Analytic surgery of the ζ-determinant of the Dirac operator

Nuclear Physics B-Proceedings Supplements

Volumn 104, Issue 1-3, 2002, Pages 89-115

  Park, Jinsung ^a   Wojciechowski, Krzysztof P ^a  

^a INDIANA UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036461863     PISSN: 09205632     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0920-5632(01)01597-3     Document Type: Article

References (52)

1. M.F. Atiyah and I.M. Singer, Dirac operators coupled to vector potentials, Proc. Nat. Acad. Sci. USA 81 (1984) 2597-2600.
2. M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Phil. Soc. 77 (1975) 43-69.
3. M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian geometry. II, Math. Proc. Cambridge Phil. Soc. 78 (1975) 405-432.
4. J.M. Bismut and D.S. Freed, The analysis of elliptic families. II. Dirac operators, eta invariants, and the holonomy theorem, Commun. Math. Phys. 107 (1986) 103-163.
5. B. Bojarski, The abstract linear conjugation problem and Fredholm pairs of subspaces, in: Volume in Memoriam I. N. Vekua: Differential and Integral Equations. Boundary Value Problems. Publications of I. N. Vekua Institute of Applied Mathematics, Tibilisi 1979, 45-60.
6. B. Booß-Bavnbek and K.P. Wojciechowski, Desuspension of splitting elliptic symbols. I, Ann. Global Anal. Geom. 3 (1985) 349-400.
7. B. Booß-Bavnbek and K.P. Wojciechowski, Desuspension of splitting elliptic symbols. II Ann. Global Anal. Geom. 4 (1986) 337-383.
8. B. Booß-Bavnbek and K.P. Wojciechowski, Pseudo-differential projections and the topology of certain spaces of elliptic boundary value problems, Commun. Math. Phys. 121 (1989) 1-9.
9. B. Booß-Bavnbek and K.P. Wojciechowski, Elliptic Boundary Problems for Dirac Operators, Birkhäuser, Boston, 1993.
10. J. Brüinning and M. Lesch, On the η-invariant of certain nonlocal boundary value problems, Duke Math. J. 96 (1999) 425-468.
11. D. Burghelea, L. Friedlander and T. Kappeler, Mayer-Vietoris type formula for determinants of elliptic differential operators, J. Funct. Anal. 107 (1992) 34-65.
12. S.E. Cappell, R. Lee and E.Y. Miller, Self-adjoint elliptic operators and manifold decompositions. I. Low eigenmodes and stretching, Commun. Pure Appl. Math. 49 (1996) 825-866.
13. J. Cheeger, Analytic torsion and the heat equation, Annals of Math. 109 (1979) 259-322.
14. J. Cheeger, Spectral geometry of singular Riemannian spaces, J. Diff. Geom. 18 (1983) 575-657.
15. J. Cheeger, η-invariants, the adiabatic approximation and conical singularities, J. Diff. Geom. 26 (1987) 175-221.
16. R.G. Douglas and K.P. Wojciechowski, Adiabatic limits of the η-invariants. The odd-dimensional Atiyah-Patodi-Singer problem, Commun. Math. Phys. 142 (1991) 139-168.
17. P.B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theory, CRC Press, Boca Raton, 1995.
18. G. Grubb, Trace expansions for pseudodifferential boundary problems for Dirac-type operators and more general systems, Arkiv f. Matematik. 37 (1999) 45-86.
19. A. Hassell, R.R. Mazzeo and R.B. Melrose, Analytic surgery and the accumulation of eigenvalues, Commun. Anal. Geom. 3 (1995) 115-222.
20. S.W. Hawking, Zeta function regularization of path integrals in curved spacetime, Commun. Math. Phys. 55 (1977) 133-148.
21. P. Kirk and M. Lesch, The η-invariant, Maslov index, and spectral flow for Dirac type operators on manifolds with boundary. Preprint DG/0012123.
22. S. Klimek and K.P. Wojciechowski, Adiabatic cobordism theorems for analytic torsion and η-invariant, J. Funct. Anal. 136 (1996) 269-293.
23. Y. Lee and K.P. Wojciechowski, Two remarks on Scott-Wojciechowski paper on Determinants (GAFA, vol. 10, 1202-1236). IUPUI Preprint 01-04.
24. M. Lesch and K.P. Wojciechowski, On the η-invariant of generalized Atiyah-Patodi-Singer problems, Illinois J. Math. 40 (1996) 30-46.
25. R.R. Mazzeo and R.B. Melrose, Analytic surgery and the eta invariant, Geom. Func. Anal. 5 (1995) 14-75.
