Cost of seven-brane gauge symmetry in a quadrillion F-theory compactifications

Abstract

We study seven-branes in O(10 15 ) four-dimensional F-theory compactifications where seven-brane moduli must be tuned in order to achieve non-abelian gauge symmetry. The associated compact spaces B are the set of all smooth weak Fano toric threefolds. By a study of fine star regular triangulations of three dimensional reflexive polytopes, the number of such spaces is estimated to be 5.8×10 14 N bases 1.8×10 17 . Typically hundreds or thousands of moduli must be tuned to achieve symmetry for h 11 (B) < 10, but the average number drops sharply into the range O(25)-O(200) as h 11 (B) increases. For some low rank groups, such as SU (2) and SU (3), there exist examples where only a few moduli must be tuned in order to achieve seven-brane gauge symmetry.

Machine learning in the string landscape

“…It is also important that our model makes accurate predictions beyond the data on which it trained. Specifically, it trained on data with h 11 ≤ 21 and the predicted values at h 11 = 22, 23, 24, 25 are in good agreement with the results of [17]. The model also makes good predictions at h 11 = 26, 27 when comparing to the extrapolation of the best fit line.…”

“…Each such FRST determines a smooth weak-Fano toric variety. These varieties and their number are interesting for at least three reasons: their anti-canonical hypersurfaces give rise to K3 surfaces that can be used for six-dimensional string compactifications, they give rise to smooth F-theory compactifications without non-Higgsable clusters [17], and they serve as a starting point for topological transitions from which the ensemble of 10 755 F-theory geometries arises.…”

“…The number of FRSTs of these polytopes was computed for low h 11 (B) in [17], where B is the toric variety associated to the FRST, and estimates were provided for the remainder based on techniques in the triangulation literature. Here we instead wish to estimate the number of FRSTs of the 3d reflexive polytopes using machine learning.…”

“…Given the accuracy of this model, we would now like to compute the average number of FRSTs per polytope as a function of h 11 (B). This was done in [17], and it will be instructive to see whether machine learning recovers those results, and also where it has difficulties. Specifically, letting n T (F ) be the number of FRTs of F computed by our model, we use the approximation discussed in [17]…”

Estimating Calabi-Yau hypersurface and triangulation counts with equation learners

“…Specifically, a decision-tree regression model was trained on known results, and used to estimate the number of FRSTs for the remaining cases [33]. The results were in good agreement with the original estimate of [40]. The objective of the current work was to obtain such an estimate for the 4d reflexive polytopes via similar techniques.…”