One of the most important developments in 3-dimensional topology over the last 25 years was the discovery of the so-called quantum invariants of links and 3-manifolds such as the Jones, HOMFLYPT and Kauffman polynomials and the Reshetikhin-Turaev invariants. These invariants were described combinatorially, their relation to classical 3-dimensional topology was mysterious, and the only conceptual explanation for their existence came from physics: Witten constructed a family of topological quantum field theories, whose path integral partition functions were conjectured to coincide with quantum invariants. Recently, Khovanov, Ozsvath ans Szabo discovered the categorification of polynomial link invariants: to a link one associates a chain complex of graded vector spaces, whose graded Euler characteristic coincides with a given polynomial invariant. Also, to a cobordism between two links one associates a chain map between the link categorification complexes. Most of the known categorification constructions are either combinatorial or `semi-combinatorial' and their intrinsic 3-dimensional origin remains mysterious. The goal of this proposal is to extend the categorification to a wider class of link polynomial invariants and, more importantly, to the Reshetikhin-Turaev invariants of 3-manifolds. Rozansky suggests two approaches to this problem. First, he intends to use combinatorial methods based upon commutative algebra (matrix factorizations) and upon the topology of virtual links. These methods have shown promise in the categorification of the HOMFLYPT and SO(2N) Kauffman polynomials. The second approach is based on the methods of quantum field theory. Categorification implies that Witten's Chern-Simons theory is a dimensional reduction of a yet unknown 4-dimensional theory. The candidate theories were considered in the papers of Gukov, Kapustin and Witten, but the correct theory has not been found yet. A construction of this topological quantum field theory may lead to combinatorial categorification constructions for link and manifold invariants. It should also provide a conceptual explanation for the existence of cobordism invariants.
3-dimensional topology deals with classification of knots, links and 3-dimensional surfaces. The main approach to this problem is to construct topological invariants, that is, the numbers or polynomials that can be assigned to a topological object, computed readily from its presentation (say, from the picture of a knot) and used to distinguish these objects. A major breakthrough it 3-dimensional topology came about 25 years ago with the discovery of a wide variety of the so-called quantum invariants. Surprisingly, these invariants were rooted in physics: they come from a special kind of a 3-dimensional quantum field theory. Thus quantum invariants provide an important bridge between topology and advanced physical theories. A more recent development came in the form of categorification: it turned out that quantum invariants were just dimensions of special vector spaces associated with topological objects. From the physics point of view, this means that the 3-dimensional quantum field theory describing quantum invariants is the dimensional reduction of the yet unknown 4-dimensional theory (the idea of dimensional reduction is familiar in string theory which requires a 9-dimensional space, 6 extra dimensions being wrapped up tightly in order to make them invisible for a general observer). Rozansky proposes to search for the 4-dimensional physical theory related to categorification and to use the topology-physics relation in order to get a better understanding of both fields of science.
