The Noncommutative Residue about Witten Deformation

  • Kai Hua BaoEmail author
  • Ai Hui Sun
  • Chao Deng


In this paper, we compute lower dimensional volumes Vol 4 (1,1) and Vol 6 (2,2) about Witten deformation for 4, 6-dimensional spin manifolds with boundary respectively, and get assosiated Kastler–Kalau–Walze type theorems. We also give theoritic explaination of the gravitational action for 4, 6 dimensional manifolds with boundary by these noncommutative residues.


Lower-dimensional volumes noncommutative residue gravitational action Witten deformation 

MR(2010) Subject Classification

58G20 53A30 46L87 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



This work is supported by Science Foundation for BS423 and Inner Mongolia Natural Science Foundation: No. 2018LHO1004. The author also thank the referee for his (or her) careful reading and helpful comments.


  1. [1]
    Ackermann, T.: A note on the Wodzicki residue. J. Geom. Phys., 20, 404–406 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators, Springer-Verlag, Berlin, 1992CrossRefzbMATHGoogle Scholar
  3. [3]
    Connes, A.: Quantized calculus and applications. XIth International Congress of Mathematical Physics (Paris, 1994), Internat Press, Cambridge, MA, 1995, 15–36Google Scholar
  4. [4]
    Connes, A.: The action functinal in Noncommutative geometry. Comm. Math. Phys., 117, 673–683 (1998)CrossRefGoogle Scholar
  5. [5]
    Fedosov, B. V., Golse, F., Leichtnam, E., et al.: The noncommutative residue for manifolds with boundary. J. Funct. Anal., 142, 1–31 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Guillemin, V. W.: A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues. Adv. Math., 55(2), 131–160 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Kastler, D.: The Dirac Operator and Gravitation. Comm. Math. Phys., 166, 633–643 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Kalau, W., Walze, M.: Gravity, Noncommutative geometry and the Wodzicki residue. J. Geom. Physics, 16, 327–344 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Ponge, R.: Noncommutative geometry and lower dimensional volumes in Riemannian geometry. Letters in Mathematical Physics, 83(1), 19–32 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Ugalde, W. J.: Differential forms and the Wodzicki residue, arXiv: Math, DG/0211361Google Scholar
  11. [11]
    Wodzicki, M.: local invariants of spectral asymmetry. Invent. Math., 75(1), 143–178 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Wang, Y.: Diffential forms and the Wodzicki residue for Manifolds with Boundary. J. Geom. Physics, 56, 731–753 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Wang, Y.: Diffential forms the Noncommutative Residue for Manifolds with Boundary in the non-product Case. Letters in Mathematical Physics, 77, 41–51 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Wang, Y.: Gravity and the Noncommutative Residue for Manifolds with Boundary. Letters in Mathematical Physics, 80, 37–56 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Wang, Y.: Lower-dimensional volumes and Kastler–Kalau–Walze type theorem for manifolds with boundary. Commun. Theor. Phys., 54, 38–42 (2010)CrossRefzbMATHGoogle Scholar
  16. [16]
    Yu, Y.: The Index Theorem and The Heat Equation Method, Nankai Tracts in Mathematics-Vol.2, World Scientific Publishing Company, Singapore City, 2001Google Scholar
  17. [17]
    Zhang, W.: Lectures on Chern-weil Theory and Witten Deformation, vol.4. World Scientific Publishing Co. Pte. Ltd., 2001. Commun. Theor. Phys., 54, 38–42 (2010)CrossRefGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science (CAS), Chinese Mathematical Society (CAS) and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsIneer Mongolia University for NationnalitiesTongliaoP. R. China
  2. 2.College of MathematicsJilin Normal UniversitySipingP. R. China
  3. 3.School of Mathematics and StatisticsNortheast Normal UniversityChangchunP. R. China

Personalised recommendations