Keeping in mind the several models of M(atrix) theory we attempt to understand the possible structure of the topological M(atrix) theory " underlying " these approaches. In particular we raise the issue about the nature of the structure of the vacuum of the topological M(atrix) theory and how this could be related to the vacuum of the electroweak theory. In doing so we are led to a simple Topol...

Introduction The notes give a survey of the basics of the following topological field theories: • The Chern-Simons gauge theory on 3-manifolds, its renormalization, geometric quantization, computation of partition functions by surgery, and relation with Jones Polynomials • Twisted N = 2 supersymmetric Yang-Mills theory on 4-manifolds, its reduction to N = 1 theory with a mass gap, and computati...

We explore some aspects of 4D supergravity theories and F-theory vacua that are parallel to structures in the space of 6D theories. The spectrum and topological terms in 4D supergravity theories correspond to topological data of F-theory geometry, just as in six dimensions. In particular, topological axion-curvature squared couplings appear in 4D theories; these couplings are characterized by v...

This paper traces the growing role of categories and n-categories in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts which manifest themselves in Feynman diagrams, spin networks, string theory, loop quantum gravity, and topological quantum field theory. Our chronology ends around 2000, with just a taste of later developments such as open...

Within the Quantum Action Principle framework we show the perturbative renormalizability of previously proposed topological lagrangianà la Witten-Fujikawa describing polymers, then we perform a 2 loop computation. The theory turns out to have the same pre-dictive power of De Gennes theory, even though its running coupling constants exhibit a very peculiar behaviour. Moreover we argue that the t...

Ideas from the theory of topological stability of smooth maps are trans ported into the controlled topological category For example the controlled topologi cal equivalence of maps is discussed These notions are related to the classi cation of manifold approximate brations and manifold strati ed approximate brations In turn these maps form a bundle theory which can be used to describe neighborho...

Topological indices are the numerical value associated with chemical constitution purporting for correlation of chemical structure with various physical properties, chemical reactivity or biological activity. Graph theory is a delightful playground for the exploration of proof techniques in Discrete Mathematics and its results have applications in many areas of sciences. One of the useful indic...