Kastler-Kalau-Walze type theorems for an even dimensional manifold with boundary
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摘要: 在任意偶数维带边Spin流形上建立了一类关于带挠率的Dirac算子的Kastler-Kalau-Walze类型定理, 为相应流形上的Einstein-Hilbert作用给出了简单的算子理论解释.
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关键词:
- 带挠率的Dirac算子 /
- 非交换留数 /
- 低维体积 /
- 偶数维带边流形
Abstract: In this paper, we establish a Kastler-Kalau-Walze type theorem for an even dimensional manifold with boundary about Dirac operators with torsion; in addition, we provide a simple theoretical explanation to the Einstein-Hilbert action for any even dimensional manifold with boundary. -
[1] ADLER M. On a trace functional for formal pseudo-differential operators and the symplectic structure of Korteweg-de Vries type equations [J]. Invent Math, 1979, 50: 219-248. [2] WODZICKI M. Local invariants of spectral asymmetry [J]. Invent Math, 1995, 75(1): 143-178. [3] GUILLEMIN V W. A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues [J]. Adv Math, 1985, 55(2): 131-160. doi: 10.1016/0001-8708(85)90018-0 [4] CONNES A. The action functinal in noncommutative geometry [J]. Comm Math Phys, 1998, 117: 673-683. [5] KASTLER D. The Dirac operator and gravitation [J]. Comm Math Phys, 1995, 166: 633-643. doi: 10.1007/BF02099890 [6] KALAU W, WALZE M. Gravity, noncommutative geometry and the Wodzicki residue [J]. J Geom Phys, 1995, 16: 327-344. doi: 10.1016/0393-0440(94)00032-Y [7] FEDOSOV B V, GOLSE F, LEICHTNAM E, et al. The noncommutative residue for manifolds with boundary [J]. J Funct Anal, 1996, 142: 1-31. doi: 10.1006/jfan.1996.0142 [8] WANG Y. Gravity and the noncommutative residue for manifolds with boundary [J]. Lett Math Phys, 2007, 80: 37-56. doi: 10.1007/s11005-007-0147-1 [9] WANG Y. Lower-dimensional volumes and Kastler-Kalau-Walze type theorem for manifolds with boundary [J]. Commun Theor Phys, 2010, 54: 38-42. doi: 10.1088/0253-6102/54/1/08 [10] WANG Y. Differential forms and the Wodzicki residue for manifolds with boundary [J]. J Geom Phys, 2006, 56: 731-753. doi: 10.1016/j.geomphys.2005.04.015 [11] PONGE R. Noncommutative geometry and lower dimensional volumes in Riemannian geometry [J]. Lett Math Phys, 2008, 83: 1-19. doi: 10.1007/s11005-007-0201-z [12] ACKERMANN T, TOLKSDORF J. A generalized Lichnerowicz formula, the Wodzicki residue and gravity [J]. J Geom Phys, 1996, 19: 143-150. doi: 10.1016/0393-0440(95)00030-5 [13] PÄFFLE F, STEPHAN C A. On gravity, torsion and the spectral action principle [J]. J Funct Anal, 2012, 262: 1529-1565. doi: 10.1016/j.jfa.2011.11.013 [14] PÄFFLE F, STEPHAN C A. Chiral asymmetry and the spectral action [J]. Comm Math Phys, 2013, 321: 283-310. doi: 10.1007/s00220-012-1641-6 [15] WANG J, WANG Y, YANG C L. Dirac operators with torsion and the noncommutative residue for manifolds with boundary [J]. J Geom Phys, 2014, 81: 92-111. doi: 10.1016/j.geomphys.2014.03.007 [16] BAO K H, SUN, A H, WANG J. A Kastler-Kalau-Walze type theorem for 7-dimensional spin manifolds with boundary about Dirac operators with torsion [J]. J Geom Phys, 2016, 110: 213-232. doi: 10.1016/j.geomphys.2016.08.005 [17] WANG J, WANG Y. A general Kastler-Kalau-Walze type theorem for manifolds with boundary [J]. International Journal of Geometric Methods in Modern Physics, 2016, 13(1): 1650003. doi: 10.1142/S0219887816500031 [18] BERLINE N, GETZLER E, VERGNE M. Heat Kernels and Dirac Operators [M]. Berlin: Springer-Verlag, 1992. [19] YU Y L. The Index Theorem and Heat Equation Method [M]. Singapore: World Scientific Publishing, 2001. [20] ACKERMANN T. A note on the Wodzicki residue [J]. J Geom Phys, 1996, 20: 404-406. doi: 10.1016/S0393-0440(95)00061-5
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