Quantum Mechanical Symmetries and Topological Invariants

TL;DR

This paper introduces the concept of topological symmetries as a generalization of supersymmetry, exploring their algebraic structure and associated topological invariants, including extensions beyond the Witten index.

Contribution

It defines topological symmetries with a grading parameter and relates them to known supersymmetries, providing a mathematical framework and interpretation for their invariants.

Findings

Identified algebraic structures of various supersymmetries as special cases of topological symmetries.

Established the connection between topological invariants and indices of Fredholm operators.

Extended the concept of the Witten index to higher-order topological invariants.

Abstract

We give the definition and explore the algebraic structure of a class of quantum symmetries, called topological symmetries, which are generalizations of supersymmetry in the sense that they involve topological invariants similar to the Witten index. A topological symmetry (TS) is specified by an integer n>1, which determines its grading properties, and an n-tuple of positive integers (m_1,m_2,...,m_n). We identify the algebras of supersymmetry, p=2 parasupersymmetry, and fractional supersymmetry of order n with those of the Z_2-graded TS of type (1,1), Z_2-graded TS of type (2,1), and Z_n-graded TS of type (1,1,...,1), respectively. We also comment on the mathematical interpretation of the topological invariants associated with the Z_n-graded TS of type (1,1,...,1). For n=2, the invariant is the Witten index which can be identified with the analytic index of a Fredholm operator. For n>2, there are n independent integer-valued invariants. These can be related to differences of the dimension of the kernels of various products of n operators satisfying certain conditions.