Categorification and Double Categorification of Quantum Topological Invariants of Links and 3-Manifolds

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An interpretation of quantum invariants of links in 3-manifolds within the framework of classical topology presented a challenge for a long time. The most convincing explanation of their nature came from Witten's work which related these invariants to quantum Chern-Simons theory. However the reason why Jones and HOMFLY-PT polynomials were polynomials in q rather than formal power series in (q-1), as a Quantum Field Theory would suggest, and what is the meaning of the coefficients of these polynomials remained a mystery. A breakthrough came from Khovanov's categorification construction: it turned out that a quantum polynomial is an Euler characteristic of a Z-graded homology associated by a combinatorial construction to a link. An extension of this result to the Witten-Reshetikhin-Turaev (WRT) invariant of links in 3-manifolds is a challenging problem, in part because the WRT invariant is not exactly a polynomial of q and is defined only for q being a root of 1. Recently Khovanov and Rozansky categorified the "stable" polynomial part of the WRT invariant of links in the product of a 2-sphere with a circle. Their construction uses the derived categories of modules over Khovanov's algebras H_n. Rozansky will try to extend this result further to general 3-manifolds. He conjectures that this might be done by deforming the algebras H_n into A-infinity algebras, thus reducing their Z-grading to a periodic Z_r grading which would correspond to the associated parameter q being the root of 1. A similar trick worked to construct a categorification of the SU(N) HOMFLY-PT polynomial from its 2-variable version. In addition to categorifying combinatorially the WRT invariant, Rozansky will try to categorify Khovanov's categorificaiton construction of the Jones polynomial. This idea is based on a similarity between the objects used in the Kamnitzer-Cautis version of Khovanov's categorification and the 2-category associated to a holomorphic symplectic manifold in the joint work of Kapustin, Rozansky and Saulina.The discovery of quantum invariants such as the Jones and HOMFLY-PT polynomials of links in a 3-sphere and the Witten-Reshetikhin-Turaev invariant of colored links in a 3-manifold opened a new chapter in 3-dimensional topology. In contrast to the Alexander polynomial which was very efficient in establishing the topological properties of links, the relation between these new invariants and classical topology is indirect. It seems that the purpose of quantum invariants is to establish deep links between 3-dimensional topology and other branches of Mathematics and Quantum Field Theory (QFT). Khovanov's categorification of the Jones polynomial was an important step in this direction: it showed that algebraic geometry could be "married" to 3-dimensional topology. Witten suggests that a superstring-inspired 6-dimensional QFT links together Khovanov homology and Langlands duality. By using the methods of homological algebra, Rozansky will try to extend Khovanov's categorification program from links in a 3-sphere to links in general 3-manifolds. He will also try to raise categorification by one level through associating a category rather than a homology to a link. If true, this might suggest that the underlying QFT is 7-dimensional rather than 6-dimensional.