The Branes Behind Generalized Symmetry Operators

TL;DR

The paper develops a brane-based framework to realize generalized $m$-form symmetries in QFTs engineered by string/M-/F-theory, identifying topological symmetry operators with magnetic dual branes wrapped on boundary cycles, and defects with branes wrapped on non-compact cycles. By reducing worldvolume couplings and leveraging twisted/cohomological data, it derives the corresponding topological field theories on symmetry operators and elucidates fusion rules, highlighting non-invertible fusion in 6D SCFTs with nontrivial duality bundles. The analysis focuses on 2-form symmetries in 6D SCFTs, detailing explicit examples for $6$D $\mathcal{N}=(2,0)$ theories (cyclic and non-abelian $\Gamma$) and $6$D $\mathcal{N}=(1,0)$ theories (NHCs), and extends to M-theory realizations and broader generalizations. The approach provides a systematic, geometry-driven method to extract operator content and TFTs on symmetry operators, offering a bridge between high-energy theory, topology, and category-theoretic perspectives on branes and their fusion.

Abstract

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 $\partial X$. The generalized symmetry operators which link with these defects arise from magnetic dual branes wrapped on cycles in $\partial X$. 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.

Figures (1)

• Figure 1: Depiction of a generalized symmetry operator realized via a wrapped brane. We consider a supersymmetric QFT (SQFT) localized at the tip of a conical geometry $X$. At the boundary $\partial X$, we have bounding cycles $\widetilde{\gamma}$ and $\gamma$ which link in $\partial X$. Wrapping a $p$-brane on the cone over $\widetilde{\gamma}$ yields a defect for an $m$-form symmetry, and wrapping the magnetic dual $q$-brane on a linking cycle $\gamma$ realizes a generalized symmetry operator. The spacetime loci $\widetilde{M}$ of the $p$-brane and $M$ of the $q$-brane also link with each other. The wrapped $q$-brane is automatically topological since it sits at infinity in the radial geometry.