Multi-monopoles and monodromy around the wobbly divisor

Abstract

The multi-monopole equations are a generalisation of the Seiberg–Witten equations where the spinor is coupled to an auxiliary vector bundle. The signed count of multi-monopoles on a 3-manifold can vary according to a new type of wall-crossing phenomenon. For products $S^1\times C$, where $C$ is a closed Riemann surface, we relate the wall and chamber structure to loops in the complement of the wobbly divisor of $C$. Such a loop induces a monodromy map on the multi-vortex moduli space, and the count of its fixed points is expected to agree with the count of multi-monopoles on $S^1\times C$ in the given chamber. By studying singularities of framed multi-vortex moduli spaces, we are able to understand these monodromy maps for various loops. Counting fixed points suggests that product 3-manifolds display incredibly rich wall-crossing behaviour.