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

TL;DR

This paper generalizes Ray and Singer's analytic torsion to complex-valued cases involving flat bundles and holomorphic bundles with (1,1) connections, extending classical results to non-self-adjoint Laplacians.

Contribution

It introduces a complex-valued analytic torsion for bundles with (1,1) connections and extends the Cheeger-Muller theorem to this setting.

Findings

Generalization of analytic torsion to complex-valued cases.

Extension of Cheeger-Muller theorem to non-self-adjoint Laplacians.

Introduction of algebraic torsion for complexes with boundary and coboundary maps.

Abstract

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.