Matter from Toric Geometry
Philip Candelas, Eugene Perevalov, Govindan Rajesh
TL;DR
This work defines a concrete algorithm to determine the six-dimensional matter content of F-theory compactifications on elliptic Calabi–Yau threefolds that are hypersurfaces in toric varieties, by reading the spectrum off a reflexive polyhedron using the generalized Green–Schwarz anomaly cancellation. The authors connect toric data to the hypermultiplet content via divisor-intersection numbers and discriminant structure, and validate the method on a broad set of mirror models with varying enhanced gauge groups, confirming anomaly cancellation in each case. Key contributions include a practical dictionary between toric geometry and 6D spectra, explicit illustrative examples, and insight into how Higgsing and mirror symmetry reflect in the spectrum. The approach enhances the geometry–physics dictionary and supports constructing large classes of $N=1$ vacua in six dimensions using toric methods.
Abstract
We present an algorithm for obtaining the matter content of effective six-dimensional theories resulting from compactification of F-theory on elliptic Calabi-Yau threefolds which are hypersurfaces in toric varieties. The algorithm allows us to read off the matter content of the theory from the polyhedron describing the Calabi-Yau manifold. This is based on the generalized Green-Schwarz anomaly cancellation condition.