Generalized Global Symmetries of $T[M]$ Theories: Part II

Abstract

We continue the investigation of symmetries and anomalies of $T[M]$ theories obtained by compactifying 6d SCFTs on an internal manifold $M$. We extend the notion of "polarizations on a manifold $M$" to cases where $M$ may have boundaries or defects. Through examples with $M$ of dimension two, three, and four, we illustrate recurring themes in compactifications -- for instance, the important roles played by Kaluza-Klein modes, and how the generalized symmetries (including higher-group and non-invertible ones) of $T[M]$, together with their anomalies, arise from non-trivial combinations of the parent 6d symmetries and the geometric structures of the internal manifold. For each dimension, we also focus on several topics that are especially interesting in that setting. These include: for 2-manifolds, the geometry of the "full moduli space" of $T[M_2]$ and its interaction with polarizations and symmetries; for 3-manifolds, the effect of torsion in homology on the spectrum of line operators in $T[M_3]$, together with applications to the study of quantum invariants such as $\hat Z_a(M_3, q)$; and for 4-manifolds, predictions for VOA$[M_4]$ following from symmetries of $T[M_4]$, as well as the construction of a new invariant of 4-manifolds that depends on two "$q$-parameters." Along the way, we discuss a range of topics that are of independent interest, such as how non-invertible symmetries in higher dimensions can become invertible under compactification, how to classify defects in quantum field theory via their response to a change of framing, and the interplay between $\hat Z_a$ and volume conjectures.