[Paper Review] Topology of the blow-up set for the Seiberg-Witten equation with multiple spinors

Andriy Haydys|arXiv (Cornell University)|Jul 6, 2016

Homotopy and Cohomology in Algebraic Topology3 citations

TL;DR

This paper introduces a multiplicity function on the blow-up set Z for the Seiberg-Witten equation with multiple spinors, proving that Z represents a homology class equal to the Poincaré dual of the first Chern class of the determinant line bundle. The construction establishes a topological invariant linking the blow-up locus to characteristic classes in spin geometry.

ABSTRACT

We construct a multiplicity function on the blow-up set Z for the Seiberg-Witten equation with multiple spinors. This is used to prove that Z determines a homology class, which is shown to be equal to the Poincar\'e dual of the first Chern class of the determinant line bundle.

Motivation & Objective

• To define a multiplicity function on the blow-up set Z for the Seiberg-Witten equation with multiple spinors.
• To establish that the blow-up set Z defines a well-defined homology class in the underlying manifold.
• To show that this homology class is equal to the Poincaré dual of the first Chern class of the determinant line bundle.

Proposed method

• The authors define a multiplicity function assigning integer weights to points in the blow-up set Z based on the order of vanishing of solutions to the Seiberg-Witten equation with multiple spinors.
• They use the structure of the Seiberg-Witten equations with multiple spinors to analyze the singular behavior of solutions near Z.
• The construction relies on the determinant line bundle associated to the spinor bundle, which carries the first Chern class.
• By analyzing the zero locus of solutions and their asymptotic behavior, the authors show that the weighted sum over Z defines a closed current.
• The current is then shown to represent a homology class via Poincaré duality.
• The key step is proving that this homology class matches the Poincaré dual of the first Chern class of the determinant line bundle.

Experimental results

Research questions

• RQ1 How can a multiplicity function be defined on the blow-up set Z for the Seiberg-Witten equation with multiple spinors?
• RQ2 Does the blow-up set Z determine a well-defined homology class in the manifold?
• RQ3 Is the homology class represented by Z equal to the Poincaré dual of the first Chern class of the determinant line bundle?
• RQ4 What topological invariants are encoded in the blow-up locus of the multiple spinor Seiberg-Witten equations?

Key findings

• A multiplicity function is successfully constructed on the blow-up set Z for the Seiberg-Witten equation with multiple spinors.
• The blow-up set Z determines a well-defined homology class in the manifold.
• This homology class is shown to be equal to the Poincaré dual of the first Chern class of the determinant line bundle.
• The result establishes a direct topological invariant linking solution singularities to characteristic classes in spin geometry.