Balancing the vacuum energy in heterotic $M$-theory

Moduli stabilisation is explored in the context of low-energy heterotic $M$-theory to show that a small value of the cosmological constant can result from a balance between the negative potential energy left over from stabilising the moduli and a positive Casimir energy from the higher dimensions. Supersymmetry breaking is induced by the fermion boundary conditions on the two branes in the theory. An explicit calculation of the Casimir energy for the gravitino reveals that the energy has the correct sign, although the size of the contribution is close to the edge of the parameter range for which the calculation is valid.