Riemann surface invariants from vortex equation

Question

The vortex equations are often pitched as a toy model of the Seiberg-Witten equations. While the SW equations are frequently referenced in the context of providing geometric invariants on the base manifold, I am finding that the vortex equations are often not framed in a context of constructing invariants of Riemann surfaces. Indeed, the two main results that I seem to have come across are:

1. The moduli of vortices on a surface $X$ is isomorphic to the symmetric power of the base Riemann surface. This result seems to give too much information, because $Sym^k(X)$ seems to be just as complicated as $X$.

2. The volume of the moduli space of vortices is a function of the area of $X$ and the genus of $X$. As an invariant, this seems like it is not providing very much geometric information at all.

Question:

Are there more granular non-topological invariants of Riemann surfaces arising from the vortex equations than just area?

Answer 1

I only have relevant comments. SW in 4d gets you for example: detection of distinct smooth structures (but there is a unique smooth structure in 2d), non-existence of codimension-2 subsets which are complex surfaces (but they are "trivial" points in 2d), and bounds on scalar curvature (but I guess this becomes the stated area bound in 2d). So we need to think beyond using SW as motivation -- here are two routes, but I am unsure if they have helped us probe the Riemannian geometry of surfaces:

1. Motion of vortices (from physics), basically relating the metric on your suface to a special metric on the vortex moduli. Papers of Manton will tell us an interesting story.

2. Surfaces bounding 3-manifolds, basically the space of abelian monopoles (on a 3-manifold) immerses as a Lagrangian in the symplectic vortex moduli space (on the boundary surface). Notes of Perutz will tell us an interesting story.