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The F-theory geometry with most flux vacua. (English) Zbl 1388.81367
Summary: 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 $$E^9_8\times F^8_4\times(G_2\times\mathrm{SU}(2))^{16}$$, of which we expect some factors to be broken by G-flux to smaller groups. It is not possible to tune an $$\mathrm{SU}(5)$$ GUT group on any further divisors in $$\mathcal M_{\max}$$, or even an $$\mathrm{SU}(2)$$ or $$\mathrm{SU}(3)$$, 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 $$\mathcal M_{\max}$$.

##### MSC:
 81T13 Yang-Mills and other gauge theories in quantum field theory
##### Keywords:
flux compactifications; F-theory; superstring vacua
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