An abelian gauge-theoretic variant of the Seiberg-Witten equations for multiple-spinors

TL;DR

This paper introduces a gauge-theoretic variant of the Seiberg-Witten equations for multiple-spinors on Kähler surfaces, linking solutions to anti-self-dual connections and defining a new numerical invariant related to bundle stability.

Contribution

It develops a new abelian gauge-theoretic formulation of the Seiberg-Witten equations for multiple-spinors and constructs a numerical invariant detecting bundle stability.

Findings

Moduli space of solutions relates to ASD connections of holomorphic bundles.

Constructed a numerical invariant for $ ext{SU}(n)$-holomorphic bundles.

Established a connection between solutions and $ ext{phi}$-stability.

Abstract

We consider a variant of the Seiberg-Witten equations for multiple-spinors. The moduli space of solutions to our generalized Seiberg-Witten equations in the setting of K\"ahler surfaces has a direct relation with ASD connections of holomorphic vector bundle. Also in K\"ahler setting, we construct a numerical invariant from the equations that detects a notion of $\phi-$stability of $SU(n)-$holomorphic vector bundles where $\phi$ is some prescribed non-trivial holomorphic section.