A Kastler-Kalau-Walze Type Theorem for 7-Dimensional Manifolds with Boundary about Witten Deformation

In the paper, we give a brute-force proof of the Kastler-Kalau-Walze type theorem for 7-dimensional manifolds with boundary about the Witten deformation and give a theoretical explanation of the gravitational action for 7 dimensional manifolds with boundary.

Mathematics Subject Classification 2010: 58G20, 53A30, 46L87
Key words: Witten deformation, noncommutative residue for manifolds with boundary, lower dimensional volumes

References

[1] T. Ackermann, A note on the Wodzicki residue, J. Geom. Phys. 20 (1996), 404–406. CrossRef

[2] 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. CrossRef

[3] 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. CrossRef

[4] 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. CrossRef

[5] A. Connes, Quantized calculus and applications, XIth International Congress of Mathematical Physics (Paris, 1994), Internat Press, Cambridge, MA, 1995, 15–36.

[6] A. Connes, The action functinal in Noncommutative geometry, Comm. Math. Phys. 117 (1998), 673–683. CrossRef

[7] B.V. Fedosov, F. Golse, E. Leichtnam, and E. Schrohe, The noncommutative residue for manifolds with boundary, J. Funct. Anal. 142 (1996), 1–31. CrossRef

[8] V.W. Guillemin, A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues, Adv. Math. 55 (1985), 131–160. CrossRef

[9] W. Kalau and M. Walze, Gravity, Noncommutative geometry and the Wodzicki residue, J. Geom. Physics. 16 (1995), 327–344. CrossRef

[10] D. Kastler, The Dirac Operator and Gravitation, Comm. Math. Phys. 166 (1995), 633–643. CrossRef

[11] R. Ponge, Noncommutative geometry and lower dimensional volumes in Riemannian geometry, Lett. Math. Phys. 83 (2008), 19–32. CrossRef

[12] W.J. Ugalde, Differential forms and the Wodzicki residue, preprint, https://arxiv. org/abs/math/0211361.

[13] 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. CrossRef

[14] 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. CrossRef

[15] Y. Wang, Diffential forms and the Wodzicki residue for Manifolds with Boundary, J. Geom. Physics 56 (2006), 731–753. CrossRef

[16] Y. Wang, Diffential forms the Noncommutative Residue for Manifolds with Boundary in the non-product Case, Lett. Math. Phys. 77 (2006), 41–51. CrossRef

[17] Y. Wang, Gravity and the Noncommutative Residue for Manifolds with Boundary, Lett. Math. Phys. 80 (2007), 37–56. CrossRef

[18] Y. Wang, Lower-Dimensional Volumes and Kastler–kalau–Walze Type Theorem for Manifolds with Boundary, Commun. Theor. Phys. 54 (2010), 38–42. CrossRef

[19] M. Wodzicki, Local invariants of spectral asymmetry, Invent. Math. 75(1) (1995), 143–178. CrossRef

[20] Y. Yu,The Index Theorem and The Heat Equation Method, Nankai Tracts in Mathematics, 2, World Scientific, Singapore, 2001. CrossRef

[21] Y. Yu, N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators, Springer-Verlag, Berlin, 1992. CrossRef

[22] W. Zhang, Lectures on chern-weil theory and witten deformation, Nankai Tracts in Mathematics, 4, World Scientific, Singapore, 2001. CrossRef