Generalized Toric Polygons, T-branes, and 5d SCFTs

TL;DR

This work builds a geometric bridge from generalized toric polygons (GTPs) to non-toric Calabi-Yau deformations, enabling a string-theoretic construction of 5d SCFTs with 7-branes. By employing T-branes and Kraft-Procesi/Kolmogorov-type transitions, the authors map boundary data (white dots) on GTPs to nilpotent Higgs vevs and Slodowy slices, which in turn induce complex-structure deformations while preserving the overall CY locus. The T_n family serves as a concrete laboratory, where the spectra of hypermultiplets and their Ext^1 data encode brane recombinations and Hanany-Witten moves, leading to deformed geometries whose crepant resolutions reproduce the expected UV flavor symmetries and Higgs-branch structures. Across diverse GTPs (including rectangles and generic triangles), the paper demonstrates consistent matching of geometric resolutions with the Higgs/Coulomb data of the corresponding 5d SCFTs, thereby validating the GTP–non-toric geometry dictionary and outlining a path to general GTPs with mutually local 7-branes. These results deepen the link between brane-web boundary conditions, non-toric Calabi-Yau deformations, and the moduli spaces of 5d SCFTs, with potential implications for classifying and understanding their Higgs branches via Slodowy slices and Hasse diagrams.

Abstract

5d Superconformal Field Theories (SCFTs) are intrinsically strongly-coupled UV fixed points, whose realization hinges on string theoretic methods: they can be constructed by compactifying M-theory on local Calabi-Yau threefold singularities or alternatively from the world-volume of 5-brane-webs in type IIB string theory. There is a correspondence between 5-brane-webs and toric Calabi-Yau threefolds, however this breaks down when multiple 5-branes are allowed to end on a single 7-brane. In this paper, we extend this connection and provide a geometric realization of brane configurations including 7-branes. A web with 7-branes defines a so-called generalized toric polygon (GTP), which corresponds to combinatorial data that is obtained by removing vertices along external edges of a toric polygon. We identify the geometries associated to GTPs as non-toric deformations of toric Calabi-Yau threefolds and provide a precise, algebraic description of the geometry, when 7-branes are introduced along a single edge. The key ingredients in our analysis are T-branes in a type IIA frame, which includes D6-branes. We show that performing Hanany-Witten moves for the 7-branes on the type IIB side corresponds to switching on semisimple vacuum expectation values on the worldvolume of D6-branes, which in turn uplifts to complex structure deformations of the Calabi-Yau geometries. We test the proposal by computing the crepant resolutions of the deformed geometries, thereby checking consistency with the expected properties of the SCFTs.

Figures (15)

• Figure 1: Example of toric polygon (left) and dual brane-web (right) in which lines denote 5-branes and circles denote 7-branes. The 7-branes on which stacks of 5-branes are spaced to emphasize how the 5-brane end, here exactly one 5-brane ends on each 7-brane. This geometry encodes a 5d SCFT of rank 3.
• Figure 2: Correspondence between white dots on GTPs (left) and boundary conditions of $(p,q)$ 5-branes on $(p,q)$ 7-branes (middle). Here we have $(p,q) = (1,0)$ and the partition $\lambda = [3,2,2,1]$ of $n=8$. When we draw brane-webs we usually ignore the detached 7-branes and separate the 7-branes on a stack of 5-branes to show the boundary conditions (right).
• Figure 3: Summary of the main question addressed in this paper. On the left hand side, one starts from a convex polygon with integral vertices. It defines a 5d SCFT in two distinct ways: from M-theory on the associated toric CY, and from the dual brane-web. If on the contrary, as shown on the right hand side, the polygon is not convex, i.e. is a generalized toric polygon (GTP), the toric description is lost. However there still exists a dual brane-web, which has non-trivial boundary conditions on 7-branes, and thus a 5d SCFT. The central goal of this paper is to develop a map (the dashed line in the diagram) from GTPs to (non-toric) geometry.
• Figure 4: White dot and brane transition for a length 2 edge of a toric polygon.
• Figure 5: Three GTPs are shown on the first line, and below the algebraic equations characterizing the associated Calabi-Yau threefold geometry. The model on the left is a toric threefold. The other two, non-toric GTPs, are characterized in terms of deformations. Each of these geometries defines a 5d SCFT. The Hasse diagrams of symplectic singularities for the Higgs branch of these 5d SCFTs are shown below. The vertices represent symplectic leaves. For transverse slices we use a standard notation where the closure of the minimal nilpotent orbit of a simple Lie algebra is denoted using the lowercase form of the name of the algebra, e.g. $e_7$ for algebra $E_7$. In red are drawn the effects of the deformations.