Heterotic-$\mathbf{F}$-theory Duality with Wilson Line Symmetry-breaking

Abstract

We begin with an $E_8\times E_8$ Heterotic model broken to an $SU(5{)}_{gauge}$ and a mirror $SU(5{)}_{gauge}$, where one $SU(5)$ and its spectrum is identified as the visible sector while the other can be identified as a hidden mirror world. In both cases we obtain the minimal supersymmetric standard model spectrum after Wilson-line symmetry-breaking enhanced by a low energy R-parity enforced by a local (or global) $U(1{)}_X$-symmetry. Using Heterotic/$F$-theory duality, we show how to eliminate the vector-like exotics which were obtained in previous constructions. In these constructions, the Calabi-Yau {[}CY{]} four-fold was defined by an elliptic fibration with section over a base $B_3$ and a GUT surface given by $K3/{\mathbb{Z}}_2=$ Enriques surface. In the present paper we construct a quotient CY four-fold fibered by tori with two elliptic structures given by a a pair of sections fibered over the Enriques surface. Using Heterotic/$F$-theory duality we are able to define the cohomologies used to derive the massless spectrum. Our model for the 'correct' $F$-theory dual of a Heterotic model with Wilson-line symmetry-breaking builds on prior literature but employs the stack-theoretic version of the dictionary between the Heterotic semi-stable $E_8$-bundles with Yang-Mills connection and the $d{P}_9$-fibrations used to construct the $F$-theory dual.

51 pages, corrected some typos

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