# Seiberg-Witten theory as a complex version of Abelian Higgs model

## TL;DR

This paper relates Seiberg-Witten equations on symplectic 4-manifolds to a complex analog of the Abelian Higgs model, extending the adiabatic limit concept to a (2+2)-dimensional setting.

## Contribution

It introduces a complex (2+2)-dimensional framework for Seiberg-Witten theory, connecting solutions to pseudoholomorphic divisors as a complex analog of geodesics.

## Key findings

- Seiberg-Witten solutions correspond to pseudoholomorphic divisors.
- The adiabatic limit links solutions to complex geometric objects.
- Extension of the Abelian Higgs model analogy to higher dimensions.

## Abstract

The adiabatic limit procedure associates with every solution of Abelian Higgs model in (2+1) dimensions a geodesic in the moduli space of static solutions. We show that the same procedure for Seiberg--Witten equations on 4-dimensional symplectic manifolds introduced by Taubes may be considered as a complex (2+2)-dimensional version of the (2+1)-dimensional picture. More precisely, the adiabatic limit procedure in the 4-dimensional case associates with a solution of Seiberg--Witten equations a pseudoholomorphic divisor which may be treated as a complex version of a geodesic in (2+1)-dimensional case.