Generalized Global Symmetries of $T[M]$ Theories. I

TL;DR

<3-5 sentence high-level summary>This work builds a universal framework for reducing 6d theories on a d-dimensional manifold $M_d$ to $(6-d)$-dimensional theories, by promoting a refined polarization on $M_d$ that fixes the spectrum of charged operators and encodes the interplay of anomalies and higher-group symmetries. It centralizes a 7d bulk TQFT based on a 3-form Chern–Simons theory to describe partition functions, boundary conditions, and the mapping class-group action, and it classifies reductions into pure and mixed polarizations tied to symmetry-protected topological phases. The paper develops the general theory for arbitrary $d$, with detailed treatment of $d=0$ (Pol$( ext{pt})$) and $d=1$ (circle compactification) and case studies in $SO(8)$-related theories, highlighting how higher-group symmetries constrain RG flows, symmetry enhancement, and discrete theta-angles. A companion work extends these insights to $d=2,3,4$, applying the framework to diverse compactifications and clarifying the role of polarization in dualities and anomaly matching across dimensions.

Abstract

We study reductions of 6d theories on a $d$-dimensional manifold $M_d$, focusing on the interplay between symmetries, anomalies, and dynamics of the resulting $(6-d)$-dimensional theory $T[M_d]$. We refine and generalize the notion of "polarization" to "polarization on $M_d$," which serves to fix the spectrum of local and extended operators in $T[M_d]$. Another important feature of theories $T[M_d]$ is that they often possess higher-group symmetries, such as 2-group and 3-group symmetries. We study the origin of such symmetries as well as physical implications including symmetry breaking and symmetry enhancement in the renormalization group flow. To better probe the IR physics, we also investigate the 't Hooft anomaly of 5d Chern-Simons matter theories. The present paper focuses on developing the general framework as well as the special case of $d=0$ and 1, while an upcoming paper will discuss the case of $d=2$, $3$ and $4$.

Figures (13)

• Figure 1: 6d theories can be viewed as boundary conditions of a 7d TQFT.
• Figure 2: 7d partition function from gluing $7$-manifolds with defects described by the Poincaré dual of $B$. Removing the pair $B',B_*$ together introduces a phase.
• Figure 3: Linking in the bulk and intersection on the boundary.
• Figure 4: An absolute theory can be constructed from a domain wall separating the original bulk and an invertible TQFT, after colliding it with the boundary.
• Figure 5: Six theories permuted by $PSL(2,\mathbb{Z}_4)$ arranged as vertices of an octahedron. $PSL(2,\mathbb{Z}_4)$ then acts as orientation preserving isometries of the octahedron.