A local equivariant index theorem for sub-signature operators

Abstract: In this paper, we prove a local equivariant index theorem for sub-signature operators which generalizes the Zhang's index theorem for sub-signature operators.

• The paper extends Zhang’s local index theorem by deriving explicit equivariant indices for sub-signature operators in both even and odd dimensions.
• It utilizes advanced heat kernel asymptotics and Volterra pseudodifferential operators to extract local index densities at fixed-point components.
• The study provides a unified analytical framework that accommodates isometries and orientation-reversing symmetries, offering new tools for equivariant geometric analysis.

A Local Equivariant Index Theorem for Sub-Signature Operators

Introduction and Context

This work presents a local equivariant index theorem for sub-signature operators, generalizing the local index theorem framework of W. Zhang for these operators. The results are situated within the context of the Atiyah-Bott-Segal-Singer index theory, extending local index theorems to include group actions and involutive symmetries, and providing formulae both in even and odd dimensions. Motivated by recent advances in heat kernel methods for index theory—including approaches via Volterra pseudodifferential operators and Getzler’s rescaling—the paper leverages these analytical tools to address equivariant phenomena for sub-signature operators, which themselves play a role intermediate between signature and Dirac-type operators.

Sub-Signature Operators: Construction and Algebraic Framework

The sub-signature operator is defined on an oriented, closed Riemannian manifold $M$, with a fixed oriented sub-bundle $E \subset TM$. The construction involves:

• Orthogonal decompositions $TM = E \oplus E^-$, metrics $g_{TM} = g_E + g_{E^-}$, and associated exterior algebraic structures.
• The identification of the Clifford action on differential forms, with grading and splitting induced by $E$.
• Definition of the sub-signature operator $\mathcal{D}_E$ as a self-adjoint, first-order elliptic operator acting on appropriately graded bundles, using the exterior derivative $d$, its adjoint $\delta$, and Clifford multiplication twisted by the sub-bundle structure.

The analysis requires precise manipulation of the Clifford algebra and the induced gradings from the sub-bundle. Explicit Lichnerowicz-type formulas for $\mathcal{D}_E^2$ are given, revealing the interactions of curvature, torsion, and the geometry of $E$ and its orthogonal complement $E^-$.

Heat Kernel Methods and Volterra Pseudodifferential Operators

Central to the local index theorem is the utilization of heat kernel asymptotics in the presence of symmetries. Using the framework developed by Greiner and subsequent refinements, the authors employ parabolic homogeneous symbol calculus tailored to Volterra-type operators, which are suited to initial-value problems and asymptotics localized at fixed points.

The key analytic tools include:

• Construction of Volterra pseudodifferential parametrices for $(\mathcal{D}_E^2 + \partial_t)$.
• Fine asymptotic expansions of heat kernels in tubular neighborhoods around components of the symmetry’s fixed-point set.
• An explicit trace formula incorporating the contributions of the symmetry, exterior algebra gradings, and the structure of sub-bundles at fixed points.

In the even-dimensional case, for an isometry $\sigma$ preserving orientation and $E$, the equivariant index in terms of the trace of the heat kernel localized at the fixed-point set is calculated, incorporating refined Clifford-algebraic traces, Berezin integrals, and curvature forms.

Main Theorems

Even Dimensional Equivariant Local Index Formula

For an even-dimensional oriented Riemannian manifold with an equivariant isometry, the local equivariant index formula for the sub-signature operator (Theorem 3.9) establishes:

• The supertrace of the composition of the symmetry and the heat kernel, localized at a fixed point component, admits an explicit limit as $t \to 0$.
• The formula involves characteristic forms (the $\widehat{A}$-form, the equivariant Euler class, and Pfaffians of curvature forms associated to $E$ and $E^-$), determinants involving the equivariant normal bundle, and the Berezin trace.
• The Getzler rescaling is essential to extract the leading term in the asymptotic expansion, corresponding to the local index density.

Explicitly, the result relates the local supertrace to the product of characteristic classes: $$\lim_{t \to 0} \operatorname{Str}\left[\sigma^* \exp(-t \mathcal{D}_E^2)\right] \propto \widehat{A}(R^{TM})\, \mathrm{Vo}(R^{N})\, \det^{1/2}\left(\frac{\cosh(\frac{R^E}{2})}{\sinh(\frac{R^{E^-}}{2})}\right) \mathrm{Pf}(R^{E^-})$$ where all forms and determinants are evaluated on the fixed-point component.

Odd Dimensional Equivariant Local Index Formula

For an odd-dimensional manifold and orientation-reversing involution, Theorem 3.18 provides:

• The equivariant local index, defined via traces of the sub-signature operator twisted by the symmetry, is computed via similar local heat kernel expansions and traces.
• The formula again involves Berezin traces, determinants of twisted curvature forms, and is sensitive to the grading induced by the involution and the orientation of $E$.
• The odd index thus obtained is dictated by the equivariant topology and the local differential geometry at the fixed-point set of the involution.

Novelty and Impact

The main accomplishment of the paper is extending Zhang's local index theorem for sub-signature operators to the equivariant setting—in particular for isometries and orientation-reversing symmetries, and in both even and odd dimensions. The proofs systematically apply the Greiner-Volterra approach, providing alternative rigorous heat kernel methods for these index problems.

Strong technical claims include:

• Local explicit geometric formulae for the equivariant index in terms of heat kernel supertraces and Clifford/Berezin trace computations.
• Unified treatment of both even and odd dimensional cases, incorporating orientation-reversing actions where traditional Dirac-type index theory fails to apply directly.

Implications and Future Directions

Practically, these results enable the computation of equivariant indices for elliptic differential operators beyond the Dirac paradigm, relevant for studying geometric quantization, positive scalar curvature questions, and rigidity theorems in the presence of symmetry. Theoretically, they offer insights for extending index theory to non-standard operators, particularly in the context of sub-signature and hypoelliptic settings. The approach and results suggest new avenues in:

• Noncommutative geometry and equivariant cyclic cohomology, particularly for spectral triples involving sub-signature operators.
• Refining analytic torsion and eta-invariant computations sensitive to sub-bundle data and group actions.
• Studying family versions of these results, potentially yielding new rigidity and vanishing theorems in equivariant index theory.

Further developments may include generalizations to the context of foliations, noncompact manifolds, or more general symmetry groups, leveraging the analytic tools established herein.

Conclusion

The paper rigorously establishes a local equivariant index theorem for sub-signature operators, utilizing advanced heat kernel and pseudodifferential operator techniques. The local formulas obtained generalize Zhang's index theorem to the equivariant context and account for both even and odd dimensions, encompassing actions by orientation-preserving isometries and orientation-reversing involutions. These analytic and algebraic results strengthen the bridge between sub-signature index theory and equivariant geometric analysis, providing robust tools for further research in index theory, global analysis, and equivariant noncommutative geometry.