What question did this study set out to answer?

May 8, 2026Open Access

A Hilbert series for generalized toric polygons

Key Points

This research aims to develop a Hilbert series for 5D superconformal field theories using generalized toric polygons.
Engineered hilbert series for 5D SCFTs from generalized toric polygons.
Proposed an extension of the Martelli-Sparks-Yau algorithm to compute hilbert series directly from GTP tessellations.
Analyzed the Ehrhart series as an invariant under Hanany-Witten transitions.
Established a new methodology for computing Hilbert series via GTP tessellations.
Showed that the Ehrhart series remains invariant under Hanany-Witten transitions, indicating robustness in the theory.
Demonstrated the generalization of minimal triangles' roles in traditional toric diagrams.

Abstract

A bstract We study the Hilbert series for 5 d 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.

Bookmark

View Full Paper