2d $\mathcal{N}=(0,1)$ Gauge Theories and Spin(7) Orientifolds

TL;DR

This paper develops a geometric framework for engineering 2d (0,1) gauge theories using Spin(7) orientifolds, connecting them to (0,2) theories via real slices of Calabi-Yau 4-folds and Joyce's Spin(7) construction.

Contribution

It introduces Spin(7) orientifolds as a new class of backgrounds for 2d (0,1) theories, linking geometric quotients of Calabi-Yau 4-folds to gauge theory realizations.

Findings

Constructed explicit examples of Spin(7) orientifolds.

Analyzed the impact of vector structure choices.

Studied partial resolutions of singularities.

Abstract

We initiate the geometric engineering of 2d $\mathcal{N}=(0,1)$ gauge theories on D1-branes probing singularities. To do so, we introduce a new class of backgrounds obtained as quotients of Calabi-Yau 4-folds by a combination of an anti-holomorphic involution leading to a Spin(7) cone and worldsheet parity. We refer to such constructions as Spin(7) orientifolds. Spin(7) orientifolds explicitly realize the perspective on 2d $\mathcal{N}=(0,1)$ theories as real slices of $\mathcal{N}=(0,2)$ ones. Remarkably, this projection is geometrically realized as Joyce's construction of Spin(7) manifolds via quotients of Calabi-Yau 4-folds by anti-holomorphic involutions. We illustrate this construction in numerous examples with both orbifold and non-orbifold parent singularities, discuss the role of the choice of vector structure in the orientifold quotient, and study partial resolutions.