Quantum mechanical symmetries and 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 a-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), Zz-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. (C) 2001 Elsevier Science B.V.