THE SYMMETRY OF spin DIRAC SPECTRUMS ON RIEMANNIAN PRODUCT MANIFOLDS

Title & Authors
THE SYMMETRY OF spin DIRAC SPECTRUMS ON RIEMANNIAN PRODUCT MANIFOLDS
HONG, KYUSIK; SUNG, CHANYOUNG;

Abstract
It is well-known that the spectrum of a $\small{spin^{\mathbb{C}}}$ Dirac operator on a closed Riemannian $\small{spin^{\mathbb{C}}}$ manifold $\small{M^{2k}}$ of dimension 2k for $\small{k{\in}{\mathbb{N}}}$ is symmetric. In this article, we prove that over an odd-dimensional Riemannian product $\small{M^{2p}_1{\times}M^{2q+1}_2}$ with a product $\small{spin^{\mathbb{C}}}$ structure for $\small{p{\geq}1}$, $\small{q{\geq}0}$, the spectrum of a $\small{spin^{\mathbb{C}}}$ Dirac operator given by a product connection is symmetric if and only if either the $\small{spin^{\mathbb{C}}}$ Dirac spectrum of $\small{M^{2q+1}_2}$ is symmetric or $\small{(e^{{\frac{1}{2}}c_1(L_1)}{\hat{A}}(M_1))[M_1}$$\small{]}$$\small{=0}$, where $\small{L_1}$ is the associated line bundle for the given $\small{spin^{\mathbb{C}}}$ structure of $\small{M_1}$.
Keywords
Dirac operator;$\small{spin^{\mathbb{C}}}$ manifold;spectrum;eta invariant;
Language
English
Cited by
References
1.
M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43-69.

2.
M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry II, Math. Proc. Cambidge Philos. Soc. 78 (1975), 405-432.

3.
H. Baum, A remark on the spectrum of the Dirac operator on pseudo-Riemannian spin manifolds, SFB 288, Preprint 136, 1994.

4.
Th. Friedrich, Dirac Operators in Riemannian Geometry, Graduate Studies in Mathematics, vol. 25, AMS, Providence, RI, 2000.

5.
Th. Friedrich and E. C. Kim, The Einstein-Dirac equation on Riemannian spin manifolds, J. Geom. Phys. 33 (2000), no. 1-2, 128-172.

6.
E. C. Kim, Die Einstein-Dirac-Gleichung uber Riemannschen Spin-Mannigfaltigkeiten, Thesis, Humboldt-Universitat zu Berlin, 1999.

7.
E. C. Kim, The A-genus and symmetry of the Dirac spectrum on Riemannian product manifolds, Differential Geom. Appl. 25 (2007), 309-321.

8.
H.-B. Lawson and M.-L.Michelsohn, Spin Geometry, Princeton University Press, Princeton, NJ, 1989.

9.
A. Lichnerowicz, Champs spinoriels et propagateurs en relativite general, Bull. Sci. Math. France 92 (1964), 11-100.

10.
L. I. Nicolaescu, Notes on Seiberg-Witten Theory, Graduate Studies in Mathematics, 28. American Mathematical Society, Providence, RI, 2000.