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The moduli space of multi-monopoles on a Riemann surface

Ollie Thakar

TL;DR

The paper analyzes the moduli spaces of solutions to a dimensional reduction of the multi-spinor Seiberg–Witten equations on a genus $g$ Riemann surface, forming a rich bridge between gauge theory and Brill–Noether geometry. It first establishes the Euler characteristic and expected dimension, then develops two complementary embeddings of the moduli spaces into finite-dimensional projective bundles, enabling a concrete description of their topology via index theory. By implementing spectral-curve methods (BNR) and a degeneracy-locus framework, the authors reduce the problem to finite-dimensional algebraic geometry and apply Fulton–Lazarsfeld theory to compute the rational cohomology, obtaining explicit formulas in terms of the genus and dimension. The results illuminate the Brill–Noether interpretation of multi-monopole moduli and provide a pathway to compute Hodge data, connecting 3-manifold invariants to higher-rank Brill–Noether geometry.

Abstract

We study the moduli space of solutions to the Seiberg-Witten equations with $N$ spinors on a compact Riemann surface. These moduli spaces arise in a program to define a new enumerative invariant of 3-manifolds. They are also of independent interest in the geometry of algebraic curves, as they parameterize generalized divisors in Brill-Noether theory for higher rank vector bundles. We compute the Euler characteristic of these spaces, completing a computation initiated by Doan, and then compute their rational homology using spectral curves and techniques of Fulton and Lazarsfeld.

The moduli space of multi-monopoles on a Riemann surface

TL;DR

The paper analyzes the moduli spaces of solutions to a dimensional reduction of the multi-spinor Seiberg–Witten equations on a genus Riemann surface, forming a rich bridge between gauge theory and Brill–Noether geometry. It first establishes the Euler characteristic and expected dimension, then develops two complementary embeddings of the moduli spaces into finite-dimensional projective bundles, enabling a concrete description of their topology via index theory. By implementing spectral-curve methods (BNR) and a degeneracy-locus framework, the authors reduce the problem to finite-dimensional algebraic geometry and apply Fulton–Lazarsfeld theory to compute the rational cohomology, obtaining explicit formulas in terms of the genus and dimension. The results illuminate the Brill–Noether interpretation of multi-monopole moduli and provide a pathway to compute Hodge data, connecting 3-manifold invariants to higher-rank Brill–Noether geometry.

Abstract

We study the moduli space of solutions to the Seiberg-Witten equations with spinors on a compact Riemann surface. These moduli spaces arise in a program to define a new enumerative invariant of 3-manifolds. They are also of independent interest in the geometry of algebraic curves, as they parameterize generalized divisors in Brill-Noether theory for higher rank vector bundles. We compute the Euler characteristic of these spaces, completing a computation initiated by Doan, and then compute their rational homology using spectral curves and techniques of Fulton and Lazarsfeld.
Paper Structure (11 sections, 15 theorems, 41 equations)

This paper contains 11 sections, 15 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Multi-monopole equations as generalized Brill-Noether theory
  3. The multi-monopole equations in dimension 3
  4. Acknowledgements
  5. Characteristic Classes of the Moduli Spaces
  6. Embedding $\mathcal{N}^d$ in a projective bundle
  7. The tangent bundle of $\mathcal{N}$
  8. Techniques from Algebraic Geometry
  9. Spectral curves
  10. Degeneracy locus construction
  11. Rational cohomology of $\mathcal{N}$

Key Result

Theorem 1.1

The Euler characteristic of the moduli space $\mathcal{N}^d_X(E)$ is:

Theorems & Definitions (34)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8: Doan
  • Remark 1.9
  • Remark 1.10
  • ...and 24 more