(Symplectic) leaves and (5d Higgs) branches in the Poly(go)nesian Tropical Rain Forest

Abstract

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.

Higher-form symmetries of 6d and 5d theories

“…Here we conjecture that there is a generalization to non-toric, generalized toric polygon (GTP). Consider a GTP, comprised of black and white vertices, and bring it into a convex form (see [39]). The 1-form symmetry is computed in the same way as (3.118), except we include all vertices that lie on the polygon -i.e.…”

S-fold magnetic quivers

“…One then mass-deforms it in Z symmetric ways, in effect decoupling s hypermultiplets for some integer s ≥ 1. • The resulting 5d theories can be brought to the SCFT point, where they develop a Higgs branch for which we can compute a magnetic quiver (see appendix B), using the brane web realization or the corresponding generalized toric polygon (GTP) [33,73,74]. • Finally, the 5d theories are compactified down to 4d theories on a circle with a Z twist.…”

“…At rank 1 these theories are well known; they were constructed through compactifications of 6d N = (1, 0) theories in [7][8][9], their Coulomb JHEP02(2021)054 branches were studied in detail [10][11][12][13][14][15], and their Higgs branches were recently studied through magnetic quivers in [16]. The concept of magnetic quivers proves useful to study Higgs branches of theories, both Lagrangian and non-Lagrangian [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. A convenient way to understand a hyper-Kähler moduli space, like the Higgs branch, of a quantum field theory with 8 supercharges is through its Hasse diagram [34][35][36][37][38][39][40][41][42].…”

(5d RG-flow) trees in the tropical rain forest

“…In general, however, it is not known how to associate a GTP with a Calabi-Yau singularity. Indeed this is an interesting question, to which a first step towards an answer was taken in [1], where a map from a general GTP to the magnetic quiver and Hasse diagram of the 5d theory was formulated, thereby approaching a description of the deformation space associated to the GTP.…”

“…In this paper we apply the framework developed in [1] to characterize the Higgs branch of 5d SCFTs that can flow to single gauge node theories with anti-symmetric (AS) and fundamental (F) matter [9], as well as SU(2) m quiver gauge theories, in the IR. We give a detailed description of the descendant trees obtained by decoupling matter multiplets in these theories, providing the associated magnetic quivers.…”