Introduction to TQFTs

The study of Topological Quantum Field Theories (TQFTs) establishes new exciting relations between mathematics and physics connecting many of the most advanced ideas in topology, geometry and physics. On the one side, the most sophisticated topological invariants of three and four-dimensional manifolds are encountered. On the other side, the most recent achievements in Quantum Field Theory play a salient role especially in explicitly calculating these topological invariants. It is remarkable to observe that precisely these low dimensions in which Topology has shown to present important features are the dimensions where many interesting quantum field theories are renormalizable.

Though connections between Quantum Physics, Topology and Geometry can be traced back to the fifties, it is in the eighties when a new and unprecedented kind of relation between them took place. In 1982 E. Witten considered so-called supersymmetric sigma models in two dimensions and rewrote Morse theory in the language of Quantum Field Theory. Furthermore, he constructed out of those models a refined version of Morse theory known nowadays known as Morse-Witten theory. Some years later A. Floer reformulated Morse-Witten theory providing a rigorous mathematical background.

This trend in which some mathematical structure is first constructed by Quantum Field Theory methods and then reformulated in a rigorous mathematical ground constitutes one of the tendencies in these new relations between Topology, Geometry and Physics. The influence of M. Atiyah on E. Witten in the fall of 1987 culminated with the construction by the latter of the first Topological Quantum Field Theory (TQFT) in January 1988. The Quantum Theory turned out to be a "twisted" version of supersymmetric Yang-Mills gauge theory. This theory, whose existence was conjectured by M. Atiyah, is related to Donaldson invariants for four-manifolds, and it is known nowadays as Donaldson-Witten theory.

In 1988 E. Witten formulated also two models which have been of fundamental importance in two and three dimensions: topological sigma models and Chern-Simons gauge theory. The first one can be understood as a twist of the supersymmetric sigma model considered by Witten in his work on Morse theory, and is related to Gromov invariants. The second one is not the result of a twist but a model whose action is the integral of the Chern-Simons form. In this case the corresponding topological invariants are knot and link invariants as the Jones polynomial and its generalizations. TQFTs provide a new point of view to study topological invariants of 3- and 4-dimensional manifolds.

Topological Quantum Field Theories are at the moment in the heart of fundamental research in mathematics being related to, among other things, knot theory and the theory of four-manifolds in Algebraic Topology, and to the theory of moduli spaces in Algebraic Geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for their work related to topological field theory and its relations to Topology and Geometry.

This advanced course emphasizes especially the interplay between Geometry, Topology and Physics and aims to engage PhD students and postdoctoral researchers in these new exciting developments together with international experts in the field.

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