Moduli and Hidden Matter in Heterotic M-Theory with an Anomalous $U(1)$ Hidden Sector

This paper discusses the dilaton, K\"ahler moduli and hidden sector matter chiral superfields of heterotic $M$-theory vacua in which the hidden sector gauge bundle is chosen to be a line bundle with an anomalous U(1) structure group. For simplicity of notation, the theory is compactified on a Calabi-Yau threefold with $h^{1,1}=1$, although all methods and results apply to more general heterotic compactifications. After introducing a non-perturbative $F$-term potential and coupling to supergravity, the canonically normalized scalar and fermion mass eigenstates, evaluated around a fixed supersymmetry breaking vacuum, are computed and the explicit expressions for their masses presented. In addition, the relevant couplings of these eigenstates to themselves and to chiral matter in the observable sector are evaluated. The decay rates of generic observable sector scalars into both moduli and hidden sector matter scalars and fermions are then presented. This opens the door to explicit calculations of the decay of an observable sector cosmological inflaton into moduli and hidden sector dark matter candidates. Finally, an explicit flux and gaugino condensate induced non-perturbative superpotential is introduced which is shown to stabilize three of the four real components of the moduli fields.