E. Brezin

Exactly Solvable Field Theories of Closed Strings

Abstract Field theories of closed strings are shown to be exactly solvable for a central charge of matter fields c=1−6/m(m+1),m=1,2, 3, …. The two-point function χ(λ,N), in which λ is the cosmological constant and N−1 is the string coupling constant, obeys a scaling law χ (λ,N=N −(m+ 1 2 ) ⨍((λ c −λ)N m/(m+ 1 2 ) ) in the limit in which N−1 goes to zero and λ goes to a critical value λc; we have determined the universal non-linear differential equation satisfied by the function ⨍. From this equation it is found that a phase transition takes place for some finite value of the scaling parameter ( λ c −λ)N m/(m+ 1 2 ); this transition is a “condensation of handles” on the world sheet, characterized by a divergence of the averaged genus of the world sheets. The cases m=2,3 are elaborated in more details, and the case m=1, which corresponds to the embedding of a bosonic string in −2 dimensions, is reduced to explicit quadratures.

Finite size effects in phase transitions

We develop a systematic approach to the calculations of finite size effects in phase transitions. The method consists of constructing an effective hamiltonian for the homogeneous modes, obtained by tracing out all other degrees of freedom. These modes are obtained by averaging the order parameter over the finite dimensions of the system. These techniques, together with the renormalization group, lead to explicit calculations of universal finite size scaling functions, under the form of (2+e) or singular (4−e) expansions. Some simple universal results above the upper critical dimension are presented. Simple and universal properties of the rounding of first order transitions are derived.

Remarks about the existence of non-local charges in two-dimensional models

Abstract A simple derivation of the classical non-local conservation laws in two dimensions discovered by Luscher and Pohlmeyer is given. Several classes of models are shown to possess the same structure.

Scaling violation in a field theory of closed strings in one physical dimension

Abstract A one dimensional field theory of closed strings is solved exactly in a special double scaling limit, in which the string coupling 1/N goes to zero, the cosmological constant λ approches a critical value λc (corresponding to the limit of an infinitely large world sheet), and some nontrivial scaling parameter ξ(λc−λ, N) is fixed. The structure of singularities of the string susceptibility χ(λ, N) is analyzed, order by order in the topological (1/N) expansion, as well as nonperturbatively, for arbitrary ξ. It is shown that the “naive” scaling, which confirmed the work of Polyakov, Knizhnik and Zamolodchikov, of David, and of Distler and Kawai when the central charge is smaller than one, is violated by logarithmic corrections at every order of the topological expansion. Nonperturbative effects in 1/N arise through vacuum instabilities of this string field theory for any finite ξ.

Perturbation theory at large order. I. Theφ2Ninteraction

A new method for calculating the large orders of perturbation theory in quantum field theories has been discussed recently by Lipatov. We show that the same method applied to anharmonic oscillators in quantum mechanics allows one to rederive and generalize results previously obtained by Bender and Wu. We have also verified and generalized Lipatovs results to the case of an internal O(n) symmetry. These results show the divergence of the Wilson-Fisher epsilon expansion and indicate its Borel summability which is used for critical exponents. Similarly, the Callan-Symanzik functions for the phi/sup 4/ theory in three dimensions are characterized.

Universality of the correlations between eigenvalues of large random matrices

Abstract The distribution of eigenvalues of random matrices appears in a number of physical situations, and it has been noticed that the resulting properties are universal, i.e. independent of specific details. Standard examples are provided by the universality of the conductance fluctuations from sample to sample in mesoscopic electronic systems, and by the spectrum of energy levels of a non-integrable classical hamiltonian (the so-called quantum chaos). The correlations between eigenvalues, measured on the appropriate scale, are described in all those cases by simple gaussian statistics. Similarly numerical experiments have revealed the universality of these correlations with respect to the probability measure of the random matrices. A simple renormalization group argument leads to a direct understanding of this universality; it is a consequence of the attractive nature of a gaussian fixed point. Detailed calculations of these correlations are given for a general probability distribution (in which the logarithm of the probability is the trace of a polynomial of the matrix); the universality is shown to follow from an explicit asymptotic form of the orthogonal polynomials with respect to a non-gaussian measure. In addition it is found that the connected correlations, when suitably smoothed, exhibit, even when the eigenvalues are not in the scaling region, a higher level of universality than the density of states.

Characteristic Polynomials of Random Matrices

Abstract: Number theorists have studied extensively the connections between the distribution of zeros of the Riemann ζ-function, and of some generalizations, with the statistics of the eigenvalues of large random matrices. It is interesting to compare the average moments of these functions in an interval to their counterpart in random matrices, which are the expectation values of the characteristic polynomials of the matrix. It turns out that these expectation values are quite interesting. For instance, the moments of order 2K scale, for unitary invariant ensembles, as the density of eigenvalues raised to the power K2; the prefactor turns out to be a universal number, i.e. it is independent of the specific probability distribution. An equivalent behaviour and prefactor had been found, as a conjecture, within number theory. The moments of the characteristic determinants of random matrices are computed here as limits, at coinciding points, of multi-point correlators of determinants. These correlators are in fact universal in Dysons scaling limit in which the difference between the points goes to zero, the size of the matrix goes to infinity, and their product remains finite.

The ising model coupled to 2D gravity. A nonperturbative analysis

Abstract We apply recently developed techniques to give an exact nonperturbative solution of the two matrix realization of the Ising model coupled to 2D lattice gravity. We demonstrate that the Ising model is different from multicritical matter. We conjecture that multicritical matter is described by certain non-unitary minimal models.

Generalized non-linear σ-models with gauge invariance

In these lectures we shall give a brief description of a family of models which generalize the non-linear σ-model, and possess in addition a local gauge invariance without containing explicitly a gauge field.

UNIVERSAL SINGULARITY AT THE CLOSURE OF A GAP IN A RANDOM MATRIX THEORY

We consider a Hamiltonian