Invariants of 3-manifolds associated with quantum groups and Verlinde's formula
Reshetikhin-Turaev invariants of closed 3-manifolds (and links in such 3- manifolds) were obtained from link invariants associated with the category generated by \(r-1\) irreducible representations of \(U_ q (\text{sl} (2))\) at \(r\)-th root of unity. The construction is based on the fact that any closed connected oriented 3-manifold is a result of Dehn surgery on \(S^ 3\) along a framed link. As an application of these invariants, the author constructs a projectively linear representation of \(\text{SL} (2, \mathbb{Z})\) realized as the mapping class group of the torus. With any basis vector \(e_ i\) of the representation space he associates a solid torus \(U_ i\) which has a link in the interior coloured by one of \(r-1\) irreducible representations of \(U_ q (\text{sl} (2))\). Gluing such two solid tori \(U_ i\) and \(U_ j\) by an element \(X\) of the mapping class group of the torus, one obtains a closed 3-manifold with a link, whose invariant is a matrix element \(X_{ij}\) of the representation. This representation was discovered by Kac and Peterson (1984) to describe the modular property of character of the affine Lie algebra and was also used by Kohno to define invariants of 3-manifolds. The Verlinde formula takes the form of a relation between the matrix elements of generator \(S \in \text{SL} (2, \mathbb{Z})\) in this representation. A proof of this formula based on the computation of invariants is given. The author also constructs invariants of link and 3-manifolds using `Ising model category' with three irreducible objects.
Reshetikhin-Turaev invariants of closed 3- manifolds
scientific article; zbMATH DE number 548926
scientific article