The Exact MSSM Spectrum from String Theory

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 E8 ×E8 heterotic superstring on a smooth Calabi-Yau threefold with an SU(4) gauge instanton and a Z3 ×Z3 Wilson line. Specifically, the observable sector is N = 1 supersymmetric with gauge group SU(3)C × SU(2)L ×U(1)Y ×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. Email: vbraun, [email protected], [email protected], [email protected]. In a number of conference talks [1], we introduced a minimal heterotic standard model whose observable sector has exactly the matter spectrum of the MSSM. This was motivated and constructed as follows. The gauge group Spin(10) is very compelling from the point of view of grand unification and string theory since a complete family of quarks and leptons plus a right-handed neutrino fits exactly into its 16 spin representation. Non-vanishing neutrino masses indicate that, in supersymmetric theories without exotic multiplets, a right-handed neutrino must be added to each family of quarks and leptons [2]. Within the context [3] of N = 1 supersymmetric E8 × E8 heterotic string vacua, a Spin(10) group can arise from the spontaneous breaking of the observable sector E8 group by an SU(4) gauge instanton on an internal Calabi-Yau threefold [4]. The Spin(10) group is then broken by discrete Wilson lines to a gauge group containing SU(3)C × SU(2)L × U(1)Y as a factor [5]. To achieve this, the Calabi-Yau manifold must have, minimally, a fundamental group Z3 × Z3. Until recently, such vacua could not be constructed since Calabi-Yau threefolds with fundamental group Z3 × Z3 and a method for building appropriate SU(4) gauge instantons on them were not known. The problem of finding elliptic Calabi-Yau threefolds with Z3 × Z3 fundamental group was rectified in [6]. That of constructing SU(4) instantons was solved in a series of papers [7], where a class of SU(4) gauge instantons on these Calabi-Yau manifolds was presented. Generalizing the results in [8, 9], these instantons were obtained as connections on rank 4 holomorphic vector bundles. In order for such connections to exist, it is necessary for the corresponding bundles to be slopestable. A number of non-trivial checks of the stability of these bundles was presented in [7]. A rigorous proof of the conjectured slope-stability recently appeared in [10]. The complete low energy spectra were computed in this context. The observable sectors were found to be almost that of the minimal supersymmetric standard model (MSSM). Specifically, the matter content of the most economical of these vacua consisted of three families of quarks/leptons, each family with a right-handed neutrino, and two Higgs– Higgs conjugate pairs. Apart from these, there were no other vector-like pairs, and no exotic particles. That is, the observable sector is almost that of the MSSM, but contains an extra pair of Higgs–Higgs conjugate fields. Additionally, there are 6 geometric moduli [6] and 19 vector bundle moduli [11]. In [12], it was shown that non-vanishing μ-terms can arise from cubic moduli-Higgs–Higgs conjugate interactions. Despite the