Summary of Research

1. Braid groups, KZ equation and quantum groups

A description of the linear representations of the braid groups appearing as the monodromy of Knizhnik-Zamolodchikov (KZ) equation in terms of quantum groups was first discovered in my work around 1986. KZ equation is a differential equation satisfied by n-point functions of conformal field theory on the Riemann sphere with the gauge symmetry of affine Lie algebras. Our result describes the commutation relation of vertex operators in conformal field theory. Later on, around 1990, Drinfeld established the relation between the monodromy of KZ equation and the universal R matrix in more general framework and introduced the important notion of quasi Hopf algebras.

2. Monodromy representations of conformal field theory and topological invariants of 3-manifolds

By generalizing the above work, an explicit description of the holonomy of the projectively flat vector bundle on the moduli space of Riemann surfaces in conformal field theory was given in terms of quantum groups. I established an expression of Witten's 3-manifold invariants based on Heegaard splittings of 3-manifolds and the above representations of mapping class groups. As an application of such invariants, we obtained a lower bound for classical topological invariants such as Heegaard genus of 3-manifolds and tunnel numbers of knots. This method was used to determine the tunnel numbers of certain knots, which had not been accecible by other methods. In the case of SU(n), these invariants are parametrized by the rank n and the level k of the conformal field theory. We showed the level-rank duality of Witten's 3-manifold invariants, which is a duality between the theory of rank n with level k and that of rank k with level n.

3. Vassiliev invariants and Chern-Simons perturbation theory

By using a method similar to the bar construction for the cohomology of loop spaces, we constructed a map from the complex of chord diagrams to the de Rham complex on the space of knots. This method was applied to investigate the integral representations of the Vassiliev invariants for knots. In the case of the pure braid group, the Vassiliev invariants are completely described by Chen's iterated integrals of logarithmic 1-forms. It was shown that the Vassiliev invariants for pure braids are complete invariants. Related to the Chern-Simons perturbation theory for 3-manifolds with boundary, I investigated the structure of Poisson algebra for the space of chord diagrams on a Riemann surface discovered by Reshetikhin and others. We have a natural Poisson algebra homomorphism from the Poisson algebra of chord diagrams on a Riemann surface to the Poisson algebra of the space of functions on the moduli space of flat G-bundles on the surface. For links in the product of a surface and the unit interval we can formulate invariants with values in a quantization of the above function space. I showed that the weight system associated with a flat connection can be integrated to give Vassiliev invariants for the above links. In particular, in the case of genus 1, I constructed an analogue of the Kontsevich integral based on elliptic KZ system.

4. Loop spaces of configuration spaces and finite type invariants

Motivated by recent works due to F. Cohen and S. Gitler on loop spaces, I proved that the space of Vassiliev invariants of braids are isomorphic to the full cohomology of the loop space of the configuration space of distinct points in the 3-space. More precisely, the above cohomology of the loop space is in one to one correspondence with the set of weight systems for Vassliev invariants and the associated topological invariant is given as the pairing with a certain homology class of the loop space constructed from a braid. Gereralizing this construction, I developed a new point of view on the construction of finite type invariants and link homotopy invariants.

5. Iterated integrals and hyperbolic volumes

Following work due to K. Aomoto, I gave an expression for the volume of hyperbolic simplices as the analytic continuation of the Schläfli functions using iterated integrals of logarithmic forms. I also described explicitly a differential equation of nilpotent type satisfied by the Schläfli functions. Based on this approach, I am worinking on the asymptotic behavior of the hyperbolic volumes on the boundary of moduli spaces.

6. Morse-Novikov theory for hyperplane arrangements

This is a joint work with A. Pajitnov. We develop the circle-valued Morse theory for the complement of a complex hyperplane arrangement in Cn Based on Morse functions constructed by a result due to P. Orlik and H. Terao, we show that the complement has the homotopy type of a space obtained from a finite n-dimensional CW complex fibered over a circle by attaching n-dimensional cells. We then focus on the Novikov homology attached to an abelian representation of the fundamental group of the complement and showed that the Novikov homology vanishes except in dimension $n$ for a large class of the above representations of the fundamental group.

7. Quantum representations and finite index subgroups of mapping class groups

This is a joint work with L. Funar. We give a description of the images of quantum representations of mapping class groups. We prove that, except for a few explicit roots of unity, the quantum image of any Johnson subgroup of the mapping class group contains an explicit free non-abelian subgroup.

8. Quantum symmetry in homological representations of braid groups

Homological representations of braid groups are defined as the action of homeomorphisms of a punctured disk on the homology of an abelian covering of its configuration space. These representations were extensively studied by Krammer and Bigelow. I described a relation between homological representations of braid groups and the monodoromy representations of KZ connections based on solutions of the KZ equation expressed by hypergeometric integrals. I also studied the case of resonance at infinity appearing in conformal field theory and investigated the structure of integration cycles. In this case I described the symmetry by quantum groups at roots of unity. I showed that the KZ equation is represented as a differential equation satisfied by period integrals for certain algebraic varieties, and is expressed as a Gauss-Manin connection.

9. Holonomy representations and their higher category extension

I developed a method to construct higher category extension of holonomy representations of homotopy path groupoid based on Chen's formal homology connection. I applied this general method to the case of braids. By means of a 2-functor from the path 2-groupoid of the configuration space, I constructed representations of the 2-category of braid cobordisms. Using this method, I investigated categorification of quantum representations of braid groups.

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