Index Theorem for the $q$-Deformed Fuzzy Sphere

TL;DR

This paper computes the index of the Dirac operator on the q-deformed fuzzy sphere, linking it to topological invariants and introducing a q-deformed chirality operator with invariant trace properties.

Contribution

It introduces a q-deformed chirality operator and demonstrates its trace invariance under the quantum group, connecting the index to topological invariants on the q-deformed fuzzy sphere.

Findings

The q-invariant trace of the chirality operator yields the topological index.

The q-dimension of zero modes relates to the topological index.

Construction of a q-deformed chirality operator with invariant trace.

Abstract

We calculate the index of the Dirac operator defined on the q-deformed fuzzy sphere. The index of the Dirac operator is related to its net chiral zero modes and thus to the trace of the chirality operator. We show that for the q-deformed fuzzy sphere, a $\uq$ invariant trace of the chirality operator gives the q-dimension of the eigenspace of the zero modes of the Dirac operator. We also show that this q-dimension is related to the topological index of the spinorial field. We then introduce a q-deformed chirality operator and show that its $\uq$ invariant trace gives the topological invariant index of the Dirac operator. We also explain the construction and important role of the trace operation which is invariant under the $\uq$, which is the symmetry algebra of the q-deformed fuzzy sphere. We briefly discuss chiral symmetry of the spinorial action on q-deformed fuzzy sphere and the possible role of this deformed chiral operator in its evaluation using path integral methods.