On a Z_3-Graded Generalization of the Witten Index

TL;DR

This paper generalizes the Witten index using Z_3-graded topological symmetry, constructing a new algebraic framework and linking topological invariants to analytic indices of specific operators.

Contribution

It introduces a Z_3-graded algebraic realization of topological symmetry and relates topological invariants to analytic indices, extending previous Z_2-based theories.

Findings

Constructed a Z_3-graded algebraic realization of topological symmetry.

Established the equality of topological invariants with analytic indices.

Linked the zero-energy subspace structure to complex formation.

Abstract

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.