Topological quantum field theories: towards the cobordism hypothesis

Abstract

Topological quantum field theories (TQFTs) have been a past century attempt to axiomatize quantum field theories from physics. While the underlying theory for quantum mechanics had been fully developed in terms of Hilbert spaces and operator theory, the analytic basis of quantum field theories remained unsettled, and several approaches are still nowadays being considered. Surprisingly or not, the introduction of these new theories was received with high interest not only by physicists but also by the mathematical community. TQFTs became a recurrent field of study in mathematics mainly because of the interest they had from a topological standpoint. While TQFTs became less popular over the time in physics, the mathematical approach has increasingly attracted the attention of researchers because of its natural drift towards the homotopy theory of higher categories, showing some outstanding results such as the Cobordism Hypothesis, formulated by John Baez and James Dolan and recently proved by Jacob Lurie. The original definition of TQFTs was first given by Michael Atiyah's in 1988 as a generalization to category theory of group representations. A TQFT was defined as a functor from the category of cobordisms (smooth manifolds with boundary and additional structure), to the category of vector spaces (originally Atiyah formulated the definition in terms of $\Lambda -modules for a ring \Lambda$). The definition, as Atiyah himself stated, was inspired in the previous work done by Edward Witten on super-symmetry and Graeme Segal on conformal theory. A successful understanding of TQFTs in low dimensions was rapidly achieved, and several theories in dimensions $\leq$ 4 were developed. Baez and Dolan, foreseeing a near future, suggested in 1995 that a more complex theory was behind the classical formulation, and although their work lacked formality, it delimited essential guidelines of study. The mathematical importance of TQFTs has not passed unnoticed, and at least four Fields Medals have been given to this date to mathematicians for research related to TQFTs: Simon Donaldson, Vaughan Jones, Edward Witten and Maxim Kontsevich.