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<p>We study the Hilbert series for 5<i>d</i> 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.</p>

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A Hilbert series for generalized toric polygons

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月5日

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  • Ignacio Carreño Bolla,
  • Sebastián Franco,
  • Diego Rodríguez-Gómez

摘要

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.