Topological invariants for 3-manifolds using representations of mapping class groups I

Abstract

LET A4 be a closed orientable 3-manifold. The purpose of this paper is to define topological invariants 4,(M) parametrized by K = 1,2,. . . using representations of the mapping class group of an orientable surface. In [27], Witten introduced new topological invariants for 3manifolds based on quantum field theory and motivated by Witten’s Dehn surgery formula, Reshetikhin and Turaev [I93 gave a formula expressed by the Jones polynomial and its relatives, by means of representations of the quantized universal enveloping algebra in the case q is a root of unity. Our approach using a Heegaard decomposition of a 3-manifold is difTerent from theirs. Let Xr denote a closed oricntablc surface of genus y. First, WC consider a pants decomposition of C, and WC dcnotc by Y its dual graph, which is a trivalent graph. Given a positive intcgcr K called a Icvel, WC introduce a finite dimensional complex vector space Z,(y) whose basis is in one-to-one correspondence with the set of admissible weights /: edge(y) + (0, l/2, * . . , K/2} satisfying the Clebsch-Gordan condition and the algebraic constraintj(c,) +/(cL) +/(cJ) IS K for any edges c,, c2 and c, meeting at a vertex. This vector space appears as a combinatorial description of the space of conformal blocks for SU(2) Wess-Zumino-Witten model at level K (see [23]). Let yr and Y2 be trivalent graphs associated with markings of C,. By means of the fusing matrices describing the holonomy of the Knizhnik-Zamolodchikov equation, we obtain a canonical isomorphism Z,(Yr) z Z,(y2). A detailed study of the monodromy of the Knizhnik-Zamolodchikov equation was pursued by Tsuchiya and Kanie [22]. It is known that the representations of the braid groups appearing as the monodromy of the Knizhnik-Zamolodchikov equation provide the Jones polynomial and its relatives (see [13], [IS] and [22]). More recently, Drinfel’d established a relation between the monodromy and the quantized universal enveloping algebras in a general situation (see [6]). using the notion of quasi-Hopf algebras. In [is], Moore and Seiberg wrote down a series of polynomial equations among fusing matrices, braiding matrices and so-called switching operators expected from the viewpoint of the consistency conditions in conformal field theory. The holonomy of the Knizhnik-Zamolodchikov equation gives solutions to these polynomial equations, and as a consequence we obtain projectively linear representations of the mapping class group