On gluing formulas for the spectral invariants of Dirac type operators
Authors:
Paul Loya and Jinsung Park
Journal:
Electron. Res. Announc. Amer. Math. Soc. 11 (2005), 1-11
MSC (2000):
Primary 58J28, 58J52
DOI:
https://doi.org/10.1090/S1079-6762-05-00141-1
Published electronically:
January 14, 2005
MathSciNet review:
2122444
Full-text PDF Free Access
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Abstract: In this note, we announce gluing and comparison formulas for the spectral invariants of Dirac type operators on compact manifolds and manifolds with cylindrical ends. We also explain the central ideas in their proofs.
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L P. Loya, Dirac operators, boundary value problems, and the b-calculus, Contemp. Math. 366 (2005), 241–280.
LP0 P. Loya and J. Park, Decomposition of the $\zeta$-determinant for the Laplacian on manifolds with cylindrical end, Illinois J. Math. to appear.
LP2 ---, On the gluing problem for the spectral invariants of Dirac operators, Advances in Math. to appear.
LP3 ---, On the gluing problem for Dirac operators on manifolds with cylindrical ends, Preprint, 2004.
LP1 ---, The comparison problem for the spectral invariants of Dirac type operators, Preprint, 2004.
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LPasymp ---, Asymptotics of $\zeta$-determinant of Dirac Laplacian under adiabatic process, In preparation.
- R. R. Mazzeo and R. B. Melrose, Analytic surgery and the eta invariant, Geom. Funct. Anal. 5 (1995), no. 1, 14–75. MR 1312019, DOI https://doi.org/10.1007/BF01928215
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PWII ---, Adiabatic decomposition of the $\zeta$-determinant and Scattering theory, MPI Preprint, 2002.
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- I. M. Singer, Families of Dirac operators with applications to physics, Astérisque Numéro Hors Série (1985), 323–340. The mathematical heritage of Élie Cartan (Lyon, 1984). MR 837207
- S. M. Vishik, Generalized Ray-Singer conjecture. I. A manifold with a smooth boundary, Comm. Math. Phys. 167 (1995), no. 1, 1–102. MR 1316501
- Krzysztof P. Wojciechowski, The additivity of the $\eta $-invariant. The case of a singular tangential operator, Comm. Math. Phys. 169 (1995), no. 2, 315–327. MR 1329198
- Krzysztof P. Wojciechowski, The $\zeta $-determinant and the additivity of the $\eta $-invariant on the smooth, self-adjoint Grassmannian, Comm. Math. Phys. 201 (1999), no. 2, 423–444. MR 1682214, DOI https://doi.org/10.1007/s002200050561
APS M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69.
BB D. Bleecker and B. Booss-Bavnbek, Spectral invariants of operators of Dirac type on partitioned manifolds, Aspects of Boundary Problems in Analysis and Geometry, Birkhäuser, Boston, 2004, pp. 1–130.
BL98 J. Brüning and M. Lesch, On the $\eta$-invariant of certain nonlocal boundary value problems, Duke Math. J. 96 (1999), 425–468.
Br V. Bruneau, Fonctions zêta et êta en présence de spectre continu, C. R. Acad. Sci. Paris Sér. I Math. 323, no. 5 (1996), 475–480.
Bu95 U. Bunke, On the gluing formula for the $\eta$-invariant, J. Differential Geometry 18 (1995), 397–448.
BFK D. Burghelea, L. Friedlander, and T. Kappeler, Mayer-Vietoris type formula for determinants of differential operators, J. Funct. Anal. 107 (1992), 34–65.
C A.-P. Calderón, Boundary value problems for elliptic equations, Outlines Joint Sympos. Partial Differential Equations (Novosibirsk, 1963), Acad. Sci. USSR Siberian Branch, Moscow, 1963, pp. 303–304.
Car G. Carron, Déterminant relatif et la fonction Xi, Amer. J. Math. 124, no. 2 (2002), 307–352.
Dfr94 X. Dai and D. Freed, $\eta$-invariants and determinant lines, J. Math. Phys. 35 (1994), 5155–5195.
F R. Forman, Functional determinants and geometry, Invent. Math. 88 (1987), 447–493.
Gr99 G. Grubb, Trace expansions for pseudodifferential boundary problems for Dirac-type operators and more general systems, Ark. Math. 37 (1999), 45–86.
Gr01 ---, Poles of zeta and eta functions for perturbations of the Atiyah-Patodi-Singer problem, Comm. Math. Phys. 215 (2001), 583–589.
Gr03 ---, Spectral boundary conditions for generalizations of Laplace and Dirac operators, Comm. Math. Phys. 240 (2003), 243–280.
Ha98 A. Hassell, Analytic surgery and analytic torsion, Comm. Anal. Geom. 6, no. 2 (1998), 255–289.
HZ A. Hassell and S. Zelditch, Determinants of Laplacians in exterior domains, IMRN 18 (1999), 971–1004.
HMaM95 A. Hassell, R. R. Mazzeo, and R. B. Melrose, Analytic surgery and the accumulation of eigenvalues, Comm. Anal. Geom. 3 (1995), 115–222.
