A general connected sum formula for the families Seiberg-Witten invariant

TL;DR

This paper establishes a comprehensive connected sum formula for families Seiberg-Witten invariants, unifying and extending previous specific results like blow-up and gluing formulas in the context of family theories.

Contribution

It introduces a general connected sum formula for families Seiberg-Witten invariants that encompasses earlier specialized formulas, advancing the theoretical framework.

Findings

Unified connected sum formula for families Seiberg-Witten invariants

Extension of previous Bauer-Furuta invariant results

Broader applicability of connected sum techniques in family theories

Abstract

In ordinary Seiberg-Witten theory, there are well known connected sum formulae such as the vanishing formula and the blow up formula. For families Seiberg-Witten theory, there are results such as Liu's families blow-up formula and Baraglia-Konno's gluing formula, but these have limited uses. In this paper, we prove a general connected sum formula which incorporates these results. This is subsequent work to a previous paper [arXiv:2510.14201] in which we proved a connected sum formula for the Bauer-Furuta invariant.