A CATEGORICAL CONSTRUCTION OF 4D TOPOLOGICAL QUANTUM FIELD THEORIES

Abstract

In recent years, it has been discovered that invariants of three dimensional topological objects, such as links and three dimensional manifolds, can be constructed from various tools of mathematical physics, such as Von Neumann algebras [1], Quantum Groups [2], and Rational Conformal Field Theories [3]. Since these different structures lead to the same 3D invariants, it is natural to wonder how they are related. A fundamental connection is that they all give rise to the same special tensor categories, which act as expressions of “quantum symmetry” in the very different physical settings [4]. It is therefore very important that the 3D invariants can all be reconstructed from a tensor category with the appropriate properties, called a “modular tensor category (MTC) [3,5].” The invariants have the property that they factorize nicely if the manifold or link is cut along a surface, which we express by saying that they form a Topological Quantum Field Theory, or TQFT. Thus the theorem proven in [3] states that a modular tensor category gives rise to a 3D TQFT. This represents a remarkable convergence with [6], in which it was realized that a suitable categorical structure would give rise to 3D topological information, although without the examples from physics. The original suggestion to look for TQFT’s appears in (not mathematically rigorous) work of Atiyah [7] and Witten [8]. Atiyah devotes more attention to the 4D than the 3D situation, and poses the question whether the two are related; or more concretely, whether the invariants of Donaldson and Jones are related. Witten actually works in the 4D situation, and formally suggests that Donaldson’s invariant can be fitted into a 4D TQFT. At this point, there is no mathematical construction of the 4D TQFT envisioned in [7] and [8]. Donaldson-Floer theory has so far eluded the efforts of the strongest of analysts [9]. It is therefore clear that a 4D construction of a TQFT paralleling the 3D categorical one would have very great implications for mathematics. As we shall mention in the conclusion section of this paper, there may very well be important physical implications also. The construction we describe in this paper is a considerable step in this direction. Following a suggestion of Ooguri [10], we offer an expression which