  The aim of this paper is to give an explicit expression for Hitchin's
connection in the case of rank 2 bundles with trivial determinant over curves
of genus 2. We recall the definition of this connection (which arose in Quantum
Field Theory) and characterize it for families of genus two curves. We then
consider a particular family (over a configuration space), the corresponding
vector bundles are (almost) trivial. The Heat equations which give the
connection these bundles are related to the Lie algebra so(6). We compute some
local monodromy representations. As a byproduct we obtain, for any
representation space V of so(2g+2), a flat connection on the trivial bundle
with fiber V over a configuration space and thus a monodromy representation of
a pure braid group.
