Ginsparg-Wilson Dirac operator in monopole backgrounds on the fuzzy 2-sphere

Abstract

In the previous papers, we studied the 't Hooft-Polyakov (TP) monopole configurations in the U(2) gauge theory on the fuzzy 2-sphere, and showed that they have nonzero topological charges in the formalism based on the Ginsparg-Wilson (GW) relation. In this paper, we will show an index theorem in the TP monopole background, which is defined in the projected space, and provide a meaning of the projection operator. We also extend the index theorem to general configurations which do not satisfy the equation of motion, and show that the configuration space can be classified into the topological sectors. We further calculate the spectrum of the GW Dirac operator in the TP monopole backgrounds, and consider the index theorem in these cases.

Dominance of a single topological sector in gauge theory on non-commutative geometry

“…For example, in the random matrix theory for QCD, the relative weight is fixed by the guiding principle that one should recover QCD [36]. Alternatively, one can introduce the Higgs sector in the NC gauge theory [37] (See [38] for a related work.). In this case, the winding number of the Higgs field plays the role of the twisted boundary condition, and all the topological sectors appear from a single theory.…”

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

“…As is shown in detail in section III of ref. [15], in addition to the (2J X + 1)-folded degeneracy, the state |J X has an extra two-folded degeneracy for J X = do not have such two-folded degeneracy. The lowest spin states are shown to be zero modes of the operator P X (Γ −Γ) X , while the highest spin states are zero modes of the operator P X (Γ +Γ) X .…”

“…In the case of the fuzzy S 2 , we constructed a 't Hooft-Polyakov monopole configuration where the gauge symmetry group U (2) is spontaneously-broken down to U (1)×U (1) [21,20,15,16]. Since the diagonal U (1) is decoupled in the commutative limit, we discuss only the SU (2) part of the gauge group in the following.…”

“…It was given in [13] and analyzed in [14]. For constructions of the GW Dirac operators in gauge field backgrounds with nontrivial topology, see [15,16] for fuzzy S 2 and [17] for noncommutative tori.…”

“…In order to construct topologically nontrivial configurations, we need a bit more modification [21,20,15,16]. Consider, for instance, U (2) gauge theory on the fuzzy S 2 .…”