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arXiv · 2509.08876

A Hilbert Series for Generalized Toric Polygons

Ignacio Carreño Bolla·Sebastián Franco·Diego Rodríguez-Gómez
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Abstract

We study the Hilbert series for $5d$ Superconformal Field Theories (SCFTs) engineered by Generalized Toric Polygons (GTPs), which extend the geometric realization of these theories from toric Calabi-Yau 3-folds to theories associated to general webs of 5- and 7-branes. Smoothed T-cones provide fundamental building blocks of GTP tessellations, generalizing the role of minimal triangles in toric diagrams. Building on this construction, we propose an extension of the Martelli-Sparks-Yau algorithm for computing Hilbert series of toric Calabi-Yau 3-folds that computes the Ehrhart series directly from GTP tessellations. The Ehrhart series is an invariant under Hanany-Witten transitions, which translate geometrically into polytope mutations.

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Ignacio Carreño Bolla, Sebastián Franco, Diego Rodríguez-Gómez. 2025-09-10. A Hilbert Series for Generalized Toric Polygons. https://arxiv.org/abs/2509.08876

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