Complex-valued analytic torsion for flat bundles and for holomorphic bundles with (1,1) connections

Sylvain E. Cappell, Edward Y. Miller

Research output: Contribution to journalArticle

Abstract

The work of Ray and Singer that introduced analytic torsion, a kind of determinant of the Laplacian operator in topological and holomorphic settings, is naturally generalized in both settings. The couplings are extended in a direct way in the topological setting to general flat bundles and in the holomorphic setting to bundles with (1,1) connections, which, by using the Newlander-Nirenberg theorem, are seen to be the bundles with both holomorphic and antiholomorphic structures. The resulting natural generalizations of Laplacians are not always self-adjoint, and the corresponding generalizations of analytic torsions are thus not always real-valued. The Cheeger-Müller theorem on equivalence in a topological setting of analytic torsion to classical topological torsion generalizes to this complex-valued torsion. On the algebraic side the methods introduced include a notion of torsion associated to a complex equipped with both boundary and coboundary maps.

Original languageEnglish (US)
Pages (from-to)133-202
Number of pages70
JournalCommunications on Pure and Applied Mathematics
Volume63
Issue number2
DOIs
StatePublished - Feb 2010

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Analytic Torsion
Torsional stress
Torsion
Bundle
Theorem
Half line
Determinant
Equivalence
Generalise
Generalization

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Complex-valued analytic torsion for flat bundles and for holomorphic bundles with (1,1) connections. / Cappell, Sylvain E.; Miller, Edward Y.

In: Communications on Pure and Applied Mathematics, Vol. 63, No. 2, 02.2010, p. 133-202.

Research output: Contribution to journalArticle

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