Spin(7)-Manifolds as Generalized Connected Sums and 3d N=1 Theories

TL;DR

This paper introduces a novel method to construct Spin(7)-holonomy manifolds using a generalized connected sum approach, enabling the geometric engineering of 3d N=1 theories in M-theory.

Contribution

It proposes a new construction technique for Spin(7)-manifolds based on generalized connected sums of Calabi-Yau four-folds and G2-manifolds, expanding the toolkit for geometric engineering.

Findings

Constructed new Spin(7)-holonomy manifolds using generalized connected sums.

Validated the spectra of these manifolds via duality with heterotic theories.

Connected the construction to twisted-connected sum methods for G2-holonomy.

Abstract

M-theory on compact eight-manifolds with $\mathrm{Spin}(7)$-holonomy is a framework for geometric engineering of 3d $\mathcal{N}=1$ gauge theories coupled to gravity. We propose a new construction of such $\mathrm{Spin}(7)$-manifolds, based on a generalized connected sum, where the building blocks are a Calabi-Yau four-fold and a $G_2$-holonomy manifold times a circle, respectively, which both asymptote to a Calabi-Yau three-fold times a cylinder. The generalized connected sum construction is first exemplified for Joyce orbifolds, and is then used to construct examples of new compact manifolds with $\mathrm{Spin}(7)$-holonomy. In instances when there is a K3-fibration of the $\mathrm{Spin}(7)$-manifold, we test the spectra using duality to heterotic on a $T^3$-fibered $G_2$-holonomy manifold, which are shown to be precisely the recently discovered twisted-connected sum constructions.