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

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.