Principal Investigator: Albert Schwarz
The first examples of topological quantum field theories were constructed by PI in the late seventies. The importance of such theories became clear in late eighties after Witten's papers. Witten suggested a more general way of construction of topological quantum field theories and used it to find new interesting models. Among the most important topological theories are Chern-Simons theory, suggested by PI and Witten and solved by Witten, and Witten's topological sigma-models (A-model and B-model). Chern-Simons theory was very useful in knot theory. Topological sigma-models, especially mirror symmetry between A-and B-models, were used to obtain striking results in enumerative algebraic geometry. Topological version of gauge theories led to remarkable developments in the theory of four-dimensional and three-dimensional manifolds. One can say that topological quantum field theories opened the way for applications of ideas borrowed from physics to pure mathematics. However, these theories were used also as a powerful instrument in string/M-theory. It was shown by Witten that starting with any N=2 superconformal theory one can construct topological quantum field theories by means of so called twisting. This means, in particular,that string theory with N=2 superconformal target has topological sectors, that can be analyzed more easily than complete theory. Topological sectors served as a testing ground for many important conjectures. From the other side, for any critical string we can obtain two-dimensional topological theory considering matter fields together with ghosts. More generally, two-dimensional topological quantum field theory obeying certain conditions can be "coupled to gravity"; the resulting theory can be considered as string theory.
The present proposal is devoted to some problems related to topological quantum field theories. In particular, we would like to relate recent developments in knot theory to the theory of topological strings; it seems that this relation should lead to essential progress both in knot theory and in string theory. We are planning to apply methods of number theory to topological string theory. Our projects are interdisciplinary; their realization will be important both for mathematics and theoretical physics.
