Abstract
We give estimates for the eigenvalues of multi-form modified Dirac operators which are constructed from a standard Dirac operator with the addition of a Clifford algebra element associated to a multi-degree form. In particular such estimates are presented for modified Dirac operators with a k-degree form 0≤k≤4, those modified with multi-degree (0,k)-form 0≤k≤3 and the horizon Dirac operators which are modified with a multi-degree (1,2,4)-form. In particular, we give the necessary geometric conditions for such operators to admit zero modes as well as those for the zero modes to be parallel with a respect to a suitable connection. We also demonstrate that manifolds which admit such parallel spinors are associated with twisted covariant form hierarchies which generalize the conformal Killing–Yano forms.
Highlights
It is a consequence of the Lichnerowicz formula and theorem that the Dirac operator on closed1 spin manifolds Mn with scalar curvature R ⪈ 0, i.e. R ≥ 0 with R= 0 somewhere, does not admit zero modes
Other key results that are centred around the Lichnerowicz formula and theorem are the estimates for the eigenvalues of the Dirac operator
These are of two kinds, one is a lower bound on the eigenvalues of the Dirac operator, see [2,3,8], and other is an upper bound for the growth of the eigenvalues of the Dirac operator [1,10]. All these results summarize some of the applications of spinors to geometry
Summary
It is a consequence of the Lichnerowicz formula and theorem that the Dirac operator on closed spin manifolds Mn with scalar curvature R ⪈ 0, i.e. R ≥ 0 with R= 0 somewhere, does not admit zero modes. The purpose of this paper is to initiate a more systematic investigation of spin bundle connections ∇ˆ and multiform Dirac operators D on a spin Riemannian manifold for which a Lichnerowicz type of formula and theorem can be formulated, and to explore some applications to geometry. The conditions satisfied by the forms of the twisted covariant hierarchy can be found in Corollary 3.2.1 and Propositions 4.6 and 5.4 Another class of D operators that will be investigated are those constructed from degree (0, k) multi-forms.
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