Abstract
We provide the first estimate of the number of fine, regular, star triangulations of the four-dimensional reflexive polytopes, as classified by Kreuzer and Skarke (KS). This provides an upper bound on the number of Calabi-Yau threefold hypersurfaces in toric varieties. The estimate is performed with deep learning, specifically the novel equation learner (EQL) architecture. We demonstrate that EQL networks accurately predict numbers of triangulations far beyond the h1,1 training region, allowing for reliable extrapolation. We estimate that number of triangulations in the KS dataset is 1010,505, dominated by the polytope with the highest h1,1 value.
Highlights
In string theory, Calabi-Yau manifolds are some of the simplest and best-studied backgrounds on which to compactify the extra dimensions of space while preserving supersymmetry in four dimensions [1]
We provide the first estimate of the number of fine, regular, star triangulations of the four-dimensional reflexive polytopes, as classified by Kreuzer and Skarke (KS)
One concrete goal is to understand the full ensemble of Calabi-Yau threefolds, networklike structures induced by transitions between them, and associated implications for cosmological dynamics and vacuum selection in compatifications of string theory
Summary
Batyrev has shown [33] that a hypersurface in a toric variety can be chosen to be Calabi-. In [39], the number of FRSTs for many of the 3d reflexive polytopes was computed, and an estimation of the total number was obtained, using standard triangulation techniques. Soon thereafter, another estimate was obtained using supervised machine learning. A neural network was employed to construct a model which predicts the natural logarithm of the number of FRTs of each facet From this we arrive at an estimate of the total number of FRSTs of the 4d reflexive polytopes.
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