Effective zero-mode hamiltonian in supersymmetric chiral non-Abelian gauge theories

The effective Hamiltonian for supersymmetric chiral gauge theories placed in the finite space box of size L much less than the confinement length Λ-1 is constructed. Two examples are discussed: the theory with the gauge group SU(3) involving 7 left-handed triplets and a right-handed sextet and the SU(5) theory involving a left-handed quintet and a right-handed decuplet. The second example is the most interesting as there are strong indications that supersymmetry is spontaneously broken in this theory due to dynamical effects. The effective hamiltonians, analogous to the hamiltonian of the chiral supersymmetric QED, display the lattice structure in the space of weights of the gauge group. Unfortunately, they are very complicated and we did not succeed in calculating reliably the Witten index for these hamiltonians and are unable to reveal supersymmetry breaking in the SU(5) model and its absence in the SU(3) model in this approach