Noncompact N=2 supergravity

A massive spin-one multiplet with central charge is coupled to N=2 supergravity. Compared to conventional gauge fields the anomalous magnetic moment of the spin-one particles is of the opposite sign. The construction of this theory is based on an N=2 supersymmetric gauge theory associated with the noncompact group SO(2,1). As a byproduct we present a convenient expression for the N=2 Einstein-Yang-Mills lagrangian.     

Large N expansion for Yukawa-type theories

4 dimensional Yukawa-type theories with an internal SU(N) symmetry group are studied in the large N limit by means of path integral techniques. For a simplified model where the bosonic field transforms as a singket under SU(N), we explicitly solve the gap equation and shiw that the symmetry is spontaneously broken.     

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.     

Finiteness, 1/N expansion and supersymmetry breaking

The possibility of spontaneous supersymmetry breaking in the limit of large N, in models with an internal O(N) symmetry, is demonstrated by constructing an explicit example in two dimensions. The model is finite and this is shown to be important for the supersymmetry breaking. A general criterion for finiteness of scalar superfield theories in two dimensions is given. Finally, the generalization of our results to three dimensions, and their relevance to four-dimensional models, is discussed.     

Migdal approximation and 1/N expansion in QCD

We show that the Migdal approximation for the renormalization group in gauge theories in exact in the leading order in $\text{1}\text{N}$, above a critical value of the coupling constant. A systematic expansion in $\text{1}\text{N}$ is proposed. The string tension is calculated in the leading order in $\text{1}\text{N}$ in a theory without fermions.     

Chiral symmetry,the 1/N expansion and the SU(N) thirring model

In the two-dimensional SU(N) Thirring model, the 1/N expansion seems to predict spontaneous breaking of the continuous chiral symmetry. This is impossible in two-dimensions. Reasoning along the lines of Berezinski, Kosterlitz and Thouless for the two-dimensional XY model, we argue that, in fact, rather than showing long-range order, $\text{〈}\text{ψ}\text{ψ(x)}\text{ψ}\text{ψ(0)〉}$ vanishes in this model as |x|^?1/N at large |x|. The 1/N expansion is, in fact, a rather good guide to the properties of this model.     

Nonconvergence of the 1/N expansion for SU(N) gauge fields on a lattice

We present specific examples that demonstrate the non-convergence of the 1/N expansion for the lattice theory of SU(N) gauge fields.     

On the large N limit of conformal field theory

Following recent advances in large N matrix mechanics, I discuss here the free (Cuntz) algebraic formulation of the large N limit of two-dimensional conformal field theories of chiral adjoint fermions and bosons. One of the central results is a new affine free algebra which describes a large N limit of affine Lie algebra. Other results include the associated free-algebraic partition functions and characters, a free-algebraic coset construction, free-algebraic construction of , free-algebraic vertex operator constructions in the large N Bose systems, and a provocative new free-algebraic factorization of the ordinary Koba-Nielsen factor.     

Non-perturbative effects and infrared renormalons within the 1/N expansion of the O(N) non-linear sigma model

We analyze the structure of the Borel transform of the two-dimensional O(N) non-linear σ model within its $\text{1}\text{N}$ expansion. We check the existence of IR singularities (IR renormalons) and the presence of non-perturbative terms which organize themselves in an operator expansion à la Shifman-Vainshtein-Zakharov. We prove that renormalons cancel between the different terms of the operator expansion, so that there is a well-defined resummation procedure of the perturbative series. We suggest that this mechanism provides a general solution of the IR renormalons problem for massless UV free field theories.     

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.     

Large N saddle formulation of quadratic building block theories

I develop a large N saddle point formulation for the broad class of “theories of quadratic building blocks”. Such theories are those in which the sums over internal indices are contained in quadratic building blocks, e.g., φ² = Σ_{a = 1}^Nφ^aφ^a. The formulation applies as well to fermions, derivative coupling and non-polynomial interactions. In a related development, closed Schwinger-Dyson equations for Green functions of the building blocks are derived and solved for large N.     

The external field problem in the large N limit of QCD

Several problems in lattice gauge theories such as mean field theory or the few plaquette problem lead to the evaluation of the properties of one link in an external matrix source. This problem is solved here in the large N limit. There are two phases characterized by a single parameter, the average value of the inverse of the modulus of the eigenvalues of the external source. The third derivative of the free energy is discontinuous at the transition point.     

Critical behavior of colored tensor models in the large N limit

Colored tensor models have been recently shown to admit a large N expansion, whose leading order encodes a sum over a class of colored triangulations of the D-sphere. The present paper investigates in details this leading order. We show that the relevant triangulations proliferate like a species of colored trees. The leading order is therefore summable and exhibits a critical behavior, independent of the dimension. A continuum limit is reached by tuning the coupling constant to its critical value while inserting an infinite number of pairs of D-simplices glued together in a specific way. We argue that the dominant triangulations are branched polymers.     

SO(2, 1) algebra and the large N expansion in quantum mechanics

We discuss the large N expansion in quantum mechanics using an algebraic procedure based on a Holstein-Primakoff representation of the well-known SO(2, 1) algebra. Both spherically and axially symmetric potentials are studied. The method is explicitly illustrated for the family of potentials $\text{V =}\text{ω}{}_0{}^2\text{r}{}^2\text{2}\text{+ 2νr}{}^{\mathrm{2\nu }}$ as well as the hydrogen atom in a uniform magnetic field. In the latter case, the first non-trivial iteration of the present perturbative scheme yields accurate results for the energy levels, even for strong magnetic field intensities. Further generalizations and applications are outlined.     

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.     

Moment method and the Schrödinger equation in the large N limit

A new technique is presented for solving the Schrödinger equation in the framework of the 1/N expansion. Based on recursion relations satisfied by moments of the coordinate operator, this method which allows to compute energy levels and wavefunctions is applied to four examples: the harmonic oscillator, the rotating harmonic oscillator, a linear plus Coulomb potential and a logarithmic one.     

N = 1 superspace formulation of N = 2 and N = 4 super-Yang-Mills models with central charge

The component models of N = 2 and N = 4 supersymmetric Yang-Mills theories of Sohnius, Stelle and West are reformulated in terms of N = 1 superfields. The non-supersymmetric constraints are supersymmetrized generalizing the linear multiplet in the presence of the non-abelian gauge superfield and (in the N = 4 case) a doublet of chiral superfields. The extended supersymmetry transformations preserving constraints are explicitly given in terms of N = 1 superfields. We are able to introduce the constraints back into the lagrangian using superfield Lagrange multipliers. The on-shell equivalence of this formulation with the formulation of Fayet with one (for N = 2) and three (for N = 4) chiral superfields is shown. The abelian N = 2 model is worked out to show the connection between full superspace treatment and the N = 1 superfield formulation.