  In this paper, we classify homomorphisms from the braid group of $n$ strands
to the mapping class group of a genus $g$ surface. In particular, we show that
when $g<n-2$, all representations are either cyclic or standard. Our result is
sharp in the sense that when $g=n-2$, a generalization of the hyperelliptic
representation appears, which is not cyclic or standard. This gives a
classification of surface bundles over the configuration space of the complex
plane. As a corollary, we partially recover the result of Aramayona-Souto,
which classifies homomorphisms between mapping class groups, with a slight
improvement.
