Skip to main content
SpringerLink
Log in

Statistical de Rham Hodge operators and general Kastler-Kalau-Walze type theorems for manifolds with boundary

  • Published:
Journal of Pseudo-Differential Operators and Applications Aims and scope Submit manuscript

Abstract

In this paper, we establish some general Kastler-Kalau-Walze type theorems for any dimensional manifolds with boundary which generalize the results in (Wei and Wang, J Nonlinear Math Phys 28: 254–275, 2021).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adler, M.: On a trace functional for formal pseudo-differential operators and the symplectic structure of Korteweg-de Vries type equations. Invent. Math. 50, 219–248 (1979)

    Article  Google Scholar 

  2. Ackermann, T.: A note on the Wodzicki residue. J. Geom. Phys. 20, 404–406 (1996)

    Article  MathSciNet  Google Scholar 

  3. Connes, A.: Quantized calculus and applications. XIth International Congress of Mathematical Physics(Paris,1994), Internat Press, Cambridge, MA, 15-36, (1995)

  4. Connes, A.: The action functinal in noncommutative geometry. Comm. Math. Phys. 117, 673–683 (1998)

    Article  Google Scholar 

  5. Connes, A.: The action functinal in noncommutative geometry. Rev. Math. Phys. 5, 477–532 (1993)

    MathSciNet  Google Scholar 

  6. Fedosov, B.V., Golse, F., Leichtnam, E., Schrohe, E.: The noncommutative residue for manifolds with boundary. J. Funct. Anal. 142, 1–31 (1996)

    Article  MathSciNet  Google Scholar 

  7. Guillemin, V.W.: A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues. Adv. Math. 55(2), 131–160 (1985)

    Article  MathSciNet  Google Scholar 

  8. Kastler, D.: The Dirac operator and gravitation. Comm. Math. Phys. 166, 633–643 (1995)

    Article  MathSciNet  Google Scholar 

  9. Kalau, W., Walze, M.: Gravity, noncommutative geometry and the Wodzicki residue. J. Geom. Phys. 16, 327–344 (1995)

    Article  MathSciNet  Google Scholar 

  10. Opozda, B.: Bochner’s technique for statistical structures. Ann. Glob. Anal. Geom. 48(4), 357–395 (2015)

    Article  MathSciNet  Google Scholar 

  11. Schrohe, E.: Noncommutative residue, Dixmier’s trace, and heat trace expansions on manifolds with boundary. Contemp. Math. 242, 161–186 (1999)

    Article  MathSciNet  Google Scholar 

  12. Wodzicki, M.: Local invariants of spectral asymmetry. Invent. Math. 75(1), 143–178 (1995)

    Article  MathSciNet  Google Scholar 

  13. Wang, J., Wang, Y.: The Kastler-Kalau-Walze type theorem for six-dimensional manifolds with boundary. J. Math. Phys. 56, 052501 (2015)

    Article  MathSciNet  Google Scholar 

  14. Wang, J., Wang, Y.: A general Kastler-Kalau-Walze type theorem for manifolds with boundary. Int. J. Geom. Methods Mod. Phys. 13(1), 1650003–16 (2016)

    Article  MathSciNet  Google Scholar 

  15. Wang, Y.: Differential forms and the Wodzicki residue for manifolds with boundary. J. Geom. Phys. 56, 731–753 (2006)

    Article  MathSciNet  Google Scholar 

  16. Wang, Y.: Gravity and the noncommutative residue for manifolds with boundary. Lett. Math. Phys. 80, 37–56 (2007)

    Article  MathSciNet  Google Scholar 

  17. Wang, Y.: Lower-dimensional volumes and Kastler-kalau-Walze type theorem for manifolds with boundary. Commun. Theor. Phys. 54, 38–42 (2010)

    Article  Google Scholar 

  18. Wei, S., Wang, Y.: Statistical de Rham Hodge operators and the Kastler-Kalau-Walze type theorem for manifolds with boundary. J. Nonlinear Math. Phys. 28(2), 254–275 (2021)

    Article  MathSciNet  Google Scholar 

  19. Yu, Y.: The index theorem and the heat equation method, nankai tracts in mathematics-Vol.2, World Scientific Publishing, (2001)

Download references

Acknowledgements

This work was supported by NSFC No.11771070. The authors thank the referee for his (or her) careful reading and helpful comments.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, People’s Republic of China

    Hongfeng Li & Yong Wang

  2. Department of Mathematics, Northeastern University, Shenyang, 110819, China

    Tong Wu

Authors
  1. Hongfeng Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tong Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by HL, TW and YW. The first draft of the manuscript was written by HL and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Yong Wang.

Ethics declarations

Conflict of interest

The authors declare no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, H., Wu, T. & Wang, Y. Statistical de Rham Hodge operators and general Kastler-Kalau-Walze type theorems for manifolds with boundary. J. Pseudo-Differ. Oper. Appl. 14, 68 (2023). https://doi.org/10.1007/s11868-023-00563-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11868-023-00563-1

Keywords

Search

Navigation