Chern-Simons Theory, and Its Limiting Geometry and Topology

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

DMS-0203129 Stavros Garoufalidis Stavros Garoufalidis plans to study applications of quantum field theory to 3-dimensional topology and geometry. This work is motivated by the Chern-Simons path integral, its exact computation (in terms of quantum invariants of knots and 3-manifolds) and its perturbative expansion. Quantum field theory predicts that invariants of knots and 3-manifolds which are defined via cut-and-paste topology carry geometric information of the underlying structures; in particular of the fundamental group of a 3-manifold and its representations to Lie groups. A key question is to recover the geometric information (described in terms of surgery theory or Riemannian hyperbolic geometry) by taking an appropriate limit of quantum invariants. The above mentioned relation between physics and mathematics has deep roots in the past century. Mathematics has strongly influenced Physics, as is evident from the development of Mechanics, Quantum Mechanics and General Relativity, which describe our world in three quite different scales. Knotedness is a fundamental property that binds fundamental particles together, but also the DNA molecule. The most recent interaction between 3-dimensional quantum field theory and topology aims at describing knotedness in terms of geometric and topological language. Topology could be a unifying language to describe the building blocks of physics, but also of life itself.