Higgs Doublets, Split Multiplets and Heterotic SU(3)_C x SU(2)_L x U(1)_Y Spectra

TL;DR

This paper develops a method to compute the massless spectrum of heterotic vacua with Wilson lines and applies it to torus-fibered Calabi-Yau threefolds $X$ with $\\pi_1(X)=\mathbb{Z}_2$ and holomorphic $SU(5)$ bundles $V$, yielding low-energy theories with gauge group $SU(3)_C \times SU(2)_L \times U(1)_Y$ and three quark/lepton families. The spectrum is determined by decomposing ${\rm ad}\tilde{V}$ under $G\times H$ and selecting $\mathbb{Z}_2$-invariant components via the Dirac operator kernel, with cohomology groups $H^q(\tilde{X}, U_i(\tilde{V}))$ and $H^q(\tilde{X}, \wedge^k \tilde{V})$ computed and the $\mathbb{Z}_2$ action tracked on both $U_i(\tilde{V})$ and the $H$-representations $R_i$. In a concrete example with $F=\mathbb{Z}_2$ and $G=SU(5)$, the resulting massless spectrum contains one vector supermultiplet in the SM adjoint, three families with representations $(3,2)_1$, $(\overline{3},1)_{-4}$, $(\overline{3},1)_2$, $(1,2)_{-3}$, $(1,1)_6$, plus additional chiral exotics and nine Higgs doublet conjugate pairs, i.e., $n_{(1,2)_3}=9$, indicating substantial non-minimality. The work provides a general, computable framework for analyzing heterotic vacua with Wilson lines and highlights the need to explore broader moduli spaces to achieve phenomenologically viable spectra.

Abstract

A methodology for computing the massless spectrum of heterotic vacua with Wilson lines is presented. This is applied to a specific class of vacua with holomorphic SU(5)-bundles over torus-fibered Calabi-Yau threefolds with fundamental group Z_2. These vacua lead to low energy theories with the standard model gauge group SU(3)_C x SU(2)_L x U(1)_Yand three families of quark/leptons. The massless spectrum is computed, including the multiplicity of Higgs doublets.