Geometric Monodromy of Mixed Braid Groups and the Multivariate Burau Representation

Abstract

We study the monodromy action of the mixed braid group $B_{n,\mathcal{P}}$ on the first cohomology of cyclic branched covers of $\mathbb{P}^1$, which are mutually determined by a partition of branch points by equal ramification. The monodromy representation splits into irreducible representations on the $t$-eigenspaces of the deck transformation. For each, we construct an explicit spanning set using lifts of Pochhammer contours and figure-eight curves, and compute the Hermitian intersection form. The representation factors through a reduced mixed braid group by dropping $t$-invisible parts of the partition (those with trivial local monodromy). In this reduced representation, each generator acts by a complex reflection when the corresponding spanning class is non-isotropic, and by a unitary transvection when it is isotropic. Provided $\infty$ has non-trivial local monodromy, the factored representation is isomorphic to the reduced multivariate Burau representation evaluated at $t$.