Toshitake Kohno

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

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

Quantum and homological representations of braid groups

By means of a description of the solutions of the KZ equation using hypergeometric integrals we show that the homological representations of the braid groups studied by Lawrence, Krammer and Bigelow are equivalent at generic complex values to the monodromy of the KZ equation with values in the space of null vectors in the tensor product of Verma modules of sl2(C).

Elliptic KZ system, braid group of the torus and Vassiliev invariants

Abstract We study finite type invariants for links in the product of the torus and the unit interval by means of the monodromy of the elliptic KZ equation. We show that the weight system for the chord diagrams on the torus defined for any representation of the Lie algebra sl(N, C ) and a certain projectively flat connection on the torus can be integrated to a link invariant by using the elliptic KZ system.

Bar complex of the Orlik–Solomon algebra

Abstract Let A be an arrangement of complex hyperplanes and M A the complement of the union of hyperplanes in A . The Orlik–Solomon algebra of A determines a subcomplex of the de Rham complex of the loop space of M A , which is called the bar complex of the Orlik–Solomon algebra. The dual of this complex is isomorphic to the tensor algebra of the homology of M A equipped with a derivation arising from the product structure of the Orlik–Solomon algebra. Based on this construction we give an explicit description of Chens iterated integrals of logarithmic forms depending only on the homotopy class of a loop.

Novikov homology, jump loci and Massey products

Let X be a finite CW complex, and ρ: π1(X) → GL(l, ℂ) a representation. Any cohomology class α ∈ H1(X, ℂ) gives rise to a deformation γt of ρ defined by γt (g) = ρ(g) exp(t〈α, g〉). We show that the cohomology of X with local coefficients γgen corresponding to the generic point of the curve γ is computable from a spectral sequence starting from H*(X, ρ). We compute the differentials of the spectral sequence in terms of the Massey products and show that the spectral sequence degenerates in case when X is a Kähler manifold and ρ is semi-simple.If α ∈ H1(X, ℝ) one associates to the triple (X, ρ, α) the twisted Novikov homology (a module over the Novikov ring). We show that the twisted Novikov Betti numbers equal the Betti numbers of X with coefficients in the local system γgen. We investigate the dependence of these numbers on α and prove that they are constant in the complement to a finite number of proper vector subspaces in H1(X, ℝ).

THE VOLUME OF A HYPERBOLIC SIMPLEX AND ITERATED INTEGRALS

We express the volume of a simplex in spherical or hyperbolic spece by iterated integrals of differential forms following Schläfli and Aomoto. We study analytic properties of the volume function and describe the differential equation satisfied by this function.

THREE-MANIFOLD INVARIANTS DERIVED FROM CONFORMAL FIELD THEORY AND PROJECTIVE REPRESENTATIONS OF MODULAR GROUPS

The purpose of this paper is to give a brief review on the authors approach to define 3-manifold invariants using representations of modular groups appearing in conformal field theory on Riemann surfaces. After the Wittens discovery of 3-manifold invariants based on Chern-Simons gauge theory. Reshetikhin and Turaev gave a Dehn surgery formula for Witten invariant, which was studied extensively by Kirby and Melvin and others. In this paper the authors describe the nature of these representations. The representations of the mapping class groups discussed in this paper are projectively linear and we write down the associated 2-cocyle in a combinatorial way. The authors also show that these representations are unitary. Combining this unitarity with the comparison of our approach and the Dehn surgery formula, the authors give a lower estimate of the Heegaard genus of a closed orientable 3-manifold by means of the Witten invariant. In this paper the authors discuss the case of SU(2) WZW model at level K.

Homological Representations of Braid Groups and the Space of Conformal Blocks

We compare homological representations of the braid groups and the monodromy representations of the KZ connection by means of hypergeometric integrals. Then we discuss a relationship to the space of conformal blocks.

Higher holonomy of formal homology connections and braid cobordisms

We construct a representation of the homotopy 2-groupoid of a manifold by means of Chen’s formal homology connections. By using the idea of this 2-holonomy map, we describe a method to obtain a representation of the category of braid cobordisms.