The Branes Behind Generalized Symmetry Operators

• Published in: Fortschritte der Physik Vol. 71; no. 1
• Main Authors: Heckman, Jonathan J., Hübner, Max, Torres, Ethan, Zhang, Hao Y.
• Format: Journal Article
• Language: English
• Published: Weinheim Wiley Subscription Services, Inc 01.01.2023
• Subjects: Branes; Defects; geometric engineering; geometry; Operators (mathematics); Quantum field theory; Quantum theory; string theory; Symmetry; Topology
• ISSN: 0015-8208; 1521-3978
• DOI: 10.1002/prop.202200180

The modern approach to m‐form global symmetries in a d‐dimensional quantum field theory (QFT) entails specifying dimension d−m−1$d-m-1$ topological generalized symmetry operators which non‐trivially link with m‐dimensional defect operators. In QFTs engineered via string constructions on a non‐compact geometry X, these defects descend from branes wrapped on non‐compact cycles which extend from a localized source / singularity to the boundary ∂X$\partial X$. The generalized symmetry operators which link with these defects arise from magnetic dual branes wrapped on cycles in ∂X$\partial X$. This provides a systematic way to read off various properties of such topological operators, including their worldvolume topological field theories, and the resulting fusion rules. We illustrate these general features in the context of 6D superconformal field theories, where we use the F‐theory realization of these theories to read off the worldvolume theory on the generalized symmetry operators. Defects of dimension 3 which are charged under a suitable 3‐form symmetry detect a non‐invertible fusion rule for these operators. We also sketch how similar considerations hold for related systems. The modern approach to m‐form global symmetries in a d‐dimensional quantum field theory (QFT) entails specifying dimension d − m − 1 topological generalized symmetry operators which non‐trivially link with m‐dimensional defect operators. In QFTs engineered via string constructions on a non‐compact geometry X, these defects descend from branes wrapped on non‐compact cycles which extend from a localized source / singularity to the boundary ∂X. The generalized symmetry operators which link with these defects arise from magnetic dual branes wrapped on cycles in ∂X. This provides a systematic way to read off various properties of such topological operators, including their worldvolume topological field theories, and the resulting fusion rules. The authors illustrate these general features in the context of 6D superconformal field theories. They use the F‐theory realization of these theories to read off the worldvolume theory on the generalized symmetry operators. Defects of dimension 3 which are charged under a suitable 3‐form symmetry detect a non‐invertible fusion rule for these operators. It will be also sketched how similar considerations hold for related systems.