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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 \(\mathcal M_{\max}\).

81T13 Yang-Mills and other gauge theories in quantum field theory
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