26. W. Müller, Analytic torsion and R-torsion of Riemannian manifolds, Adv. in Math. 28 (1978) 233-305.
27. W. Müller, Eta invariants and manifolds with boundary, J. Diff. Geom. 40 (1994) 311-377.
28. W. Müller, Relative zeta functions, relative determinants, and scattering theory, Commun. Math. Phys. 192 (1998) 309-347.
29. L. Nicolaescu, The Maslov index, the spectral flow, and decompositions of manifolds, Duke Math. J. 80 (1995) 485-533.
30. R.S. Palais, Seminar on the Atiyah-Singer Index Theorem, Princeton University Press, Princeton, 1965.
31. J. Park and K.P. Wojciechowski, Relative ζ determinant and Adiabatic decomposition of the ζ-determinat of the Dirac Laplacian, Lett. Math. Phys. 52 (2000) 329-337.
32. J. Park and K.P. Wojciechowski, with Appendix by Y. Lee, 2001, Adiabatic Decomposition of the ζ-determinant of the Dirac Laplacian I. The Case of Invertible Tangential Operator, Commun. Part. Diff. Eq. (to appear).
33. J. Park and K.P. Wojciechowski, Scattering Theory and Adiabatic Decomposition of the ζ-determinant of the Dirac Laplacian, IUPUI Preprint 01-02.
34. J. Park and K.P. Wojciechowski, Adiabatic decomposition of the ζ-determinant of the Dirac Laplacian II. The case of non-invertible tangential operator. In preparation (2001).
35. P. Piazza, Determinant bundles, manifolds with boundary and surgery. I, Commun. Math. Phys. 178 (1996) 597-626.
36. P. Piazza, Determinant bundles, manifolds with boundary and surgery. II, Commun. Math. Phys. 193 (1998) 105-124.
37. D.G. Quillen, Determinants of Cauchy-Riemann operators over a Riemann surface, Funkcionalnyi Analiz i ego Prilozhenya 19 (1985) 37-41.
38. D. Ray and I.M. Singer, R-torsion and the Laplacian on Riemannian manifolds, Adv. Math. 7 (1971) 145-210.
39. S.G. Scott, Determinants of Dirac boundary value problems over odd-dimensional manifolds, Commun. Math. Phys. 173 (1995) 43-76.
40. S.G. Scott and K.P. Wojciechowski, ζ-determinant and the Quillen determinant on the Grassmannian of elliptic self-adjoint boundary conditions, C. R. Acad. Sci., Serie I, 328 (1999) 139-144.
41. S.G. Scott and K.P. Wojciechowski, The ζ-determinant and Quillen determinant for a Dirac operator on a manifold with boundary, Geom. Funct. Anal. 10 (1999) 1202-1236.
42. S.G. Scott and K.P. Wojciechowski, Heat Kernels, Determinants and Elliptic Boundary Problems, to appear in: Noncommutative Differential Geometry and Its Applications to Physics. Proceedings of the Shonan Kokusaimura June 1999 Workshop.
43. R.T. Seeley, Complex powers of an elliptic operator, AMS Proc. Symp. Pure Math. 10 (1967) 288-307.
44. R.T. Seeley, Topics in pseudodifferential operators. In: CIME Conference on Pseudo-Differential operators (Stresa 1968), pp. 167-305. Cremonese 1969.
45. G.B. Segal, The definition of conformal field theory, Oxford preprint (1990).
46. I.M. Singer, Families of Dirac operators with applications to physics, Asterisque, hors série (1985) 323-340.
47. I.M. Singer, The η-invariant and the index, in: Mathematical Aspects of String Theory, ed. S.T. Yau, World Scientific Press, Singapore, 1988, pp. 239-258.
48. M.E. Taylor, Partial Differential Equations. Qualitative Studies of Linear Equations, Springer-Verlag, New York, 1996.
49. K.P. Wojciechowski, Elliptic operators and relative K-homology groups on manifolds with boundary, C. R. Math. Rep. Acad. Sci. Canada 7 (1985) 149-154.
50. K.P. Wojciechowski, The additivity of the η-invariant: The case of an invertible tangential operator, Houston J. Math. 20 (1994) 603-621.
51. K.P. Wojciechowski, The additivity of the η-invariant. The case of a singular tangential operator, Commun. Math. Phys. 169 (1995) 315-327.
52. K.P. Wojciechowski, The ζ-determinant and the additivity of the η-invariant on the smooth, self-adjoint Grassmannian, Commun. Math. Phys. 201 (1999) 423-444.