Non-perturbative double scaling limits

Recently, the author has proposed a generalization of the matrix and vector models approach to the theory of random surfaces and polymers. The idea is to replace the simple matrix or vector (path) integrals by gauge theory or non-linear sigma model (path) integrals. We explain how this solves one of the most fundamental limitation of the classic approach: we automatically obtain non-perturbative definitions in non-Borel summable cases. This is exemplified on the simplest possible examples involving O(N) symmetric non-linear sigma models with N-dimensional target spaces, for which we construct (multi)critical metrics. The non-perturbative definitions of the double scaled, manifestly positive, partition functions rely on remarkable identities involving (path) integrals.Comment: 18 pages, one figur

Super Yang-Mills, Matrix Models and Geometric Transitions

I explain two applications of the relationship between four dimensional N=1 supersymmetric gauge theories, zero dimensional gauged matrix models, and geometric transitions in string theory. The first is related to the spectrum of BPS domain walls or BPS branes. It is shown that one can smoothly interpolate between a D-brane state, whose weak coupling tension scales as Nc or 1/gs, and a closed string solitonic state, whose weak coupling tension scales as Nc^2 or 1/gs^2. This is part of a larger theory of N=1 quantum parameter spaces. The second is a new purely geometric approach to sum exactly over planar diagrams in zero dimension. It is an example of open/closed string duality.Comment: 11 pages, 2 figures, .cls files included; to appear in the proceedings of the Strings 2004 conference in Pari

Topologically Linked Polymers are Anyon Systems

We consider the statistical mechanics of a system of topologically linked polymers, such as for instance a dense solution of polymer rings. If the possible topological states of the system are distinguished using the Gauss linking number as a topological invariant, the partition function of an ensemble of N closed polymers coincides with the 2N point function of a field theory containing a set of N complex replica fields and Abelian Chern-Simons fields. Thanks to this mapping to field theories, some quantitative predictions on the behavior of topologically entangled polymers have been obtained by exploiting perturbative techniques. In order to go beyond perturbation theory, a connection between polymers and anyons is established here. It is shown in this way that the topological forces which maintain two polymers in a given topological configuration have both attractive and repulsive components. When these opposite components reach a sort of equilibrium, the system finds itself in a self-dual point similar to that which, in the Landau-Ginzburg model for superconductors, corresponds to the transition from type I to type II superconductivity. The significance of self-duality in polymer physics is illustrated considering the example of the so-called $4-plat$ configurations, which are of interest in the biochemistry of DNA processes like replication, transcription and recombination. The case of static vortex solutions of the Euler-Lagrange equations is discussed.Comment: 7 pages, 1 figure, LaTeX +Revtex

Using textual clues to improve metaphor processing

In this paper, we propose a textual clue approach to help metaphor detection, in order to improve the semantic processing of this figure. The previous works in the domain studied the semantic regularities only, overlooking an obvious set of regularities. A corpus-based analysis shows the existence of surface regularities related to metaphors. These clues can be characterized by syntactic structures and lexical markers. We present an object oriented model for representing the textual clues that were found. This representation is designed to help the choice of a semantic processing, in terms of possible non-literal meanings. A prototype implementing this model is currently under development, within an incremental approach allowing step-by-step evaluations. \footnote{This work takes part in a research project sponsored by the AUPELF-UREF (Francophone Agency For Education and Research)}Comment: 3 pages, single LaTeX file, uses aclap.st

Field Theories on the Poincar\'e Disk

The massive scalar field theory and the chiral Schwinger model are quantized on a Poincar\'e disk of radius $\rho$. The amplitudes are derived in terms of hypergeometric functions. The behavior at long distances and near the boundary of some of the relevant correlation functions is studied. The exact computation of the chiral determinant appearing in the Schwinger model is obtained exploiting perturbation theory. This calculation poses interesting mathematical problems, as the Poincar\'e disk is a noncompact manifold with a metric tensor which diverges approaching the boundary. The results presented in this paper are very useful in view of possible extensions to general Riemann surfaces. Moreover, they could also shed some light in the quantization of field theories on manifolds with constant curvature scalars in higher dimensions.Comment: 22 pages, Plain TeX+harvma