  The work of Ray and Singer which 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 using the
Newlander-Nirenberg Theorem are seen to be the bundles with both holomorphic
and anti-holomorphic 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-Muller 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 coboundry maps.
