On possible values of the signature of flat symplectic bundles over surfaces with boundary

TL;DR

This paper characterizes the possible signature values of flat symplectic bundles over surfaces with boundary, showing all integers in a specific interval are realizable, and discusses boundary holonomy types for the case p=1.

Contribution

It establishes the range of signature values for flat symplectic bundles over surfaces with boundary and allows boundary holonomy prescriptions when p=1.

Findings

All integers in the interval are achievable as signatures.

The signature interval depends on the Euler characteristic and rank.

Boundary holonomy types can be prescribed for p=1.

Abstract

We show that every integer in the interval $[2p\chi(\Sigma), -2p\chi(\Sigma)]$ is achieved by the signature of a rank $2p$ flat symplectic bundle over a surface with boundary $\Sigma$. When $p=1$, one can prescribe the type (elliptic, parabolic, hyperbolic) of the holonomy along the boundary.