(−1)-form symmetries from M-theory and SymTFTs

Abstract

We explore (−1)-form symmetries within the framework of geometric engineering in M-theory. By constructing the Symmetry Topological Field Theory (SymTFT) for selected 5d \( \mathcal{N} \) = 1, 4d \( \mathcal{N} \) = 2 and 4d \( \mathcal{N} \) = 1 theories, we formalize the geometric origin of these symmetries and compute the mixed anomaly polynomials involving (−1)-form and higher- form symmetries. Our findings consistently reveal both discrete and continuous (−1)-form symmetries, aligning with established field theory results, while also uncovering new (−1)-form symmetry factors and structural insights. In particular, we study the SymTFT of 4d \( \mathcal{N} \) = 1 theories from M-theory on a class of spaces with G₂ holonomy, and obtain properties such as modified instanton sums and 4-group structures observed in other 4d gauge theories. Additionally, we systematically construct symmetry operators for continuous abelian symmetries, refining existing proposals, and providing an M-theory origin for them.