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The Hodge Numbers of Divisors of Calabi‐Yau Threefold Hypersurfaces
Author(s) -
Braun Andreas P.,
Long Cody,
McAllister Liam,
Stillman Michael,
Sung Benjamin
Publication year - 2020
Publication title -
fortschritte der physik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.469
H-Index - 71
eISSN - 1521-3978
pISSN - 0015-8208
DOI - 10.1002/prop.202000087
Subject(s) - calabi–yau manifold , divisor (algebraic geometry) , hypersurface , mathematics , pure mathematics , cohomology , euclidean geometry , toric variety , geometry
We prove a formula for the Hodge numbers of square‐free divisors of Calabi‐Yau threefold hypersurfaces in toric varieties. Euclidean branes wrapping divisors affect the vacuum structure of Calabi‐Yau compactifications of type IIB string theory, M‐theory, and F‐theory. Determining the nonperturbative couplings due to Euclidean branes on a divisor D requires counting fermion zero modes, which depend on the Hodge numbersh i ( O D ) . Suppose that X is a smooth Calabi‐Yau threefold hypersurface in a toric variety V , and let D be the restriction to X of a square‐free divisor of V . We give a formula forh i ( O D )in terms of combinatorial data. Moreover, we construct a CW complex P D such thath i ( O D ) = h i ( P D ) . We describe an efficient algorithm that makes possible for the first time the computation of sheaf cohomology for such divisors at large h 1, 1 . As an illustration we compute the Hodge numbers of a class of divisors in a threefold withh 1 , 1 = 491 . Our results are a step toward a systematic computation of Euclidean brane superpotentials in Calabi‐Yau hypersurfaces.