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

TL;DR

This work extends the framework of generalized global symmetries to T[M] theories obtained from compactifying 6d SCFTs on manifolds M, emphasizing the bulk/boundary viewpoint and the crucial role of polarizations in defining absolute theories. It develops a comprehensive calculus of polarizations on closed and open manifolds, including quadratic refinements and defects, and studies how higher-group and non-invertible symmetries emerge and transform under compactification, cutting/gluing, and mapping class group actions. The manuscript details symmetry structures across 4d, 3d, and 2d reductions, uncovering how KK modes, fibrations, and torsion in homology influence spectra of line operators, domain walls, and anomalies, and connects these structural insights to invariants like $ ext{Zhat}$ and to VOA/CFT correspondences. A key advancement is the proposal of two-parameter (two-$q$) 4-manifold invariants arising from deformed partition functions that separate gauge and KK contributions, offering refined tools for distinguishing topological data in higher-dimensional theories. Overall, the paper provides a unified, bulk-topological approach to a broad class of symmetries and anomalies in compactifications and lays groundwork for new 4-manifold invariants and their relationships to VOAs and quantum invariants.

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.

Figures (14)

• Figure 1: Choosing a polarization in a relative theory with boundary is equivalent to colliding the topological boundary conditions of the corresponding TQFT, that are labelled by (1) a topological boundary condition (purple) of the TQFT (grey) for the relative theory (black), which is Morita equivalent to the TQFT (green) for the boundary relative theory (red), and (2) topological domain wall (blue) between the topological boundary condition (purple) and the TQFT (green) for the boundary relative theory. Here topological boundary conditions refer more generally to the topological domain walls that separate the theory from an invertible TQFT. For the picture to be consistent, the invertible TQFT is the same for the purple and the green boundaries, since otherwise there would be another branch cut extending from the blue corner.
• Figure 2: In general, there are two classes of bulk operators in the presence of a boundary---these that can end on the boundary (red) and these that cannot. One can also classify them by how they behave when moved to the boundary. They can either stay non-trivial (blue) or become the identity operator (green, which is in fact a subclass of red ones). There are also operators (orange) that only live on the boundary, i.e. ending on the trivial bulk operator. From the perspective of polarization on a manifold with boundary, these correspond to the following groups: $\partial(L_\delta)$, $L/\partial(L_\delta)$, $L\cap\mathrm{im}(\partial)$, and $L_\delta\cap\mathrm{ker} (\partial)$.
• Figure 3: The boundary $S^2$ of $B^3$ obtained by gluing two disks $D^2$ along a great circle.
• Figure 4: This illustrates how to "blow up" along $M_d\subset M_D$ in the case of $M_d$ being a point in $M_D=T^2$. After compactifying on this geometry, one obtains a codimension-$(D-d)$ defect, which we denote as $T[M_D\backslash M_d]$, in the theory $T[M_d]$.
• Figure 5: Bulk operator with topological boundary can always end on the boundary to give topological defect on the boundary.