A Kastler-Kalau-Walze Type Theorem for 7-Dimensional Manifolds with Boundary about Witten Deformation
DOI:
https://doi.org/10.15407/mag17.02.119Анотація
У цій роботі ми методом повного перебору доводимо теорему типу Кастлера-Калау-Вальце про деформацію Віттена для $7$-вимірних многовидів з межею і даємо теоретичне пояснення гравітаційної дії для $7$-вимірних многовидів з межею.Mathematics Subject Classification: 58G20, 53A30, 46L87
Ключові слова:
деформація Віттена, некомутативний залишок для многовидів з межею, низьковимірний об'ємПосилання
T. Ackermann, A note on the Wodzicki residue, J. Geom. Phys. 20 (1996), 404–406. https://doi.org/10.1016/S0393-0440(95)00061-5
Kai Hua Bao and Ying Lei, The Kastler–Kalau–Walze type theorem about Witten deformation for manifolds with boundary, J. Geom. Phys. 131 (2018), 170–181. https://doi.org/10.1016/j.geomphys.2018.04.013
Kai Hua Bao, Ai Hui Sun, and Chao Deng, The Noncommutative Residue about Witten Deformation, Acta Math. Appl. Sin. Engl. Ser. 35 (2019), 550–568. https://doi.org/10.1007/s10114-018-8069-7
Kai Hua Bao, Ai Hui Sun, and Jian Wang, A Kastler–Kalau–Walze Type Theorem for 7-dimensional spin manifolds with boundary about Dirac operators with torsion, Zh. Mat. Fiz. Anal. Geom 110 (2016), 213–232. https://doi.org/10.1016/j.geomphys.2016.08.005
A. Connes, Quantized calculus and applications, XIth International Congress of Mathematical Physics (Paris, 1994), Internat Press, Cambridge, MA, 1995, 15–36.
A. Connes, The action functinal in Noncommutative geometry, Comm. Math. Phys. 117 (1998), 673–683. https://doi.org/10.1007/BF01218391
B.V. Fedosov, F. Golse, E. Leichtnam, and E. Schrohe, The noncommutative residue for manifolds with boundary, J. Funct. Anal. 142 (1996), 1–31. https://doi.org/10.1006/jfan.1996.0142
V.W. Guillemin, A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues, Adv. Math. 55 (1985), 131–160. https://doi.org/10.1016/0001-8708(85)90018-0
W. Kalau and M. Walze, Gravity, Noncommutative geometry and the Wodzicki residue, J. Geom. Physics. 16 (1995), 327–344. https://doi.org/10.1016/0393-0440(94)00032-Y
D. Kastler, The Dirac Operator and Gravitation, Comm. Math. Phys. 166 (1995), 633–643. https://doi.org/10.1007/BF02099890
R. Ponge, Noncommutative geometry and lower dimensional volumes in Riemannian geometry, Lett. Math. Phys. 83 (2008), 19–32. https://doi.org/10.1007/s11005-007-0199-2
W.J. Ugalde, Differential forms and the Wodzicki residue, preprint, https://arxiv. org/abs/math/0211361.
J. Wang, Y. Wang, and C.L. Yang, A Kastler–Kalau–Walze Type Theorem for 7Dimensional Manifolds with Boundary, Abstr. Appl. Anal. 2014 (2014), Art. ID: 465782. https://doi.org/10.1155/2014/465782
J. Wang, Y. Wang, and C.L. Yang, Dirac operators with torsion and the noncommutative residue for manifolds with boundary, J. Geom. Physics. 81 (2014), 92–111. https://doi.org/10.1016/j.geomphys.2014.03.007
Y. Wang, Diffential forms and the Wodzicki residue for Manifolds with Boundary, J. Geom. Physics 56 (2006), 731–753. https://doi.org/10.1016/j.geomphys.2005.04.015
Y. Wang, Diffential forms the Noncommutative Residue for Manifolds with Boundary in the non-product Case, Lett. Math. Phys. 77 (2006), 41–51. https://doi.org/10.1007/s11005-006-0078-2
Y. Wang, Gravity and the Noncommutative Residue for Manifolds with Boundary, Lett. Math. Phys. 80 (2007), 37–56. https://doi.org/10.1007/s11005-007-0147-1
Y. Wang, Lower-Dimensional Volumes and Kastler–kalau–Walze Type Theorem for Manifolds with Boundary, Commun. Theor. Phys. 54 (2010), 38–42. https://doi.org/10.1088/0253-6102/54/5/03
M. Wodzicki, Local invariants of spectral asymmetry, Invent. Math. 75(1) (1995), 143–178. https://doi.org/10.1007/BF01403095
Y. Yu,The Index Theorem and The Heat Equation Method, Nankai Tracts in Mathematics, 2, World Scientific, Singapore, 2001. https://doi.org/10.1142/4691
Y. Yu, N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators, Springer-Verlag, Berlin, 1992. https://doi.org/10.1007/978-3-642-58088-8
W. Zhang, Lectures on chern-weil theory and witten deformation, Nankai Tracts in Mathematics, 4, World Scientific, Singapore, 2001. https://doi.org/10.1142/4756