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A symmetrized canonical determinant on odd-class pseudodifferential operators. (English) Zbl 1223.58022
Ocampo, Hernán (ed.) et al., Geometric and topological methods for quantum field theory. Papers based on the presentations at the summer school ‘Geometric and topological methods for quantum field theory’, Villa de Leyva, Colombia, July 2–20, 2007. Cambridge: Cambridge University Press (ISBN 978-0-521-76482-7/hbk; 978-0-511-71797-0/ebook). 381-393 (2010).
Summary: Inspired by M. Braverman’s symmetrized determinant [J. Geom. Phys. 59, No. 4, 459–474 (2009; Zbl 1175.58007)], we introduce a symmetrized logarithm logsym for certain elliptic pseudodifferential operators. The symmetrized logarithm of an operator lies in the odd class whenever the operator does. Using the canonical trace extended to log-polyhomogeneous pseudodifferential operators, we define an associated canonical symmetrized determinant DET\(^{\text{sym}}\) on odd-class classical elliptic operators in odd dimensions: DET\(^{\text{sym}}=\exp\circ\,\text{TR}\circ\log^{\text{sym}}\) which provides a canonical description of Braverman’s symmetrized determinant. Using the cyclicity of the canonical trace on odd-class operators, one then easily infers multiplicative properties of this canonical symmetrized determinant.
For the entire collection see [Zbl 1195.81006].
MSC:
58J40 Pseudodifferential and Fourier integral operators on manifolds
35S05 Pseudodifferential operators as generalizations of partial differential operators
47G30 Pseudodifferential operators
58J32 Boundary value problems on manifolds
58J28 Eta-invariants, Chern-Simons invariants
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