[Paper Review] Eta forms and the Chern Character

Simon Scott|ArXiv.org|Nov 22, 2002

Advanced Operator Algebra Research25 references3 citations

TL;DR

This paper establishes geometric index theorems for families of first-order elliptic operators on manifolds with boundary by constructing eta form representatives for Chern character classes of the index bundle. Using regularized superconnection traces and relative eta invariants, it proves a non-local, general index formula without requiring spin structures or dimensional restrictions, extending the Atiyah-Singer families index theorem to boundary settings via spectral sections and transgression forms.

ABSTRACT

We prove two geometric index theorems for a family of first-order elliptic operators over a manifold with boundary by computing eta form representatives for the Chern character classes of the index bundle. The eta forms occur as relative and regularized traces on infinite-dimensional vector bundles realized as the limiting values of superconnection character forms.

Motivation & Objective

• To extend the families index theorem to manifolds with boundary by incorporating non-local boundary corrections via eta forms.
• To construct explicit representatives of Chern character classes for the index bundle using relative and regularized superconnection traces.
• To generalize the Atiyah-Singer families index theorem to settings without spin structures or dimensional restrictions.
• To provide a conceptual and geometric derivation of the APS index theorem for families using spectral sections and transgression forms.
• To establish a cohomological formula for the Chern character of the index bundle in terms of boundary eta forms and superconnection curvature.

Proposed method

• Derives the eta form as the regularized trace of the superconnection curvature: $ \eta = \frac{1}{\sqrt{\pi}} \int_0^\infty \mathrm{Str}(\dot{\mathbb{A}}_t e^{-\mathbb{A}_t^2}) \, dt $.
• Uses the Bismut superconnection $ \mathbb{A}_t $ on the total space and its restriction $ \mathbb{B}_t $ to the boundary to define boundary eta forms.
• Applies the $ b $-calculus and spectral section theory to generalize the boundary projection $ \Pi_{>} $ to arbitrary spectral sections $ \mathcal{P} $.
• Constructs the relative eta form $ \widehat{\eta}_{\mathcal{P}} $ as a transgression form satisfying $ d\widehat{\eta}_{\mathcal{P}} = 2 \int_{\partial M/B} \widehat{A}(\partial M/B) \mathrm{ch}'(\mathbb{E}) $.
• Employs complex contour integration and resolvent analysis to compute asymptotic expansions of superconnection traces and extract index contributions.
• Uses the Diagonalization Lemma to identify the limit of the Chern character form $ \mathrm{ch}(\mathbb{A}_{t,\mathcal{P}}) $ as $ t \to \infty $ with the Chern character of the index bundle.

Experimental results

Research questions

• RQ1 How can the families index theorem be extended to manifolds with boundary using non-local boundary corrections?
• RQ2 What is the geometric and cohomological meaning of the eta form in the context of superconnections and index bundles?
• RQ3 How do spectral sections generalize the standard boundary projection $ \Pi_{>} $ in the construction of index formulas?
• RQ4 What is the precise relationship between the relative eta form and the Chern character of the index bundle?
• RQ5 Can the Chern character of the index bundle be represented by a superconnection form without assuming spin or dimensional constraints?

Key findings

• The eta form $ \widehat{\eta}_{\mathcal{P}} $ is a canonical transgression form satisfying $ d\widehat{\eta}_{\mathcal{P}} = 2 \int_{\partial M/B} \widehat{A}(\partial M/B) \mathrm{ch}'(\mathbb{E}) $, correcting the non-closedness of the bulk Chern character form.
• The cohomological APS families index theorem holds as $ \mathrm{ch}(\mathrm{Ind}(\textsf{D}_{\mathcal{P}})) = \int_{M/B} \widehat{A}(M/B) \mathrm{ch}'(\mathbb{E}) - \frac{1}{2} \widehat{\eta}_{\mathcal{P}} $ in $ H^\bullet(B) $.
• The relative eta form $ \widehat{\eta}_{{\mathcal{P}}_1} - \widehat{\eta}_{{\mathcal{P}}_2} $ is closed and represents the relative Chern character $ \mathrm{ch}(\mathrm{Ind}(\mathcal{P}_2, \mathcal{P}_1)) $.
• The limit of the superconnection Chern character $ \mathrm{ch}(\mathbb{A}_{t,\mathcal{P}}) $ as $ t \to \infty $ equals $ \mathrm{ch}(\nabla^0) $, the Chern character of the index bundle.
• The degree-zero index is recovered as $ \mathrm{ind}(D_{P_1}) - \mathrm{ind}(D_{P_2}) = \lim_{t \to 0} \mathrm{Str}(e^{-t\Delta_1}) - \mathrm{Str}(e^{-t\Delta_2}) $, confirming consistency with the supertrace formula.
• The construction is general and does not require spin structures, Clifford compatibility, or dimensional restrictions, making it broadly applicable to families of elliptic operators.