Complex Valued Analytic Torsion for Flat Bundles and for Holomorphic   Bundles with (1,1) Connections
  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. 
Sylvain E. Cappell and Edward Y. Miller