## Abstract and Applied Analysis

### A Kastler-Kalau-Walze Type Theorem and the Spectral Action for Perturbations of Dirac Operators on Manifolds with Boundary

Yong Wang

#### Abstract

We prove a Kastler-Kalau-Walze type theorem for perturbations of Dirac operators on compact manifolds with or without boundary. As a corollary, we give two kinds of operator-theoretic explanations of the gravitational action on boundary. We also compute the spectral action for Dirac operators with two-form perturbations on 4-dimensional compact manifolds.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 619120, 13 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412273245

Digital Object Identifier
doi:10.1155/2014/619120

Mathematical Reviews number (MathSciNet)
MR3186972

Zentralblatt MATH identifier
07022738

#### Citation

Wang, Yong. A Kastler-Kalau-Walze Type Theorem and the Spectral Action for Perturbations of Dirac Operators on Manifolds with Boundary. Abstr. Appl. Anal. 2014 (2014), Article ID 619120, 13 pages. doi:10.1155/2014/619120. https://projecteuclid.org/euclid.aaa/1412273245

#### References

• V. Guillemin, “A new proof of Weyl's formula on the asymptotic distribution of eigenvalues,” Advances in Mathematics, vol. 55, no. 2, pp. 131–160, 1985.
• M. Wodzicki, “Noncommutative residue. Chapter I. Fundamentals,” in K-Theory, Arithmetic and Geometry, Lecture Notes in Mathematics, pp. 320–399, 1987.
• A. Connes, “Quantized calculus and applications,” in 11th International Conference on Mathematical P Conference, pp. 15–36, Internat Press, Cambridge, Mass, USA, 1995.
• A. Connes, “The action functional in noncommutative geometry,” Communications in Mathematical Physics, vol. 117, no. 4, pp. 673–683, 1988.
• D. Kastler, “The Dirac operator and gravitation,” Communications in Mathematical Physics, vol. 166, no. 3, pp. 633–643, 1995.
• W. Kalau and M. Walze, “Gravity, non-commutative geometry and the Wodzicki residue,” Journal of Geometry and Physics, vol. 16, no. 4, pp. 327–344, 1995.
• T. Ackermann, “A note on the Wodzicki residue,” Journal of Geometry and Physics, vol. 20, no. 4, pp. 404–406, 1996.
• T. Ackermann and J. Tolksdorf, “A generalized Lichnerowicz formula, the Wodzicki residue and gravity,” Journal of Geometry and Physics, vol. 19, no. 2, pp. 143–150, 1996.
• R. Ponge, “Noncommutative residue for Heisenberg manifolds: applications in CR and contact geometry,” Journal of Functional Analysis, vol. 252, no. 2, pp. 399–463, 2007.
• U. Battisti and S. Coriasco, “A note on the Einstein-Hilbert action and Dirac operators on ${R}^{n}$,” Journal of Pseudo-Differential Operators and Applications, vol. 2, no. 3, pp. 303–315, 2011.
• U. Battisti and S. Coriasco, “Wodzicki residue for operators on manifolds with cylindrical ends,” Annals of Global Analysis and Geometry, vol. 40, no. 2, pp. 223–249, 2011.
• F. Nicola, “Trace functionals for a class of pseudo-differential operators in ${R}^{n}$,” Mathematical Physics, Analysis and Geometry, vol. 6, no. 1, pp. 89–105, 2003.
• B. V. Fedosov, F. Golse, E. Leichtnam, and E. Schrohe, “The noncommutative residue for manifolds with boundary,” Journal of Functional Analysis, vol. 142, no. 1, pp. 1–31, 1996.
• E. Schrohe, “Noncommutative residue, Dixmier's trace, and heat trace expansions on manifolds with boundary,” Contemporary Mathematics, vol. 242, pp. 161–186, 1999.
• Y. Wang, “Gravity and the noncommutative residue for manifolds with boundary,” Letters in Mathematical Physics, vol. 80, no. 1, pp. 37–56, 2007.
• Y. Wang, “Lower-dimensional volumes and Kastler-Kalau-Walze type theorem for manifolds with boundary,” Communications in Theoretical Physics, vol. 54, pp. 38–42, 2010.
• A. Sitarz and A. Zajac, “Spectral action for scalar perturbations of Dirac operators,” Letters in Mathematical Physics, vol. 98, no. 3, pp. 333–348, 2011.
• B. Iochum and C. Levy, “Tadpoles and commutative spectral triples,” Journal of Noncommutative Geometry, vol. 5, no. 3, pp. 299–329, 2011.
• F. Hanisch, F. Pfäffle, and C. A. Stephan, “The spectral action for Dirac operators with skew-symmetric torsion,” Communications in Mathematical Physics, vol. 300, no. 3, pp. 877–888, 2010.
• A. Connes and H. Moscovici, “Type III and spectral triples,” in Traces in Number Theory, Geometry and Quantum Fields, Aspects of Mathematics E38, pp. 57–71, Vieweg, 2008.
• A. Connes and A. H. Chamseddine, “Inner fluctuations of the spectral action,” Journal of Geometry and Physics, vol. 57, no. 1, pp. 1–21, 2006.
• P. B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, vol. 11 of Mathematics Lecture Series, 1984. \endinput