Supersymmetric Cycles in Exceptional Holonomy Manifolds and Calabi-Yau 4-Folds

TL;DR

This work classifies supersymmetric cycles in exceptional holonomy manifolds and Calabi–Yau 4-folds by combining SCFT boundary conditions with low-energy effective actions. It identifies Cayley submanifolds in Spin(7) and associative/coassociative cycles in G2 as preserving half of space-time supersymmetry, while Calabi–Yau 4-folds exhibit holomorphic and special Lagrangian cycles also at half-SUSY, with Cayley 4-cycles being novel in preserving only 1/4. The analysis shows that Cayley calibrations arise from a specific boundary condition that preserves a single linear combination of spectral-flow sectors, leading to new 1/4-BPS brane configurations, and provides explicit examples. The results have implications for higher-dimensional mirror symmetry, including how Cayley cycles may populate $H^{2,2}$ and how mirror symmetry might extend via a higher-dimensional SYZ framework, potentially yielding a supersymmetric $T^n$-fibration.

Abstract

We derive in the SCFT and low energy effective action frameworks the necessary and sufficient conditions for supersymmetric cycles in exceptional holonomy manifolds and Calabi-Yau 4-folds. We show that the Cayley cycles in $Spin(7)$ holonomy eight-manifolds and the associative and coassociative cycles in $G_2$ holonomy seven-manifolds preserve half of the space-time supersymmetry. We find that while the holomorphic and special Lagrangian cycles in Calabi-Yau 4-folds preserve half of the space-time supersymmetry, the Cayley submanifolds are novel as they preserve only one quarter of it. We present some simple examples. Finally, we discuss the implications of these supersymmetric cycles on mirror symmetry in higher dimensions.