Spinorially Twisted Spin Structures. II: Twisted Pure Spinors, Special Riemannian Holonomy and Clifford Monopoles

TL;DR

This paper introduces twisted pure spinors to characterize special Riemannian holonomy groups and proposes Clifford monopole equations as a geometric generalization of Seiberg-Witten equations, revealing new solutions.

Contribution

It defines twisted pure spinors for a unified characterization of special holonomy groups and introduces Clifford monopole equations extending Seiberg-Witten theory.

Findings

Clifford monopole equations reduce to Seiberg-Witten equations in 4D

Existence of non-trivial solutions on manifolds with special holonomy

Unified framework for characterizing special Riemannian holonomy

Abstract

We introduce a notion of twisted pure spinor in order to characterize, in a unified way, all the special Riemannian holonomy groups just as a classical pure spinor characterizes the special K\"ahler holonomy. Motivated by certain curvature identities satisfied by manifolds admitting parallel twisted pure spinors, we also introduce the Clifford monopole equations as a natural geometric generalization of the Seiberg-Witten equations. We show that they restrict to the Seiberg-Witten equations in 4 dimensions, and that they admit non-trivial solutions on manifolds with special Riemannian holonomy.