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

Abstract

AbstractThe 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. © 2009 Wiley Periodicals, Inc.