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A comparative study of Laplacians and Schrödinger– Lichnerowicz–Weitzenböck identities in Riemannian and antisymplectic geometry

### Abstract

We introduce an antisymplectic Dirac operator and antisymplectic gamma matrices. We explore similarities between, on one hand, the Schrödinger–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 operator in antisymplectic geometry, which, in general, has a zeroth-order term proportional to the odd scalar curvature of an arbitrary antisymplectic and torsion-free 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.

© 2009 American Institute of Physics

Received 12 December 2008
Accepted 15 May 2009
Published online 09 July 2009

Acknowledgments:
We would like to thank Poul Henrik Damgaard, the Niels Bohr Institute, and the Niels Bohr International Academy for warm hospitality. I.A.B. would also like to thank Michal Lenc and the Masaryk University for warm hospitality. The work of I.A.B. and K.B. is supported by the Ministry of Education of the Czech Republic under the Project No. MSM 0021622409. The work of I.A.B. is supported by Grant Nos. RFBR 08-01-00737, RFBR 08-02-01118, and LSS-1615.2008.2.

Article outline:

I. INTRODUCTION
A. General remarks about notation
II. GENERAL THEORY
A. Connection
B. Torsion
C. Divergence
D. The Riemann curvature
E. The Ricci tensor
F. The Ricci two-form
G. Covariant tensors
H. Coordinate transformations
III. RIEMANNIAN GEOMETRY
A. Metric
B. Laplacian
C. Two-cocycle
D. Scalar
E. and
F. Levi–Civita connection
G. The Riemann curvature
H. Scalar curvature
I. The operator at
J. Particle in curved space
K. First-order matrices
L. matrices
M. versus
N. Hodge ∗ operation
O. Hodge–Dirac operator
P. IS THERE A SECOND-ORDER FORMALISM?
IV. ANTISYMPLECTIC GEOMETRY
A. Metric
B. Odd Laplacian
C. Odd scalar
D. and
E. Antisymplectic connection
F. The Riemann curvature
G. Odd scalar curvature
H. First-order matrices
I. matrices
J. Dirac operator
K. IS THERE A SECOND-ORDER FORMALISM?
L. WHAT IS AN ANTISYMPLECTIC CLIFFORD ALGEBRA?
V. GENERAL SPIN THEORY
A. Spin manifold
B. Spin connection
C. Spin curvature
D. Covariant tensors with flat indices
E. Local gauge transformations
VI. RIEMANNIAN SPIN GEOMETRY
A. Spin geometry
B. Levi–Civita spin connection
C. First-order matrices
D. matrices and Clifford algebras
E. Dirac operator
F. Second-order matrices
G. Lichnerowicz’ formula
H. Clifford representations
I. Intertwining operator
J. Schrödinger–Lichnerowicz’ formula
VII. ANTISYMPLECTIC SPIN GEOMETRY
A. Spin geometry
B. First-order matrices
C. matrices
D. Dirac operator
E. SHIFTED MATRICES
VIII. CONCLUSIONS

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2009-07-09

2016-09-28

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