field are shown to be equivalent to the leading order of the 1/N expansion. In supersymmetric theories, these two lagrangians are equivalent to all orders of the 1/N expansion. The lagrangian with an auxiliary field is solvable in the 1/N expansion.     

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 φ4 lattice field theory as an asymptotic expansion about the Ising limit

For a d-dimensional φ⁴ lattice field theory consisting of N spins, an asymptotic expansion of expectations about the Ising limit is established in inverse powers of the bare coupling constant λ. In the thermodynamic limit (N → ∞), the expansion is expected to be valid in the noncritical region of the Ising system.     

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.