Towards $N=2\mathrm{}$ SUSY homogeneous quantum cosmology: Einstein-Yang-Mills systems

The application of $N=2\mathrm{}$ supersymmetric quantum mechanics for the quantization of homogeneous systems coupled with gravity is discussed. Starting with the superfield formulation of an $N=2\mathrm{}$ SUSY sigma model, Hermitian self-adjoint expressions for quantum Hamiltonians and Lagrangians for any signature of a sigma-model metric are obtained. This approach is then applied to coupled SU(2) Einstein-Yang-Mills (EYM) systems in axially symmetric Bianchi-type I, II, VIII, IX, Kantowski-Sachs, and closed Friedmann-Robertson-Walker cosmological models. It is shown that all these models admit the embedding into the $N=2\mathrm{}$ SUSY sigma model with the explicit expressions for superpotentials being direct sums of gravitational and Yang-Mills (YM) parts. In addition, the YM parts of superpotentials exactly coincide with the corresponding Chern-Simons terms. The spontaneous SUSY breaking caused by YM instantons in EYM systems is discussed in a number of examples.