Harmonic spinors on a family of Einstein manifolds

Abstract

The purpose of this paper is to study harmonic spinors defined on a 1-parameter family of Einstein manifolds which includes Taub-NUT, Eguchi-Hanson and P 2 (C) with the Fubini-Study metric as particular cases. We discuss the existence of and explicitly solve for spinors harmonic with respect to the Dirac operator twisted by a geometrically preferred connection. The metrics examined are defined, for generic values of the parameter, on a non-compact manifold with the topology of C 2 and extend to P 2 (C) as edge-cone metrics. As a consequence, the subtle boundary conditions of the Atiyah-Patodi-Singer index theorem need to be carefully considered in order to show agreement between the index of the twisted Dirac operator and the result obtained by counting the explicit solutions.

Harmonic forms and spinors on the Taub-bolt space

“…Harmonic spinors on TN have been studied in [14,10,5] so we only recall briefly the relevant results. As discussed in Section 3.2, any harmonic L 2 form on TN is self-dual and can be written…”

“…The operator P A is essentially the twisted Dirac operator on the squashed 3-sphere, which has been considered in [15]. A review of its main properties following the notation used here can be found in [5]. We recall in particular that, because of the spherical symmetry of the problem, the operators / D A , P A commute with the scalar Laplacian on the round 3-sphere…”

“…Since F is self-dual, only T A has non-trivial L 2 zero modes. It can be shown [10,5] that these zero modes are of the form…”

Axial anomaly in nonlinear conformal electrodynamics

“…Moreover, if the spinors are charged under some gauge group, the index theorem receives additional contributions coming from their second Pontryagin class. For example, in four dimensions, the index theorem of the Dirac operator coupled to U (1) gauge fields, i.e., / D = γ a E µ a (∂ µ + 1 4 ω ab µ γ ab + iA µ ), yields [78,107]…”

The Asymptotic Structure of the Centred Hyperbolic 2-Monopole Moduli Space

“…Following the analysis of [26,27], we reinterpret particle dynamics restricted to antipodal configurations as geodesic motion on an S 1 bundle over this 3-dimensional configuration space. By doing so we obtain a Riemannian metric, which we like to call hyperbolic Taub-NUT [12] due to its manifest similarities with the Taub-NUT metric, first constructed in [24]. The hyperbolic Taub-NUT metric, just like the Taub-NUT one, depends on one effective parameter M called mass, is complete for M ≥ 0 and becomes singular in the interior if M < 0.…”