T-dualizing D5/D7 junctions yields smooth D6 curves whose spectral data (from coherent sheaves) give explicit complex-structure deformations of the dual M-theory threefold, with s-rule violations appearing as poles.
A Hilbert Series for Generalized Toric Polygons
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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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M-theory geometries from five-brane webs, seven-branes, and T-branes
T-dualizing D5/D7 junctions yields smooth D6 curves whose spectral data (from coherent sheaves) give explicit complex-structure deformations of the dual M-theory threefold, with s-rule violations appearing as poles.