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The Geometry of GTPs and 5d SCFTs
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abstract
We make progress in understanding the geometry associated to the Generalized Toric Polygons (GTPs) encoding the Physics of 5d Superconformal Field Theories (SCFTs), by exploiting the connection between Hanany-Witten transitions and the mathematical notion of polytope mutations. From this correspondence, it follows that the singular geometry associated to a GTP is identical to that obtained by regarding it as a standard toric diagram, but with some of its resolutions frozen in way that can be determined from the invariance of the so-called period under mutations. We propose the invariance of the period as a new criterion for distinguishing inequivalent brane webs, which allows us to resolve a puzzle posed in the literature. A second mutation invariant is the Hilbert Series of the geometry. We employ this invariant to perform quantitative checks of our ideas by computing the Hilbert Series of the BPS quivers associated to theories related by mutation. Lastly, we discuss the physical interpretation of a mathematical result ensuring the existence of a flat fibration over $\mathbb{P}^1$ interpolating between geometries connected by mutation, which we identify with recently introduced deformations of the corresponding BPS quivers.
Forward citations
Cited by 6 Pith papers
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Towards Generalized Dimers for GTPs: $\mathcal{N}=2$ Fractional Branes at Infinite Coupling
N=2 strip condensation — collapsing parallel zig-zag strips in brane tilings — reproduces the expected number of gauge groups and yields GTP quivers related to toric ones by relevant deformations.
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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.
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BPS Invariants for Generalized Toric Calabi-Yau Threefolds
Gopakumar–Vafa invariants are transported across Hanany–Witten transitions after removing a universal parallel-brane sector, giving first-time high-degree invariants for local dP4.
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On the Origin of Toric Diagrams
A field's scaling dimension equals the number of perfect matchings at the chosen origin that contain it, making the gauge-theory Hilbert series equal the Ehrhart series of the dual polytope.
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Quiver-Invariant Dualities between Brane Tilings
A tilting mutation of brane tilings yields distinct superpotentials on the same quiver with identical mesonic moduli space, equivalent to a sequence of Seiberg dualities.
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Birational Transformations and 2d (0,2) Quiver Gauge Theories beyond Toric Fano 3-folds
Mass deformations of 2d (0,2) brane brick models are shown in four explicit examples to implement birational transformations of toric Calabi-Yau 4-folds, including non-reflexive toric diagrams.
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