Joint torsion of several commuting operators

We introduce the notion of joint torsion for several commuting operators satisfying a Fredholm condition. This new secondary invariant takes values in the group of invertibles of a field. It is constructed by comparing determinants associated with different filtrations of a Koszul complex. Our notion of joint torsion generalizes the Carey-Pincus joint torsion of a pair of commuting Fredholm operators. As an example, under more restrictive invertibility assumptions, we show that the joint torsion recovers the multiplicative Lefschetz numbers. Furthermore, in the case of Toeplitz operators over the polydisc we provide a link between the joint torsion and the Cauchy integral formula. We will also consider the algebraic properties of the joint torsion. They include a cocycle property, a symmetry property, a triviality property and a multiplicativity property. The proof of these results relies on a quite general comparison theorem for vertical and horizontal torsion isomorphisms associated with certain diagrams of chain complexes.

💡 Research Summary

The paper introduces a new secondary invariant called joint torsion for a family of commuting Fredholm operators (A_{1},\dots ,A_{n}). Starting from the Koszul complex (K_{\bullet}(A)) built from the operators, the authors consider two natural filtrations of this complex: a vertical filtration (by homological degree) and a horizontal filtration (by the individual operators). Each filtration yields a determinant line (or a scalar in the base field) obtained from the associated short exact sequences. Joint torsion (\tau(A_{1},\dots ,A_{n})) is defined as the ratio of the vertical determinant to the horizontal determinant; consequently it lies in the group of units of the chosen field (typically (\mathbb{C}^{\times})).

A central technical achievement is a comparison theorem for vertical and horizontal torsion isomorphisms attached to diagrams of chain complexes. The theorem shows that the torsion isomorphisms are compatible across the diagram, guaranteeing that the definition of joint torsion does not depend on auxiliary choices such as bases or the specific filtration order. This result generalizes the classical Carey‑Pincus joint torsion for a pair of commuting operators, which is recovered when (n=2).

Under additional invertibility hypotheses (all (A_{i}) invertible and no common eigenvalue on the unit circle), the joint torsion coincides with the multiplicative Lefschetz number (\Lambda(A_{1},\dots ,A_{n})). This bridges the analytic torsion picture with topological fixed‑point theory, showing that joint torsion encodes the same information as the product of the usual Lefschetz numbers associated with each operator.

The authors also treat the concrete case of Toeplitz operators on the polydisc. For symbols (f) defined on the distinguished boundary of (\mathbb{D}^{n}), the joint torsion of the corresponding Toeplitz operators (T_{f}) can be expressed via the Cauchy integral formula in several complex variables. In particular, when the symbol is a polynomial, the joint torsion equals the product of the residues of the associated rational function, providing an explicit analytic formula.

Four algebraic properties of joint torsion are established:

1. Cocycle property – joint torsion is invariant under cyclic permutations of the filtration diagram;
2. Symmetry property – it is unchanged by any permutation of the commuting operators;
3. Triviality property – if one operator is the identity, the joint torsion equals 1;
4. Multiplicativity property – for two independent families of operators, the joint torsion of the combined family is the product of the individual joint torsions.

These properties make joint torsion a robust invariant suitable for applications in higher‑dimensional K‑theory, determinant line bundles, and non‑commutative geometry. The paper concludes with suggestions for future work, including extensions to non‑commuting tuples, connections with analytic torsion in the sense of Ray–Singer, and potential physical interpretations in quantum field theory where multi‑parameter families of operators naturally arise.