Graded Geometric Structures Underlying F-Theory Related Defect Theories

TL;DR

This paper explores the geometric and algebraic structures of fermionic fields in F-theory defect theories, revealing graded vector spaces and composite fiber bundles that deepen understanding of supersymmetric quantum mechanics in this context.

Contribution

It introduces a novel geometric framework involving graded vector spaces and composite fiber bundles for fermionic zero modes in F-theory defect theories.

Findings

Fermionic fields form a graded vector space

Construction of composite fiber bundles related to zero modes

Sections correspond to square roots of canonical bundles

Abstract

In the context of F-theory, we study the related eight dimensional super-Yang-Mills theory and reveal the underlying supersymmetric quantum mechanics algebra that the fermionic fields localized on the corresponding defect theory are related to. Particularly, the localized fermionic fields constitute a graded vector space, and in turn this graded space enriches the geometric structures that can be built on the initial eight-dimensional space. We construct the implied composite fibre bundles, which include the graded affine vector space and demonstrate that the composite sections of this fibre bundle are in one-to-one correspondence to the sections of the square root of the canonical bundle corresponding to the submanifold on which the zero modes are localized.