A New Construction of Calabi-Yau Manifolds: Generalized CICYs

TL;DR

The paper introduces generalized CICYs (gCICYs) by permitting negative entries in configuration matrices, enabling complete intersections defined via nested rational sections in ambient products of projective spaces with an effective anticanonical class. It develops a concrete construction and a robust topological toolkit based on Koszul sequences, adjunction, and ambient-space data to compute Hodge numbers, Chern classes, and intersection numbers, while detailing smoothness and reducedness considerations. A substantial classification program is started, revealing eight genuinely new Calabi–Yau threefolds at codimension (1,1) and a broad landscape at codimension (2,1), including both finite and apparently infinite families that are reconciled through redundancies. The work highlights rich physics potential, with many gCICYs displaying elliptic and K3 fibrations suitable for F-theory and duality studies, and discusses implications for discrete symmetries, Wilson lines, and potential GLSM realizations, marking a scalable path for exploring string vacua beyond traditional CICY datasets.

Abstract

We present a generalization of the complete intersection in products of projective space (CICY) construction of Calabi-Yau manifolds. CICY three-folds and four-folds have been studied extensively in the physics literature. Their utility stems from the fact that they can be simply described in terms of a `configuration matrix', a matrix of integers from which many of the details of the geometries can be easily extracted. The generalization we present is to allow negative integers in the configuration matrices which were previously taken to have positive semi-definite entries. This broadening of the complete intersection construction leads to a larger class of Calabi-Yau manifolds than that considered in previous work, which nevertheless enjoys much of the same degree of calculational control. These new Calabi-Yau manifolds are complete intersections in (not necessarily Fano) ambient spaces with an effective anticanonical class. We find examples with topology distinct from any that has appeared in the literature to date. The new manifolds thus obtained have many interesting features. For example, they can have smaller Hodge numbers than ordinary CICYs and lead to many examples with elliptic and K3-fibration structures relevant to F-theory and string dualities.