HMaM97 ---, A signature formula for manifolds with corners of codimension two, Topology 36, no. 5 (1997), 1055–1075.
KL01 P. Kirk and M. Lesch, The eta invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary, Forum Math. 16 (2004), 553–629.
Lee Y. Lee, Burghelea-Friedlander-Kappeler’s gluing formula for the zeta-determinant and its applications to the adiabatic decompositions of the zeta-determinant and the analytic torsion, Trans. Amer. Math. Soc. 355, no. 10 (2003), 4093–4110.
LW96 M. Lesch and K. P. Wojciechowski, On the $\eta$-invariant of generalized Atiyah-Patodi-Singer boundary value problems, Illinois J. Math. 40, no. 1 (1996), 30–46.
L P. Loya, Dirac operators, boundary value problems, and the b-calculus, Contemp. Math. 366 (2005), 241–280.
LP0 P. Loya and J. Park, Decomposition of the $\zeta$-determinant for the Laplacian on manifolds with cylindrical end, Illinois J. Math. to appear.
LP2 ---, On the gluing problem for the spectral invariants of Dirac operators, Advances in Math. to appear.
LP3 ---, On the gluing problem for Dirac operators on manifolds with cylindrical ends, Preprint, 2004.
LP1 ---, The comparison problem for the spectral invariants of Dirac type operators, Preprint, 2004.
LPnote ---, The $\zeta$-determinant of generalized APS boundary problems over the cylinder, J. Phys. A. 37, no. 29 (2004), 7381–7392.
LPasymp ---, Asymptotics of $\zeta$-determinant of Dirac Laplacian under adiabatic process, In preparation.
MM95 R. Mazzeo and R. B. Melrose, Analytic surgery and the eta invariant, Geom. Funct. Anal. 5, no. 1 (1995), 14–75.
MP98 R. Mazzeo and P. Piazza, Dirac operators, heat kernels and microlocal analysis. II. Analytic surgery, Rend. Mat. Appl. (7) 18, no. 2 (1998), 221–288.
BMeR93 R. B. Melrose, The Atiyah-Patodi-Singer index theorem, A. K. Peters, Wellesley, 1993.
Mu94 W. Müller, Eta invariants and manifolds with boundary, J. Differential Geometry 40 (1994), 311–377.
Mu96 ---, On the $L^{2}$-index of Dirac operators on manifolds with corners of codimension two. I, J. Differential Geometry 44 (1996), 97–177.
Mu98 ---, Relative zeta functions, relative determinants and scattering theory, Comm. Math. Phys. 192 (1998), 309–347.
PWI J. Park and K. P. Wojciechowski, Scattering theory and adiabatic decomposition of the $\zeta$-determinant of the Dirac Laplacian, Math. Res. Lett. 9, no. 1 (2002), 17–25.
PWII ---, Adiabatic decomposition of the $\zeta$-determinant and Scattering theory, MPI Preprint, 2002.
RS D. B. Ray and I. M. Singer, $R$-torsion and the Laplacian on Riemannian manifolds, Advances in Math. 7 (1971), 145–210.
ScS02 S. Scott, Zeta determinants on manifolds with boundary, J. Funct. Anal. 192, no. 1 (2002), 112–185.
ScKW99 S. Scott and K. P. Wojciechowski, The $\zeta$-determinant and Quillen determinant for a Dirac operator on a manifold with boundary, Geom. Funct. Anal. 10 (1999), 1202–1236.
Se69 R. T. Seeley, Topics in pseudo-differential operators, Pseudo-Diff. Operators (C.I.M.E., Stresa, 1968), 1969, pp. 167–305.
S1 I .M. Singer The eta invariant and the index, Mathematical aspects of string theory, World Scientific, Singapore, 1988, pp. 239–258.
S2 ---, Families of Dirac operators with applications to physics, Astérisque, Numero Hors Serie, The mathematical heritage of Élie Cartan (Lyon, 1984), 1985, pp. 323–340.
V S. M. Vishik, Generalized Ray-Singer conjecture. I. A manifold with a smooth boundary, Comm. Math. Phys. 167, no. 1 (1995), 1–102.
KPW95 K. P. Wojciechowski, The additivity of the $\eta$-invariant. The case of a singular tangential operator, Comm. Math. Phys. 169 (1995), 315–327.
WoK99 ---, The $\zeta$-determinant and the additivity of the $\eta$-invariant on the smooth, self-adjoint Grassmannian, Comm. Math. Phys. 201, no. 2 (1999), 423–444.
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Additional Information
Paul Loya
Affiliation:
Department of Mathematics, Binghamton University, Binghamton, New York 13902
Email:
paul@math.binghamton.edu
Jinsung Park
Affiliation:
Mathematisches Institut, Universität Bonn, Beringstraße 1, D-53115 Bonn, Germany
Email:
jpark@math.uni-bonn.de
Received by editor(s):
October 6, 2004
Published electronically:
January 14, 2005
Communicated by:
Michael Taylor
Article copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
