Enhanced gauge symmetry in 6D F‐theory models and tuned elliptic Calabi‐Yau threefolds

We systematically analyze the local combinations of gauge groups and matter that can arise in 6D F‐theory models over a fixed base. We compare the low‐energy constraints of anomaly cancellation to explicit F‐theory constructions using Weierstrass and Tate forms, and identify some new local structures in the “swampland” of 6D supergravity and SCFT models that appear consistent from low‐energy considerations but do not have known F‐theory realizations. In particular, we classify and carry out a local analysis of all enhancements of the irreducible gauge and matter contributions from “non‐Higgsable clusters,” and on isolated curves and pairs of intersecting rational curves of arbitrary self‐intersection. Such enhancements correspond physically to unHiggsings, and mathematically to tunings of the Weierstrass model of an elliptic CY threefold. We determine the shift in Hodge numbers of the elliptic threefold associated with each enhancement. We also consider local tunings on curves that have higher genus or intersect multiple other curves, codimension two tunings that give transitions in the F‐theory matter content, tunings of abelian factors in the gauge group, and generalizations of the “E₈” rule to include tunings and curves of self‐intersection zero. These tools can be combined into an algorithm that in principle enables a finite and systematic classification of all elliptic CY threefolds and corresponding 6D F‐theory SUGRA models over a given compact base (modulo some technical caveats in various special circumstances), and are also relevant to the classification of 6D SCFT's. To illustrate the utility of these results, we identify some large example classes of known CY threefolds in the Kreuzer‐Skarke database as Weierstrass models over complex surface bases with specific simple tunings, and we survey the range of tunings possible over one specific base.     

Calabi–Yau threefolds with non-Gorenstein involutions

The concept of non-Gorenstein involutions on Calabi–Yau threefolds is a higher dimensional generalization of non-symplectic involutions on K3 surfaces. We present some elementary facts about Calabi–Yau threefolds with non-Gorenstein involutions. We give a classification of the Calabi–Yau threefolds of Picard rank one with non-Gorenstein involutions, whose fixed locus is not zero-dimensional.     

Modular quintics in ℙ4

We will examine the arithmetic of some of the members of a pencil of symmetric quintics in projective 4‐space. We will give evidence for the modularity of some of the exceptional members (even the non‐rigid ones) and give a proof in one rigid case. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Calibrated Subbundles in Noncompact Manifolds of Special Holonomy

This paper is a continuation of Math. Res. Lett. 12 (2005), 493–512. We first construct special Lagrangian submanifolds of the Ricci-flat Stenzel metric (of holonomy SU(n)) on the cotangent bundle of Sⁿ by looking at the conormal bundle of appropriate submanifolds of Sⁿ. We find that the condition for the conormal bundle to be special Lagrangian is the same as that discovered by Harvey–Lawson for submanifolds in Rⁿ in their pioneering paper, Acta Math. 148 (1982), 47–157. We also construct calibrated submanifolds in complete metrics with special holonomy G₂ and Spin(7) discovered by Bryant and Salamon (Duke Math. J. 58 (1989), 829–850) on the total spaces of appropriate bundles over self-dual Einstein four manifolds. The submanifolds are constructed as certain subbundles over immersed surfaces. We show that this construction requires the surface to be minimal in the associative and Cayley cases, and to be (properly oriented) real isotropic in the coassociative case. We also make some remarks about using these constructions as a possible local model for the intersection of compact calibrated submanifolds in a compact manifold with special holonomy. Mathematics Subject Classification (2000): 53-XX, 58-XX.     

On the existence of stable bundles with prescribed Chern classes on Calabi–Yau threefolds

We prove a case of the conjecture of Douglas, Reinbacher and Yau about the existence of stable vector bundles with prescribed Chern classes on a Calabi–Yau threefold. For this purpose we prove the existence of certain stable vector bundle extensions over elliptically fibered Calabi–Yau threefolds.     

对称等仿射球和极小对称Lagrange子流形的对应

利用对称空间的对偶性,本文建立局部强凸对称等仿射球之集与某复空间形式中的极小对称Lagrange子流形之集间的对应关系,在自然定义的等价意义下,这是一一对应关系.作为这种对应关系的直接应用,本文用完全不同的方法重新证明胡泽军等人最近建立的一个重要定理.该定理对具有平行Fubini-Pick形式的局部强凸等仿射球进行了完全分类.