Abstract
Using equivariant Toeplitz operator calculus, we give a new proof of the Atiyah–Weinstein conjecture on the index of Fourier integral operators and the relative index of CR structures.
Keywords
Mathematics Subject Classification (2010): Primary: 58J20; Secondary: 19L47, 32A45, 53D10, 58J40
Research partially supported by NSF Grant 0703775
Research partially supported by NSF Grant 0707137
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Atiyah, M. F. K-theory. W. A. Benjamin, Inc., New York-Amsterdam, 1967.
Atiyah, M. F.; Singer; I. M. The index of elliptic operators. I, II, III. Ann. of Math. (2) 87 (1968) 484–530, 531–545, 546–604.
Atiyah, M. F. Bott periodicity and the index of elliptic operators. Quart. J. Math. Oxford Ser. (2) 19 (1968), 113–140.
Atiyah, M. F. Elliptic operators and compact groups. Lecture Notes in Mathematics, Vol. 401. Springer-Verlag, Berlin-New York, 1974.
Berezin, F. A. General concept of quantization. Comm. Math. Phys. 40 (1975), 153–174.
Boutet de Monvel, L. Convergence dans le domaine complexe des séries de fonctions propres. C.R.A.S. 287 (1978), 855–856.
Boutet de Monvel, L. On the index of Toeplitz operators of several complex variables. Inventiones Math. 50 (1979), 249–272.
Boutet de Monvel, L. Symplectic cones and Toeplitz operators (actes du congrès en l’honneur de Trèves, São Carlos). Contemporary Math. vol. 205 (1997), 15–24.
Boutet de Monvel, L. Vanishing of the logarithmic trace of generalized Szegő projectors. arXiv:math.AP/0604166 v1, Proceedings of the Conference “Algebraic Analysis of Differential Equations” in honor of Prof. T. Kawai, Springer-Verlag, 2007.
Boutet de Monvel, L.; Guillemin, V. The Spectral Theory of Toeplitz Operators. Ann. of Math Studies no. 99, Princeton University Press, 1981.
Boutet deMonvel, L.;Malgrange, B. Le théorème de l’indice relatif. Ann. Sci. ENS. 23 (1990), 151–192.
Boutet de Monvel, L.; Sjöstrand, J. Sur la singularité des noyaux de Bergman et de Szegő. Astérisque 34–35 (1976), 123–164.
Duistermaat, J.J.; H¨ormander, L. Fourier integral operators II. Acta Math. 128 (1972), 183–269.
Engliš, M. Berezin quantization and reproducing kernels on complex domains. Trans. Amer. Math. Soc. 349, 411–479, 1996.
Engliš, M. Weighted Bergman kernels and quantization. Comm. Math. Phys. 227 (2002), 211–241.
Engliš, M. Toeplitz operators and weighted Bergman kernels. J. Funct. Anal. 255 (2008), 1419–1457.
Epstein, C. Subelliptic Spinc Dirac operators, I. Ann. of Math. (2) 166 (2007), no. 1, 183–214.
Epstein, C. Subelliptic Spinc Dirac operators, II. Ann. of Math. (2) 166 (2007), no. 3, 723–777.
Epstein, C. Subelliptic Spinc Dirac operators, III, the Atiyah–Weinstein conjecture. Ann. of Math. 168 (2008), 299–365.
Epstein, C. Cobordism, relative indices and Stein fillings. J. Geom. Anal. 18 (2008), no. 2, 341–368.
Epstein, C.; Melrose, R. Contact degree and the index of Fourier integral operators. Math. Res. Lett. 5 (1998), no. 3, 363–381.
H¨ormander, L. Fourier integral operators I. Acta Math. 127 (1971), 79–183.
Leichtnam, E.; Nest, R.; Tsygan, B. Local formula for the index of a Fourier integral operator J. Differential Geom. 59 (2001), no. 2, 269–300.
Leichtnam, E.; Tang, X.; Weinstein, A. Poisson geometry and deformation quantization near a strictly pseudoconvex boundary. J. Eur. Math. Soc. 9 (2007), no. 4, 681–704.
Melin, A.; Sjöstrand, J. Fourier Integral operators with complex valued phase functions. Lecture Notes 459 (1975), 120–223.
Melin, A.; Sjöstrand, J. Fourier integral operators with complex phase functions and parametrix for an interior boundary value problem. Comm. P.D.E. 1:4 (1976), 313–400.
Weinstein, A. Some questions about the index of quantized contact transformations. RIMS Kokyuroku No. 1014, 1–14, 1997.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Additional information
This paper is dedicated to J.J. Duistermaat for his 65th birthday
Rights and permissions
Copyright information
© 2011 Springer Science+Buisness Media, LLC
About this chapter
Cite this chapter
de Monvel, L.B., Leichtnam, E., Tang, X., Weinstein, A. (2011). Asymptotic equivariant index of Toeplitz operators and relative index of CR structures. In: Kolk, J., van den Ban, E. (eds) Geometric Aspects of Analysis and Mechanics. Progress in Mathematics, vol 292. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-8244-6_2
Download citation
DOI: https://doi.org/10.1007/978-0-8176-8244-6_2
Published:
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-8243-9
Online ISBN: 978-0-8176-8244-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)