[Paper Review] A Canonical Quadratic Form on the Determinant Line of a Flat Vector Bundle

Maxim Braverman, Thomas Kappeler|ArXiv.org|Oct 5, 2007

Advanced Operator Algebra Research11 references3 citations

TL;DR

This paper introduces a canonical torsion quadratic form on the determinant line of a flat vector bundle over an odd-dimensional closed oriented manifold, constructed via analytic torsion and related to both the Burghelea-Haller and Cappell-Miller torsions. It proves that this form equals the Cappell-Miller torsion up to a sign and a factor depending on the Euler structure, and confirms a conjecture relating analytic and combinatorial torsions up to a unitary constant in each connected component of the flat connection space.

ABSTRACT

We introduce and study a canonical quadratic form, called the torsion quadratic form, of the determinant line of a flat vector bundle over a closed oriented odd-dimensional manifold. This quadratic form caries less information than the refined analytic torsion, introduced in our previous work, but is easier to construct and closer related to the combinatorial Farber-Turaev torsion. In fact, the torsion quadratic form can be viewed as an analytic analogue of the Poincare-Reidemeister scalar product, introduced by Farber and Turaev. Moreover, it is also closely related to the complex analytic torsion defined by Cappell and Miller and we establish the precise relationship between the two. In addition, we show that up to an explicit factor, which depends on the Euler structure, and a sign the Burghelea-Haller complex analytic torsion, whenever it is defined, is equal to our quadratic form. We conjecture a formula for the value of the torsion quadratic form at the Farber-Turaev torsion and prove some weak version of this conjecture. As an application we establish a relationship between the Cappell-Miller and the combinatorial torsions.

Motivation & Objective

• To define a canonical quadratic form on the determinant line of a flat vector bundle over an odd-dimensional closed oriented manifold.
• To establish a precise analytic analogue of the Farber-Turaev Poincaré-Reidemeister scalar product.
• To relate the new torsion quadratic form to the Burghelea-Haller and Cappell-Miller torsions.
• To prove a weak version of a conjecture relating the quadratic form to the Farber-Turaev torsion via monodromy and Euler structures.
• To provide a partial solution to the Cappell-Miller conjecture on the relationship between Cappell-Miller and Reidemeister torsions.

Proposed method

• The torsion quadratic form τ∇ is defined using the refined analytic torsion ρan, which is constructed via spectral theory of non-self-adjoint Laplace-type operators.
• The construction uses the Atiyah-Patodi-Singer odd signature operator B and its square B², with spectral projections onto intervals [0,λ] and (λ,∞) to define the Cappell-Miller torsion T∇.
• The form τ∇ is shown to satisfy τ∇(T∇) = 1, linking it directly to the Cappell-Miller torsion.
• The method involves Agmon angles θ for the operator B² on (λ,∞), ensuring independence of λ in the definition of T∇.
• The connection to the Farber-Turaev torsion is established via the monodromy class Arg∇ ∈ H¹(M, ℂ/ℤ) and the Euler structure class c(ε) ∈ H₁(M, ℤ).
• The duality operator D on determinant lines is used to relate T∇ to the Reidemeister torsion of E ⊕ E* via the Cappell-Miller conjecture.

Experimental results

Research questions

• RQ1 How can a canonical quadratic form on the determinant line of a flat vector bundle be constructed in the absence of a non-degenerate bilinear form?
• RQ2 What is the precise relationship between the new torsion quadratic form and the Cappell-Miller torsion?
• RQ3 To what extent does the torsion quadratic form recover the Farber-Turaev combinatorial torsion?
• RQ4 How does the torsion quadratic form relate to the Burghelea-Haller complex analytic torsion?
• RQ5 Under what conditions does the Cappell-Miller conjecture hold exactly or up to a sign and unitary factor?

Key findings

• The torsion quadratic form τ∇ satisfies τ∇(T∇) = 1, establishing a direct duality between τ∇ and the Cappell-Miller torsion T∇.
• The conjecture τ∇(ρε,ₒ) = R𝒞 · e^{2πi⟨Arg∇, c(ε)⟩} holds weakly, with R𝒞 ∈ ℂ of unit modulus, for each connected component 𝒞 of the flat connection space.
• The Cappell-Miller conjecture (1⊗D)T∇ = (−1)^z ρᴿ(∇⊕∇*) holds up to the constant R𝒞, and exactly when ∇ lies in a component containing an acyclic Hermitian connection.
• When a non-degenerate bilinear form b exists, the conjecture holds up to sign, confirming a partial result for the Cappell-Miller conjecture.
• The torsion quadratic form is shown to be an analytic analogue of the Poincaré-Reidemeister scalar product, refining the Farber-Turaev invariant.
• The precise relationship between the Cappell-Miller torsion and the Farber-Turaev torsion is established via the monodromy class Arg∇ and the Euler structure c(ε), with a sign factor (−1)^z.