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The work of Ray and Singer which introduced analytic torsion, a kind ofdeterminant of the Laplacian operator in topological and holomorphic settings,is naturally generalized in both settings. The couplings are extended in adirect way in the topological setting to general flat bundles and in theholomorphic setting to bundles with (1,1) connections, which using theNewlander-Nirenberg Theorem are seen to be the bundles with both holomorphicand anti-holomorphic structures. The resulting natural generalizations ofLaplacians are not always self-adjoint and the corresponding generalizations ofanalytic torsions are thus not always real-valued. The Cheeger-Muller theorem,on equivalence in a topological setting of analytic torsion to classicaltopological torsion, generalizes to this complex-valued torsion. On thealgebraic side the methods introduced include a notion of torsion associated toa complex equipped with both boundary and coboundry maps.

Complex Valued Analytic Torsion for Flat Bundles and for Holomorphic Bundles with (1,1) Connections