On a Z₃-Graded Generalization of the Witten Index

We construct a realization of the algebra of the Z_3-graded topological symmetry of type (1,1,1) in terms of a pair of operators D_1: H_1 -> H_2, and D_2: H_2 -> H_3 satisfying [D_1D_1^\dagger,D_2^\dagger D_2]=0. We show that the sequence of the restriction of these operators to the zero-energy subspace forms a complex and establish the equality of the corresponding topological invariants with the analytic indices of these operators.