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Explicit functional determinants in four dimensions. (English) Zbl 0762.47019

Let \((M,g)\) be a compact 4-dimensional manifold without boundary with a Riemannian, locally symmetric, Einstein metric \(g\) and let \(A\) be a formally selfadjoint, geometric partial differential operator with positive definite leading symbol, which is a positive integral power of a conformally covariant operator. Explicit formulas for the functional determinant \(\log|\text{det }A|\) are given. Some cases of special manifolds and special operators are also considered.

MSC:

47F05 General theory of partial differential operators
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[1] Thomas P. Branson, Conformally covariant equations on differential forms, Comm. Partial Differential Equations 7 (1982), no. 4, 393 – 431. · Zbl 0532.53021 · doi:10.1080/03605308208820228
[2] Thomas P. Branson, Differential operators canonically associated to a conformal structure, Math. Scand. 57 (1985), no. 2, 293 – 345. · Zbl 0596.53009 · doi:10.7146/math.scand.a-12120
[3] Thomas P. Branson, Group representations arising from Lorentz conformal geometry, J. Funct. Anal. 74 (1987), no. 2, 199 – 291. · Zbl 0643.58036 · doi:10.1016/0022-1236(87)90025-5
[4] -, Second-order conformal covariants, preprint. · Zbl 0890.47030
[5] Thomas P. Branson, Harmonic analysis in vector bundles associated to the rotation and spin groups, J. Funct. Anal. 106 (1992), no. 2, 314 – 328. · Zbl 0778.58066 · doi:10.1016/0022-1236(92)90050-S
[6] T. Branson, S.-Y. A. Chang, and P. Yang, Estimates and extremal problems for the zeta function determinant on four-manifolds, preprint.
[7] Thomas P. Branson and Peter B. Gilkey, The asymptotics of the Laplacian on a manifold with boundary, Comm. Partial Differential Equations 15 (1990), no. 2, 245 – 272. · Zbl 0721.58052 · doi:10.1080/03605309908820686
[8] -, Residues of the eta function for an operator of Dirac type, preprint.
[9] Thomas P. Branson and Bent Ørsted, Conformal indices of Riemannian manifolds, Compositio Math. 60 (1986), no. 3, 261 – 293. · Zbl 0608.58039
[10] Thomas P. Branson and Bent Ørsted, Conformal deformation and the heat operator, Indiana Univ. Math. J. 37 (1988), no. 1, 83 – 110. · Zbl 0646.53041 · doi:10.1512/iumj.1988.37.37004
[11] -, Conformal geometry and global invariants, Differential Geometry and Applications (to appear). · Zbl 0785.53025
[12] Robert Brooks, Peter Perry, and Paul Yang, Isospectral sets of conformally equivalent metrics, Duke Math. J. 58 (1989), no. 1, 131 – 150. · Zbl 0667.53037 · doi:10.1215/S0012-7094-89-05808-0
[13] Sun-Yung A. Chang and Paul C. Yang, Compactness of isospectral conformal metrics on \?³, Comment. Math. Helv. 64 (1989), no. 3, 363 – 374. · Zbl 0679.53038 · doi:10.1007/BF02564682
[14] Sun-Yung A. Chang and Paul C.-P. Yang, Isospectral conformal metrics on 3-manifolds, J. Amer. Math. Soc. 3 (1990), no. 1, 117 – 145. · Zbl 0701.58056
[15] Michael Eastwood and Michael Singer, A conformally invariant Maxwell gauge, Phys. Lett. A 107 (1985), no. 2, 73 – 74. · Zbl 1177.83074 · doi:10.1016/0375-9601(85)90198-7
[16] Howard D. Fegan and Peter Gilkey, Invariants of the heat equation, Pacific J. Math. 117 (1985), no. 2, 233 – 254. · Zbl 0584.58041
[17] Peter B. Gilkey, The spectral geometry of a Riemannian manifold, J. Differential Geometry 10 (1975), no. 4, 601 – 618. · Zbl 0316.53035
[18] Peter B. Gilkey, The spectral geometry of the higher order Laplacian, Duke Math. J. 47 (1980), no. 3, 511 – 528. · Zbl 0448.58026
[19] Peter B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Mathematics Lecture Series, vol. 11, Publish or Perish, Inc., Wilmington, DE, 1984. · Zbl 0565.58035
[20] S. W. Hawking, Zeta function regularization of path integrals in curved spacetime, Comm. Math. Phys. 55 (1977), no. 2, 133 – 148. · Zbl 0407.58024
[21] Yvette Kosmann, Sur les degrés conformes des opérateurs différentiels, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), no. 4, Aiii, A229 – A232 (French, with English summary). · Zbl 0297.53008
[22] B. Osgood, R. Phillips, and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), no. 1, 148 – 211. · Zbl 0653.53022 · doi:10.1016/0022-1236(88)90070-5
[23] S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, preprint. · Zbl 1145.53053
[24] Joseph Polchinski, Evaluation of the one loop string path integral, Comm. Math. Phys. 104 (1986), no. 1, 37 – 47. · Zbl 0606.58014
[25] A. M. Polyakov, Quantum geometry of bosonic strings, Phys. Lett. B 103 (1981), no. 3, 207 – 210. , https://doi.org/10.1016/0370-2693(81)90743-7 A. M. Polyakov, Quantum geometry of fermionic strings, Phys. Lett. B 103 (1981), no. 3, 211 – 213. · doi:10.1016/0370-2693(81)90744-9
[26] A. M. Polyakov, Quantum geometry of bosonic strings, Phys. Lett. B 103 (1981), no. 3, 207 – 210. , https://doi.org/10.1016/0370-2693(81)90743-7 A. M. Polyakov, Quantum geometry of fermionic strings, Phys. Lett. B 103 (1981), no. 3, 211 – 213. · doi:10.1016/0370-2693(81)90744-9
[27] Richard Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), no. 2, 479 – 495. · Zbl 0576.53028
[28] R. T. Seeley, Complex powers of an elliptic operator, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) Amer. Math. Soc., Providence, R.I., 1967, pp. 288 – 307. · Zbl 0159.15504
[29] William I. Weisberger, Normalization of the path integral measure and the coupling constants for bosonic strings, Nuclear Phys. B 284 (1987), no. 1, 171 – 200. · doi:10.1016/0550-3213(87)90032-0
[30] William I. Weisberger, Conformal invariants for determinants of Laplacians on Riemann surfaces, Comm. Math. Phys. 112 (1987), no. 4, 633 – 638. · Zbl 0629.58031
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