×

zbMATH — the first resource for mathematics

Periodicity and the determinant bundle. (English) Zbl 1130.58013
This lovely article relates a number of seemingly different topics in global analysis. The first section deals with the determinant line bundle of a self-adjoint elliptic operator (bundles of groups, classifying principle bundles, associated bundles, \(\det(P)\), Quillen’s definition, metric on \(\det(P)\), primitivity). The second section then discusses various classes of pseudo-differential operators. The adiabatic determinant is treated in the third section (isotropic determinant, asymptotics of \(\det_\varepsilon\), star product). The periodicity of the numerical index (product-suspended index, periodicity) and the periodicity of the determinant line bundle (adiabatic determinant bundle, isotropic determinant bundle, adiabitic limit) are treated in the fourth and fifth section, respectively. The eta invariant (product suspended eta, \(\eta(A+\sqrt{-1}\tau)=\eta(\tau)\)), the universal \(\eta\) and \(\tau\) invariants, the geometric \(\eta\) and \(\tau\) invariants, and the adiabatic \(\eta\) are dealt with in the sixth through the ninth section. The article concludes with appendices on symbols and products, on product suspended operators, and on mixed isotropic operators.
The authors relate two distinct extensions of the eta invariant to self-adjoint elliptic operators and to elliptic invertible suspended families. They show the corresponding \(\tau\) invariant is a determinant. They show the higher even Schwarz loop groups, which classify odd \(k\)-theory, do not carry multiplicative determinants which generate the first Chern class. However, ‘dressed’ extensions, corresponding to a star product, do carry such a function. These are used to discuss Bott periodicity for the determinant bundle and the eta invariant.

MSC:
58J28 Eta-invariants, Chern-Simons invariants
58J52 Determinants and determinant bundles, analytic torsion
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Atiyah M.F., Patodi V.K. and Singer I.M. (1975). Spectral asymmetry and Riemannian geometry. II. Math. Proc. Camb. Phils. Soc. 78(3): 405–432 · Zbl 0314.58016 · doi:10.1017/S0305004100051872
[2] Berline N., Getzler E. and Vergne M. (1992). Heat kernels and Dirac operators. Springer-Verlag, Berlin · Zbl 0744.58001
[3] Bismut J.-M. and Freed D. (1986). The analysis of elliptic families, II. Commun. Math. Phys. 107: 103–163 · Zbl 0657.58038 · doi:10.1007/BF01206955
[4] Bismut J.-M. and Freed D. (1986). The analysis of elliptic families: Metrics and connections on determinant bundles. Commun. Math. Phys. 106: 159–176 · Zbl 0657.58037 · doi:10.1007/BF01210930
[5] Bott R. and Seeley R. (1978). Some remarks on the paper of Callias. Commun. Math. Phys. 62: 235–245 · Zbl 0409.58019 · doi:10.1007/BF01202526
[6] Dai X. and Freed D.S. (1994). and determinant lines. J. Math. Phys. 35(10): 5155–5194 · Zbl 0822.58048 · doi:10.1063/1.530747
[7] Epstein, C.L., Melrose, R.B.: The Heisenberg algebra, index theory and homology. This became [8] without Mendoza as coauthor
[8] Epstein, C.L., Melrose, R.B., Mendoza, G.: The Heisenberg algebra, index theory and homology. In preparation · Zbl 0758.32010
[9] Grieser, D., Gruber, M.J.: Singular asymptotics lemma and push-forward theorem. In: Approaches to singular analysis (Berlin, 1999), Oper. Theory Adv. Appl., Vol. 125, Basel: Birkhäuser, 2001, pp. 117–130 · Zbl 0985.35119
[10] Hörmander L. (1979). The Weyl calculus of pseudo-differential operators. Comm. Pure Appl. Math. 32: 359–443 · Zbl 0396.47029 · doi:10.1002/cpa.3160320304
[11] Melrose, R.B.: Analysis on manifolds with corners. In preparation · Zbl 0754.58035
[12] Melrose R.B. (1992). Calculus of conormal distributions on manifolds with corners. Internat. Math. Res. Notices 1992(3): 51–61 · Zbl 0754.58035 · doi:10.1155/S1073792892000060
[13] Melrose R.B. (1995). The eta invariant and families of pseudodifferential operators. Math. Res. Lett. 2(5): 541–561 · Zbl 0934.58025
[14] Melrose, R.B., Rochon, F.: Boundaries, eta invariant and the determinant bundle. Preprint, http://arxiv.org/list/math.DG/0607480, 2006 · Zbl 1196.58015
[15] Melrose R.B., Rochon F. (2004) Families index for pseudodifferential operators on manifolds with boundary. IMRN (22): 1115–1141 · Zbl 1086.58011
[16] Melrose R.B. and Rochon F. (2006). Index in K-theory for families of fibred cusp operators. K-theory 37: 25–104 · Zbl 1126.58010 · doi:10.1007/s10977-006-0003-6
[17] Pressley A. and Segal G. (1986). Loop groups. Oxford Science publications, Oxford Univ. Press, Oxford · Zbl 0618.22011
[18] Quillen D. (1985). Determinants of Cauchy-Riemann operators over a Riemann surface. Funct. Anal. Appl. 19: 31–34 · Zbl 0603.32016 · doi:10.1007/BF01086022
[19] Seeley, R.T.: Complex powers of an elliptic operator. Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Providence, R.I.: Amer. Math. Soc., 1967, pp. 288–307
[20] Shubin, M.A.: Pseudodifferential operators and spectral theory. Berlin-Heidelberg-New York: Springer-Verlag, 1987, Moscow, Nauka: 1978 · Zbl 0451.47064
[21] Wodzicki M. (1982). Spectral asymmetry and zeta functions. Invent. Math. 66: 115–135 · Zbl 0489.58030 · doi:10.1007/BF01404760
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.