3-manifold Invariants from Cosets

We construct unitary modular categories for a general class of coset conformal field theories based on our previous study of these theories in the algebraic quantum field theory framework using subfactor theory. We also consider the calculations of the corresponding 3-manifold invariants. It is shown that under certain index conditions the link invaraints colored by the representations of coset factorize into the products of the the link invaraints colored by the representations of the two groups in the coset. But the 3-manifold invariants do not behave so simply in general due to the nontrivial branching and selection rules of the coset. Examples in the parafermion cosets and diagonal cosets show that 3-manifold invariants of the coset may be finer than the products of the 3-manifold invariants associated with the two groups in the coset, and these two invariants do not seem to be simply related in some cases, for an example, in the cases when there are issues of “fixed point resolutions”. In the later case our framework provides a mathematical understanding of the underlying unitary modular categories which has not been obtained by other methods. Introduction This is the fourth paper in a series of papers on algebraic coset conformal field theories (cf. [X1], [X2] and [X3]), devoted to the construction and calculation of of 3-manifold invariants associated with a general class of coset conformal field theories. The history of quantum invariants started from the striking discovery of Vaughan Jones of a new polynomial invariant of classical knots and links (cf. [J2]) by using his subfactor theory (cf. [J1]). The quantum 3-manifold invariants were predicted by E. Witten (cf. [Wi]) motivated by a quantum field theory interpretation of Jones polynomial. There are several mathematical approaches to the construction of quantum 3-manifold invariants. The original approach [RT] (also cf. [KM]) is based on theory of quantum groups at roots of unity, which has a subtle tensor product structure (cf. [A]) among a distinguished class of finite dimensional modules. The work of [TW1], [TW2] and [W2] relies on ideas from quantum groups and subfactors. There are also work (cf. [EK1], [EK2] and references therein) based on subfactors but there are no general calculations of the invariants. The approach we take in this paper (cf. [R] and [FRS]) is very different from that of [RT], [TW1], [TW2] and 1991 Mathematics Subject Classification. 46S99, 81R10. Typeset by AMS-TEX 1