SymTFTs and Duality Defects from 6d SCFTs on 4-manifolds

TL;DR

The work develops a SymTFT-based framework linking 6d $(2,0)$ SCFTs compactified on 4-manifolds to 2d theories $T_N[M_4]$, where the 3d bulk encodes symmetries and dualities and the 2d absolute theories arise from a choice of polarization. By analyzing the intersection form of $M_4$ and the associated automorphism data, the authors classify global variants, dualities, and topological defects, including non-invertible duality defects at special coupling fixed points. They provide detailed realizations for $M_4=\mathbb{P}^1\times\mathbb{P}^1$, $\mathbb{F}_1$, del Pezzo surfaces, and general $M_4$, uncovering rich networks of orbifold structures, SPT stacking, and boundary conditions that organize the defect content. The approach highlights how geometric data and topological manipulations combine to produce a comprehensive network of 0-form and higher-form symmetries, with potential extensions to broader 6d (2,0) and (1,0) theories and other compactification manifolds.

Abstract

In this work we study particular TQFTs in three dimensions, known as Symmetry Topological Field Theories (or SymTFTs), to identify line defects of two-dimensional CFTs arising from the compactification of 6d $(2,0)$ SCFTs on 4-manifolds $M_4$. The mapping class group of $M_4$ and the automorphism group of the SymTFT switch between different absolute 2d theories or global variants. Using the combined symmetries, we realize the topological defects in these global variants. Our main example is $\mathbb{P}^1 \times \mathbb{P}^1$. For $N$ M5-branes the corresponding 2d theory inherits $\mathbb{Z}_N$ $0$-form symmetries from the SymTFT. We reproduce the orbifold groupoid for theories with $\mathbb{Z}_N$ $0$-form symmetries and realize the duality defects at fixed points of the coupling constant under elements of the mapping class group. We also study other Hirzebruch surfaces, del Pezzo surfaces, as well as the connected sum of $\mathbb{P}^1 \times \mathbb{P}^1$. We find a rich network of global variants connected via automorphisms and realize more interesting topological defects. Finally, we derive the SymTFT on more general 4-manifolds and provide two examples.

Figures (16)

• Figure 1: Compactification of 7d/6d coupled system on $M_4$ with maximal isotropic sublattice $L$ lead to a 2d theory $T_{N}[M_4]$ on $\Sigma_2$ and its SymTFT on $\Sigma_2 \times I_{(0,\epsilon)}$ with topological boundary condition $\langle L(B)|$.
• Figure 2: The 2d absolute theory is obtained by shrinking the interval.
• Figure 3: Orbifold groupoids for $T_2[\mathbb{P}^1 \times \mathbb{P}^1]$ with $\mathbb{Z}_2$ symmetry. The map $g$ represents the topological manipulation gauging $\mathbb{Z}_2$ up to an SPT phase.
• Figure 4: Web of transformations for $T_2[\mathbb{P}^1 \times \mathbb{P}^1]$. The transformations in orange are the duality transformations. The transformations in blue are topological manipulations.
• Figure 5: Web of transformations for $T_p[\mathbb{P}^1 \times \mathbb{P}^1]$. The transformations in orange are the duality transformations. The transformations in blue are topological manipulations.