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During the past 15 years, quantum field theory and classical statistical mechanics have merged into a single field, and the need for nonperturbative methods for the description of critical phenomena in statistical mechanics as well as for problems in elementary particle physics are generally acknowledged. Such methods formed the central theme of the 1987 Cargese Advanced Study Institut. e on "Nonpert. urbat. ive Quantum Field Theory. " The use of conformal symmet. ry has been of central interest in recent years, and was a main subject at. t. he ASI. Conformal invariant quantum field theory describes statistical mechanical systems exactly at a critical point, and can be analysed to a remarkable ext. ent. by group t. heoretical methods. Very strong results have been obtained for 2-dimensional systems. Conformal field theory is also the basis of string theory, which offers some hope of providing a unified t. heory of all interactions between elementary particles. Accordingly, a number of lectures and seminars were presented on these two topics. After syst. ematic introductory lectures, conformal field theory on Riemann surfaces, orbifolds, sigma models, and application of loop group theory and Grassmannians were discussed, and some ideas on modular geometry were presented. Other lectures combined' traditional techniques of constructive quant. um field theory with new methods such as the use of index-t. heorems and infinite dimensional (Kac Moody) symmetry groups. The problems encountered in a quantum mechanical description of black holes were discussed in detail.
Sommario
Lectures Operator Methods in String Theory.- On the Fixed Point Formula and the Rigidity Theorem of Witten.- Non-Commutative Geometry.- Statistics of Fields,the Yang Baxter Equation And the Theory of Knots and Links.- Topological Actions in Two-Dimensional Quantum Field Theories.- Issues in Superstring Theory.- Lectures on Heterotic Strings and Orbifold Compactifications.- Quantum Gravity and Black Holes.- Supersymmetric quantum Fields and Infinite Dimensional Analysis.- Solution of the Lattice ?4Theory in 4 Dimensions.- Multigrid Methods in Quantum Field Theory.- to Conformal Invariant Quantum Field Theory in Two or More Dimensions.- Continuous Wilson Renormalization Group And the 2-D 0(n) Non-linear a-Model.- Numerical Simulations:Old and New Problems.- The Role of Locality in String Quantization.- SEMINARS On the BRST Structure of the Closed String And Superstring Theory.- The A-D-E Classification of Conformal Invariant Field Theories in Two Dimensions.- The Break down of Dimensional Reduction.- Vanishing Vacuum Energies for Nonsupersymmetric Strings.- Torsion Constraints and Super RiemannS urfaces.- Renormalization Theory for Use in Convergent Expansions of Euclidean Quantum Field Theory.- Power Counting and Renormalization in Lattice Field Theory.- Open Strings and their Symmetry Groups.- New Methods and Results in Conformal QFT2 and the String Idea.- On V. Pasquier A-D-E Models.- Generalization of the Sugawara Construction.- Conformal Field Theory at c=1.- Geometric realization of Conformal Field Theory on Riemann Surfaces.
