The s-Rule from Geometry
Abstract
String Theory provides powerful tools for constructing and studying five-dimensional Quantum Field Theories, notably through geometric engineering and five-brane webs. Understanding how these two descriptions are related in the presence of 7-branes requires incorporating the s-rule into the geometry. To address this question, we focus on basic three-legged five-brane webs ending on 7-branes. Building on the local M-theory description of five-branes ending on 7-branes, we consider triangular toric diagrams with a single non-primitive edge. We show that the corresponding local complex-structure deformation is well defined on the overlaps with the neighboring toric patches, while the corresponding gluing map remains well behaved as a function of the holomorphic coordinates throughout the neighboring patches precisely when the s-rule is satisfied. At saturation, the geometry is a T-cone and the 7-brane deformation becomes its Q-Gorenstein smoothing. Our results thus identify the s-rule with the obstruction to globalizing the local 7-brane deformation and provide a direct geometric link between T-cones and five-branes ending on 7-branes.
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BibTeX
@misc{franco2026srule,
title = {{The s-Rule from Geometry}},
author = {Sebastian Franco and Diego Rodriguez-Gomez},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07154},
eprint = {2610.07154},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07154},
}
RIS
TY - UNPB
TI - The s-Rule from Geometry
AU - Sebastian Franco
AU - Diego Rodriguez-Gomez
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.07154
UR - https://arxiv.org/abs/2610.07154
AB - String Theory provides powerful tools for constructing and studying five-dimensional Quantum Field Theories, notably through geometric engineering and five-brane webs. Understanding how these two descriptions are related in the presence of 7-branes requires incorporating the s-rule into the geometry. To address this question, we focus on basic three-legged five-brane webs ending on 7-branes. Building on the local M-theory description of five-branes ending on 7-branes, we consider triangular toric diagrams with a single non-primitive edge. We show that the corresponding local complex-structure deformation is well defined on the overlaps with the neighboring toric patches, while the corresponding gluing map remains well behaved as a function of the holomorphic coordinates throughout the neighboring patches precisely when the s-rule is satisfied. At saturation, the geometry is a T-cone and the 7-brane deformation becomes its Q-Gorenstein smoothing. Our results thus identify the s-rule with the obstruction to globalizing the local 7-brane deformation and provide a direct geometric link between T-cones and five-branes ending on 7-branes.
ER -