State Sum Invariants of 3 Manifolds and Quantum 6j Symbols

IN THE 1980s the topology of low dimensional manifolds has experienced the most remarkable intervention of ideas developed in rather distant areas of mathematics. In the 4dimensional topology this process was initiated by S. Donaldson. He applied the theory of the Yang-Mills equation and instantons to study 4-manifolds. In dimension 3 a similar breakthrough was made by V. Jones. He discovered his famous polynomial of links in 3-sphere S3 via an astonishing use of von Neumann algebras. It has been soon understood that deep notions of statistical mechanics and quantum field theory stay behind the Jones polynomial (see [8], [16], [18]). The relevant basic algebraic structures turn out to be the Yang-Baxter equation, the R-matrices, and the quantum groups (see [S], [6], [7]). This viewpoint, in particular, enables one to generalize the Jones polynomial to links in arbitrary compact oriented 3-manifolds (see [ 131). In this paper we present a new approach to constructing “quantum” invariants of 3-manifolds. Our approach is intrinsic and purely combinatorial. The invariant of a manifold is defined as a certain state sum computed on an arbitrary triangulation of the manifold. The state sum in question is based on the so-called quantum 6j-symbols associated with the quantized universal enveloping algebra U,&(C)) where CJ is a complex root of 1 of a certain degree z > 2 (see [9]). The state sum on a triangulation X of a compact 3-manifold M is defined, roughly speaking, as follows. Assume for simplicity that M is closed, i.e. 8M = @. We consider “colorings” of X which associate with edges of X elements of the set of colors (0, l/2, 1, . . . , (I 2)/2). H avm a coloring of X we associate . g with each 3-simplex of X the q-6j-symbol