Construction of a topological charge on fuzzyS2×S2via a Ginsparg-Wilson relation

Abstract

We construct a topological charge of gauge field configurations on a fuzzy ${S}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{2}$ by using a Dirac operator satisfying the Ginsparg-Wilson relation. The topological charge defined on the fuzzy ${S}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{2}$ can be interpreted as a noncommutative (or matrix) generalization of the 2nd Chern character on ${S}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{2}$. We further calculate the number of chiral zero modes of the Dirac operator in topologically nontrivial gauge configurations. Generalizations of our formulation to fuzzy $({S}^{2}{)}^{k}$ are also discussed.