Geometric representation theory of braid groups related to quantum groups and hypergeometric integrals
Toshitake Kohno
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 Gauss-Manin connection.