A Hilbert Series for Generalized Toric Polygons

2025/09/10 by Bolla, Ignacio Carreño, Franco, Sebastián, Rodríguez-Gómez, Diego
#FOS: Physical sciences #High Energy Physics - Theory (hep-th)

paper · doi:10.48550/arxiv.2509.08876

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.

Citations

• Futaki Invariants and Reflexive Polygons
• The Geometry of GTPs and 5d SCFTs
• Can you hear the shape of dual geometries?

Related

• On the Origin of Toric Diagrams
• The Geometry of GTPs and 5d SCFTs
• Towards Generalized Dimers for GTPs: N=2 Fractional Branes at Infinite Coupling
• The 5d Tangram: Brane Webs, 7-Branes and Primitive T-cones
• Non-toric brane webs, Calabi-Yau 3-folds, and 5d SCFTs