Geometric representation theory of braid groups
related to quantum groups and hypergeometric integrals
The idea of constructing representations of fundamental groups
by the monodromy of logarithmic connections goes back to
Poincaré and Lappo-Danilevsky. In 1970's a relationship between
nilpotent completions of fundamental groups and iterated integrals
was established by K. T. Chen. Subsequently, Aomoto described the
unipotent monodromy of the fundamental group of the complement
of a complex hypersurface by iterated integrals of logarithmic forms.
After reviewing these historical aspects, I will apply such technique
to representations of braid groups. For braid groups there is an
important flat connection called KZ connection.
On the other hand, there is a topological way to construct
representations of braid groups, namely homological representations.
These representations of braid groups are defined as the action of the
mapping class group of a punctured disk on the homology of an abelian
covering of its configuration space. They were extensively studied by
Krammer and Bigelow.
We show that specializations of the homological representationsof braid
groups are equivalent to the monodromy of the KZ equation with values in
the space of null vectors in the tensor product of Verma modules when
the parameters are generic. Here the representations of the
solutions of the KZ equation by hypergeometric integrals due to
Schechtman, Varchenko and others play an important role.
By this construction we recover quantum symmetry of the
monodromy of KZ connection due to Drinfel'd and myself by
means of the action of the quantum groups on twisted cycles.
In the case of special parameters corresponding to conformal
field theory, we show that KZ connection can be regarded as