Super-Higher-Form Symmetries (2503.16182v1)

Abstract: We generalize the study of higher-form-symmetries to theories with supersymmetry. Using a supergeometry formulation, we find that ordinary higher-form-symmetries nicely combine with supersymmetry to give rise to a much larger spectrum of topological conserved (super)currents. These can be classified as a supersymmetric version of Chern-Weil symmetries, and a brand new set of geometric-Chern-Weil symmetries whose generators are constructed using invariant differential forms in super-manifolds. For N=1 super-Maxwell theory in various dimensions, we build the topological operators generating these super-higher-form symmetries and construct defects carrying non-trivial charges. Notably, the charge is proportional to the super-linking number between the super-hypersurface supporting the symmetry generator and the one supporting the defect.

An Analysis of the Paper "Super-Higher-Form Symmetries"

The paper "Super-Higher-Form Symmetries" by P. A. Grassi and S. Penati presents a rigorous extension of higher-form symmetries into the domain of supersymmetry, leveraging the formalism of supergeometry. This investigation opens new vistas in the theoretical framework of higher-form symmetries, which have been a focal point due to their implications in fields and defects in quantum field theory (QFT). The paper meticulously delineates the construction of these super-higher-form symmetries and explores their practical implementations in various supersymmetric theories.

Key Contributions

1. Generalization to Supersymmetry: The paper extends the traditional higher-form symmetries into supersymmetric theories by proposing a new class of symmetries termed super-higher-form symmetries. This includes supersymmetric Chern-Weil symmetries and new geometric-Chern-Weil symmetries characterized by the use of invariant super-differential forms.
2. Supergeometry Framework: Using the formalism of supergeometry, the work translates geometric tools from classical gauge theory into their supersymmetric counterparts. It demonstrates how differential geometry and the calculus of differential forms can serve as the backdrop for constructing conserved supercurrencies in superspaces.
3. Examples in Various Dimensions: The authors provide detailed constructions of super-higher-form symmetries for specific cases, particularly in $N=1$ super-Maxwell theories across different dimensions (e.g., three, four, six, and ten dimensions). The investigation extends to include an $N=1$ super-Chern-Simons theory.
4. Topological Operators and Charges: The paper constructs topological operators that generate these super-higher-form symmetries, calculating defect charges. Notably, it highlights the role of super-linking numbers, a generalization of linking numbers to supermanifolds, in determining these charges.
5. Coupling to Background Fields: By examining the interaction of conserved currents with background gauge fields, the authors offer new insights into the symmetry actions on charged states, providing a practical computational framework which could be crucial for further theoretical developments in QFTs.

Implications and Future Research

The paper offers significant practical and theoretical implications. Practically, the generalization of higher-form symmetries can enhance our understanding of topological phases of matter in supersymmetric quantum field theories and may find applications in string theory and quantum gravity formulations.

The theoretic implications involve deeper insights into symmetry constraints in supersymmetry, especially concerning anomalies, breaking, and gauging within gauge theories. The intersection with non-invertible symmetries and the cohomological aspects of QFT invites further scholarly exploration.

Speculative Outlook

The path opened by this research holds promise for the development of richer symmetry structures in supergravity and related quantum field theories. The use of supergeometry and the paper of pseudo-differential forms could lead to identifying new invariant cohomologies and uncovering hidden symmetries, illuminating our understanding of the fabric of modern theoretical physics.

The extensions of this work could engage with non-abelian groups and scalar components in superspace, contributing to a broader conceptual milieu where geometry and supersymmetry coalesce to provide novel solutions to longstanding problems in theoretical physics.

In conclusion, this paper forms an integral contribution to the paper of symmetries within supersymmetric frameworks, significantly expanding the lattice of theoretical tools available to the field of high-energy physics.