The trinions TN are a class of 5d $$ \mathcal{N} $$ N = 1 superconformal field theories (SCFTs) realized as M-theory on ℂ3/ℤN× ℤN . We apply to TN , as well as closely-related SCFTs that are obtained by mass deformations, a multitude of recently developed approaches to studying 5d SCFTs and their IR gauge theory descriptions. Thereby we provide a complete picture of the theories both on the Coulomb branch and Higgs branch, from various geometric points of view — toric and gluing of compact surfaces as well as combined fiber diagrams — to magnetic quivers and Hasse diagrams.

By incorporating higher-form symmetries, we propose a refined definition of the theories obtained by compactification of the 6d (2, 0) theory on a three-manifold M 3 . This generalization is applicable to both the 3d N = 2 and N = 1 supersymmetric reductions. An observable that is sensitive to the higher-form symmetries is the Witten index, which can be computed by counting solutions to a set of Bethe equations that are determined by M 3 . This is carried out in detail for M 3 a Seifert manifold, where we compute a refined version of the Witten index.In the context of the 3d-3d correspondence, we complement this analysis in the dual topological theory, and determine the refined counting of flat connections on M 3 , which matches the Witten index computation that takes the higher-form symmetries into account.for which such 3d-3d duals have been discussed are the squashed three-sphere S 3 b [1, 2, 6], the superconformal index on S 2 × S 1 [3], and the twisted index on Σ g × S 1 [7]. The special case W 3 = T 3 was considered in [5,7] and computes the (regularized) Witten index [8] I = Tr(−1) F . 1 However, an important characteristic of the theories has so far been largely ignored: their higher-form symmetries [12]. Namely, the theory T [M 3 ] has a higher-form symmetry, which, as with other properties of T [M 3 ], is determined by the topology of M 3 . In fact, the theory T [M 3 ] is not fully specified by the manifold, M 3 , but requires additional topological data. This is related to the fact that the 6d N = (2, 0) theory itself is a relative QFT, i.e., it is only well-defined as the boundary of a 7d TQFT [13,14]. Equivalently, its observables depend on a choice of polarization, i.e., a choice of maximal isotropic subgroup of H 3 (M 3 , Z G ), where Z G is the center of the simply connected group, G, with Lie algebra g. Naturally, we expect that T [M 3 ] also depends on the polarization, as is the case for 4d theories [15,16]. In fact, we will see that this additional information translates into the residual 0-and 1-form symmetry of the 3d theory. We propose therefore a refined definition of the theories, which specifies this dataThis theory has a discrete 0-form (ordinary) symmetry group H. Its residual 1-form symmetry is given by the complementary subgroup, Υ H , 2 inside H 1 (M 3 , Z G ). We show that the choice of H can indeed be detected by the Witten index or, more generally, by the partition function on any W 3 with non-trivial homology. Thus, the different theories in (1.2) are indeed physically distinct.The main interest of this paper is to develop a sound definition of the theories T [M 3 , g, H]in (1.2) for M 3 a graph manifold [17], a class of three-manifolds we review in section 2.1. These manifolds, which also occur as the boundary of plumbed four-manifolds, are sometimes called plumbed three-manifolds, a special case of which are Seifert manifolds. Similar Lagrangians for the 3d theories associated to these manifolds were studied in [6,[18][19][20][21][22][23].In the following we point out new feat...

We derive the structure of the Higgs branch of 5d superconformal field theories or gauge theories from their realization as a generalized toric polygon (or dot diagram). This approach is motivated by a dual, tropical curve decomposition of the (p, q) 5-brane-web system. We define an edge coloring, which provides a decomposition of the generalized toric polygon into a refined Minkowski sum of sub-polygons, from which we compute the magnetic quiver. The Coulomb branch of the magnetic quiver is then conjecturally identified with the 5d Higgs branch. Furthermore, from partial resolutions, we identify the symplectic leaves of the Higgs branch and thereby the entire foliation structure. In the case of strictly toric polygons, this approach reduces to the description of deformations of the Calabi-Yau singularities in terms of Minkowski sums.

5d superconformal field theories (SCFTs) can be obtained from 6d SCFTs by circle compactification and mass deformation. Successive decoupling of hypermultiplet matter and RG-flow generates a decoupling tree of descendant 5d SCFTs. In this paper we determine the magnetic quivers and Hasse diagrams, that encode the Higgs branches of 5d SCFTs, for entire decoupling trees. Central to this undertaking is the approach in [1], which, starting from the generalized toric polygons (GTPs) dual to 5-brane webs/tropical curves, provides a systematic and succinct derivation of magnetic quivers and their Hasse diagrams. The decoupling in the GTP description is straightforward, and generalizes the standard flop transitions of curves in toric polygons. We apply this approach to a large class of 5d KK-theories, and compute the Higgs branches for their descendants. In particular we determine the decoupling tree for all rank 2 5d SCFTs. For each tree, we also identify the flavor symmetry algebras from the magnetic quivers, including non-simply-laced flavor symmetries.

M5-branes on an associative three-cycle M 3 in a G 2 -holonomy manifold give rise to a 3d N = 1 supersymmetric gauge theory, T N =1 [M 3 ]. We propose an N = 1 3d-3d correspondence, based on two observables of these theories: the Witten index and the S 3 -partition function. The Witten index of a 3d N = 1 theory T N =1 [M 3 ] is shown to be computed in terms of the partition function of a topological field theory, a super-BFmodel coupled to a spinorial hypermultiplet (BFH), on M 3 . The BFH-model localizes on solutions to a generalized set of 3d Seiberg-Witten equations on M 3 . Evidence to support this correspondence is provided in the abelian case, as well as in terms of a direct derivation of the topological field theory by twisted dimensional reduction of the 6d (2, 0) theory. We also consider a correspondence for the S 3 -partition function of the T N =1 [M 3 ] theories, by determining the dimensional reduction of the M5-brane theory on S 3 . The resulting topological theory is Chern-Simons-Dirac theory, for a gauge field and a twisted harmonic spinor on M 3 , whose equations of motion are the generalized 3d Seiberg-Witten equations. For generic G 2 -manifolds the theory reduces to real Chern-Simons theory, in which case we conjecture that the S 3 -partition function of T N =1 [M 3 ] is given by the Witten-ReshetikhinTuraev invariant of M 3 .