Math and Physics

It is argued that quantum gravity has an interpretation as a topological field theory provided a certain constraint from the path intergral measure is respected. The constraint forces us to couple gauge and matter fields to gravity for space - time dimensions different from 3. We then discuss possible models which may be relevant to our universe.

We give an introductory review of topological strings and their application to various aspects of superstrings and supersymmetric gauge theories. This review includes developing the necessary mathematical background for topological strings, such as the notions of Calabi-Yau manifold and toric geometry, as well as physical methods developed for solving them, such as mirror symmetry, large N dualities, the topological vertex and quantum foam. In addition, we discuss application...

What follows is a broad-brush overview of the recent synergistic interactions between mathematics and theoretical physics of quantum field theory and string theory. The discussion is forward-looking, suggesting potentially useful and fruitful directions and problems, some old, some new, for further development of the subject. This paper is a much extended version of the Snowmass whitepaper on physical mathematics [1].

This is a non-technical talk given at the Sixth Marcel Grossman Meeting on General Relativity, Kyoto, Japan in June 1991. Some developments in string theory over the last six years are discussed together with their qualitative implications for issues in quantum gravity.

In these lecture notes we explain a connection between Yang-Mills theory on arbitrary Riemann surfaces and two types of topological field theory, the so called $BF$ and cohomological theories. The quantum Yang-Mills theory is solved exactly using path integral techniques. Explicit expressions, in terms of group representation theory, are obtained for the partition function and various correlation functions. In a particular limit the Yang-Mills theory devolves to the topologic...

Witten introduced classical Chern-Simons theory to topology in 1989, when he defined an invariant for knots in 3-manifolds by an integral over a certain infinite-dimensional space, which up to today have not been entirely understood. However, they motivated lots of interesting questions and results in knot theory and low-dimensional topology, as well as the development of entirely new fields. These are lecture notes for a course in Lie groups and Chern-Simons Theory aimed a...

Topological quantum field theories can be used to probe topological properties of low dimensional manifolds. A class of these theories known as Schwarz type theories, comprise of Chern-Simons theories and BF theories. In three dimensions both capture the properties of knots and links leading to invariants characterising them. These can also be used to construct three-manifold invariants. Three dimensional gravity is described by these field theories. BF theories exist also in...

We discuss physical systems with topologies more complicated than simple gaussian linking. Our examples of these higher topologies are in non-relativistic quantum mechanics and in QCD.

Contribution to the Proceedings of the International Congress of Mathematicians 1994. We review recent developments in the physics and mathematics of Yang-Mills theory in two dimensional spacetimes. This is a condensed version of a forthcoming review by S. Cordes, G. Moore, and S. Ramgoolam.

We study a relation between topological quantum field theory and the Kodama (Chern-Simons) state. It is shown that the Kodama (Chern-Simons) state describes a topological state with unbroken diffeomorphism invariance in Yang-Mills theory and Einstein's general relativity in four dimensions. We give a clear explanation of "why" such a topological state exists.