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
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. Our result also leads to a short proof of a recent
result by M. Davis, T. Janusckiewicz and I. J. Leary on the
vanishing of the $L^2$-homology.
7. Quantum representations and finite index subgroups of mapping class
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
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
my means of 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.