The F-theory geometry with most flux vacua

TL;DR

This paper estimates that a specific elliptic Calabi-Yau fourfold yields an astronomically large number of F-theory flux vacua, making it a dominant configuration and providing insights into the emergence of the standard model in string theory.

Contribution

It identifies the fourfold with the most flux vacua and analyzes its geometric and gauge properties, highlighting its uniqueness and implications for string phenomenology.

Findings

Single fourfold ${ m M}_{ m max}$ dominates flux vacua count

Standard model gauge group likely arises from broken $E_8$

Other vacua are exponentially suppressed compared to ${ m M}_{ m max}$

Abstract

Applying the Ashok-Denef-Douglas estimation method to elliptic Calabi-Yau fourfolds suggests that a single elliptic fourfold ${\cal M}_{\rm max}$ gives rise to ${\cal O} (10^{272,000})$ 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 ${\cal O} (10^{-3000})$. The fourfold ${\cal M}_{\rm max}$ arises from a generic elliptic fibration over a specific toric threefold base $B_{\rm max}$, 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 ${\cal M}_{\rm max}$, 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 ${\cal M}_{\rm max}$.