The Exact MSSM Spectrum from String Theory

TL;DR

The paper demonstrates the existence of realistic vacua in string theory whose observable sector has exactly the MSSM matter content. By compactifying the $E_8 imes E_8$ heterotic string on a smooth Calabi–Yau threefold with an $SU(4)$ gauge instanton and a $\mathbb{Z}_3\times\mathbb{Z}_3$ Wilson line, the authors realize an $N=1$ supersymmetric observable sector with gauge group $SU(3)_C\times SU(2)_L\times U(1)_Y\times U(1)_{B-L}$, three quark/lepton families with right-handed neutrinos, and a single Higgs–Higgs conjugate pair, with no exotic matter and no vector-like pairs. They construct a slope-stable $SU(4)$ bundle on a quotient Calabi–Yau $X=\tilde X/(\mathbb{Z}_3\times\mathbb{Z}_3)$ with $h^{1,1}=h^{2,1}=3$ (six geometric moduli) and compute the exact massless spectrum after the Wilson line, including 13 vector-bundle moduli and a single MSSM-like Higgs pair. The work also addresses Yukawa couplings, anomaly cancellation, and the existence of a consistent hidden sector bundle $V'$ meeting stability and Bogomolov-type constraints, demonstrating a concrete realization of minimal heterotic standard models in both weakly and strongly coupled regimes. The results suggest these vacua are the closest string-theoretic realizations to the MSSM and may have significant phenomenological implications.

Abstract

We show the existence of realistic vacua in string theory whose observable sector has exactly the matter content of the MSSM. This is achieved by compactifying the E_8 x E_8 heterotic superstring on a smooth Calabi-Yau threefold with an SU(4) gauge instanton and a Z_3 x Z_3 Wilson line. Specifically, the observable sector is N=1 supersymmetric with gauge group SU(3)_C x SU(2)_L x U(1)_Y x U(1)_{B-L}, three families of quarks and leptons, each family with a right-handed neutrino, and one Higgs-Higgs conjugate pair. Importantly, there are no extra vector-like pairs and no exotic matter in the zero mode spectrum. There are, in addition, 6 geometric moduli and 13 gauge instanton moduli in the observable sector. The holomorphic SU(4) vector bundle of the observable sector is slope-stable.