Abstract
The trinions TN are a class of 5d mathcal{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.
Highlights
The trinions TN are a class of 5d 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
Summary
We briefly review toric Calabi-Yau threefold singularities and the geometry of their divisors. A toric flop among compact surfaces changes the 2d and 3d cones in Σ, while leaving the rays unchanged, see figure 4 for an example After this flop, the triple intersection numbers change as:. In the M-theory-SCFT dictionary, (−2)-curves inside the compact surfaces ∪iSi correspond to flavor symmetries These are encoded in the marked subgraph of the CFD. The complete intersection curve D1 · ( i ξi,αSi) always has normal bundle O(0) + O(−2), and the M2 brane wrapping mode over it exactly gives rise to a W-boson of the non-Abelian flavor symmetry From this definition of CFD vertices, it is straight forward to see that any interior point in a boundary line gives rise to a flavor (marked) vertex with (n, g) = (−2, 0) in the CFD. To keep with the generic analogy, we determine the key characteric of these theories, by computing their BPS states using the CFDs and toric resolutions
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