Abstract This paper is devoted to the classification of 4-dimensional Riemannian spin manifolds carrying skew Killing spinors. A spinor $$\psi $$ ψ a that satisfies equation $$\nabla _X\psi =AX\cdot \psi ∇ X = A</mml:...

We show that four-dimensional systems may exhibit a topological phase transition analogous to the well-known Berezinskii-Kosterlitz-Thouless vortex unbinding in two-dimensional systems. The realisation of an engineered quantum system, where predicted shall occur, is also presented. study suitable generalization sine-Gordon model four dimensions and renormalization group flow equation its coupli...