2020/10/14 by Shamit Kachru, Richard Nally, Kachru, Shamit +3
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Black Holes and Theoretical Physics #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2010.07285
openalex publication_date 2020/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In recent work, we conjectured that Calabi-Yau threefolds defined over ℚ and admitting a supersymmetric flux compactification are modular, and associated to (the Tate twists of) weight-two cuspidal Hecke eigenforms. In this work, we will address two natural follow-up questions, of both a physical and mathematical nature, that are surprisingly closely related. First, in passing from a complex manifold to a rational variety, as we must do to study modularity, we are implicitly choosing a "rational model" for the threefold; how do different choices of rational model affect our results? Second, the same modular forms are associated to elliptic curves over ℚ; are these elliptic curves found anywhere in the physical setup? By studying the F-theory uplift of the supersymmetric flux vacua found in the compactification of IIB string theory on (the mirror of) the Calabi-Yau hypersurface X in ℙ(1,1,2,2,2), we find a one-parameter family of elliptic curves whose associated eigenforms exactly match those associated to X. Actually, we find two such families, corresponding to two different choices of rational models for the same family of Calabi-Yaus.