Abstract
We carry out a systematic analysis of Calabi-Yau threefolds that are elliptically fibered with section ("EFS") and have a large Hodge number h^{2, 1}. EFS Calabi-Yau threefolds live in a single connected space, with regions of moduli space associated with different topologies connected through transitions that can be understood in terms of singular Weierstrass models. We determine the complete set of such threefolds that have h^{2, 1} >= 350 by tuning coefficients in Weierstrass models over Hirzebruch surfaces. The resulting set of Hodge numbers includes those of all known Calabi-Yau threefolds with h^{2, 1} >= 350, as well as three apparently new Calabi-Yau threefolds. We speculate that there are no other Calabi-Yau threefolds (elliptically fibered or not) with Hodge numbers that exceed this bound. We summarize the theoretical and practical obstacles to a complete enumeration of all possible EFS Calabi-Yau threefolds and fourfolds, including those with small Hodge numbers, using this approach.
Highlights
Since the early days of string theory, the geometry of Calabi-Yau (CY) threefolds has played an important role in understanding compactifications of the theory that give rise to four-dimensional effective physics [1,2,3,4]
We carry out a systematic analysis of Calabi-Yau threefolds that are elliptically fibered with section (“EFS”) and have a large Hodge number h2,1
EFS Calabi-Yau threefolds live in a single connected space, with regions of moduli space associated with different topologies connected through transitions that can be understood in terms of singular Weierstrass models
Summary
Since the early days of string theory, the geometry of Calabi-Yau (CY) threefolds has played an important role in understanding compactifications of the theory that give rise to four-dimensional effective physics [1,2,3,4]. A constructive context from the finite number of topologically distinct tunings (strata) of the class of Weierstrass models over the minimal bases P2 and Fm (the Enriques surface is not as interesting since, up to torsion, the canonical class K vanishes, so the Weierstrass model is essentially trivial) From this point of view, in principle all EFS CY threefolds can be constructed by starting with the base surfaces P2 and Fm, and tuning the Weierstrass models over these bases in all possible ways consistent with the existence of a Calabi-Yau elliptic fibration. A similar analysis can be used to systematically construct all elliptically fibered Calabi-Yau threefolds with section at increasingly small values of h2,1 While this procedure becomes computationally intensive at lower values of h2,1, and there are a number of practical and theoretical issues that must be resolved before a complete classification is possible, this program could in principle be pursued to enumerate all EFS CY3s.
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