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

TL;DR

This paper characterizes supersymmetric cycles in exceptional holonomy manifolds and Calabi-Yau 4-folds, revealing their supersymmetry preservation properties and implications for mirror symmetry in higher dimensions.

Contribution

It provides necessary and sufficient conditions for supersymmetric cycles in these manifolds within SCFT and effective action frameworks, highlighting novel Cayley cycles in Calabi-Yau 4-folds.

Findings

Cayley cycles in Spin(7) preserve half supersymmetry

Associative and coassociative cycles in G2 preserve half supersymmetry

Cayley submanifolds in Calabi-Yau 4-folds preserve one quarter supersymmetry

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.