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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.

58J28 Eta-invariants, Chern-Simons invariants
58J52 Determinants and determinant bundles, analytic torsion
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