The F-theory geometry with most flux vacua

Abstract

Applying the Ashok-Denef-Douglas estimation method to elliptic Calabi-Yau fourfolds suggests that a single elliptic fourfold ${\mathcal{M}}_{\max }$ gives rise to $\mathcal{O}(10^{272,000})$ F-theory flux vacua, and that the sum total of the numbers of flux vacua from all other F-theory geometries is suppressed by a relative factor of $\mathcal{O}(10^{-3000})$. The fourfold ${\mathcal{M}}_{\max }$ arises from a generic elliptic fibration over a specific toric threefold base $B_{\max }$, and gives a geometrically non-Higgsable gauge group of $E_8^9\times F_4^8\times (G_2\times SU(2){)}^{16}$, of which we expect some factors to be broken by G-flux to smaller groups. It is not possible to tune an $SU(5)$ GUT group on any further divisors in ${\mathcal{M}}_{\max }$, or even an $SU(2)$ or $SU(3)$, so the standard model gauge group appears to arise in this context only from a broken $E_8$ factor. The results of this paper can either be interpreted as providing a framework for predicting how the standard model arises most naturally in F-theory and the types of dark matter to be found in a typical F-theory compactification, or as a challenge to string theorists to explain why other choices of vacua are not exponentially unlikely compared to F-theory compactifications on ${\mathcal{M}}_{\max }$.

19 pages, 2 figures, v3: minor corrections, clarifications, references added