A Comparative Study of Laplacians and Schroedinger-Lichnerowicz-Weitzenboeck Identities in Riemannian and Antisymplectic Geometry
| Authors | |
|---|---|
| Year of publication | 2009 |
| Type | Article in Periodical |
| Magazine / Source | Journal of Mathematical Physics |
| MU Faculty or unit | |
| Citation | |
| Web | http://arxiv.org/abs/0809.4269 |
| Doi | http://dx.doi.org/10.1063/1.3152575 |
| Field | Theoretical physics |
| Keywords | Dirac Operator; Spin Representations; BV Field Antifield Formalism; Antisymplectic Geometry; Odd Laplacian |
| Description | We introduce an antisymplectic Dirac operator and antisymplectic gamma matrices. We explore similarities between, on one hand, the Schroedinger--Lichnerowicz formula for spinor bundles in Riemannian spin geometry, which contains a zeroth--order term proportional to the Levi--Civita scalar curvature, and, on the other hand, the nilpotent, Grassmann--odd, second--order \Delta operator in antisymplectic geometry, which in general has a zeroth--order term proportional to the odd scalar curvature of an arbitrary antisymplectic and torsionfree connection that is compatible with the measure density. Finally, we discuss the close relationship with the two--loop scalar curvature term in the quantum Hamiltonian for a particle in a curved Riemannian space. |
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