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arxiv: 2512.22517 · v3 · pith:G5LTEK65new · submitted 2025-12-27 · 🧮 math.FA · math.DG· math.KT

The L^p-index of the Hodge-Dirac operator on compact Riemannian manifolds

C\'edric Arhancet This is my paper

Pith reviewed 2026-05-16 19:38 UTC · model grok-4.3

classification 🧮 math.FA math.DGmath.KT
keywords Hodge-Dirac operatorL^p spacesBanach spectral triplesFredholm indicesEuler characteristicHirzebruch signatureRiemannian manifoldsfunctional calculus
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The pith

The Hodge-Dirac operator on compact manifolds is bisectorial with bounded H^∞ calculus, yielding p-independent topological indices.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the Hodge-Dirac operator remains bisectorial and possesses a bounded H infinity functional calculus on L^p spaces of differential forms for any p, relying solely on the compactness of the manifold. This property turns the continuous functions on the manifold together with this operator into a compact Banach spectral triple. Pairings in this setting produce Fredholm indices that correspond to standard topological quantities and do not vary with p. As a result the classical Euler characteristic and Hirzebruch signature appear as these L^p indices.

Core claim

The Hodge-Dirac operator D = d + d* acting on L^p differential forms over a compact Riemannian manifold M is bisectorial and admits a bounded H^∞ functional calculus without curvature assumptions. This makes the triple (C(M), L^p(Ω^•(M)), D) a compact Banach spectral triple. Consistent pairings between Banach K-homology and K-theory of C(M) produce Fredholm indices that coincide with classical topological invariants and are independent of p, recovering the Euler characteristic and the Hirzebruch signature as L^p-indices.

What carries the argument

The bisectoriality and bounded H^∞ functional calculus of the Hodge-Dirac operator D = d + d* on L^p spaces of differential forms.

If this is right

  • The triple (C(M), L^p(Ω^•(M)), D) forms a compact Banach spectral triple for every p.
  • The resulting Fredholm indices are independent of p.
  • The L^p-index recovers the classical Euler characteristic.
  • The L^p-index recovers the Hirzebruch signature.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same functional calculus properties could apply to other first-order differential operators on manifolds.
  • The p-independence indicates that the topological content survives the passage from Hilbert to general Banach settings.
  • This method might support index theory on manifolds with reduced regularity where curvature bounds are unavailable.

Load-bearing premise

Compactness of the manifold suffices to establish bisectoriality of the Hodge-Dirac operator without curvature assumptions.

What would settle it

A computation on a specific compact manifold such as the circle where the L^p-index differs from the topological Euler characteristic for some p would disprove the claim.

Figures

Figures reproduced from arXiv: 2512.22517 by C\'edric Arhancet.

Figure 1
Figure 1. Figure 1: open sector Σω of angle ω bounded operators acting on X, where R(z, A) def = (z − A) −1 is the resolvent operator. The operator A is said to be sectorial if it sectorial operator of type ω for some angle ω ∈ (0, π). In this case, we can introduce the angle of sectoriality (2.7) ωsec(A) def = inf{ω ∈ (0, π) : A is sectorial of type ω}. Moreover, by [HvNVW18, Example 10.1.3 p. 362], A is sectorial of type < … view at source ↗
Figure 2
Figure 2. Figure 2: the spectrum σ(D) of a bisectorial operator D By [HvNVW18, p. 447], a linear operator D is bisectorial if and only if (2.12) iR ∗ ⊂ ρ(D) and if the set {tR(it, D) : t ∈ R ∗ +} is bounded. If D is a bisectorial operator of type ω on a Banach space X then by [HvNVW18, Proposition 10.6.2 (2) p. 448] its square D2 is sectorial of type 2ω and we have (2.13) Ran D2 = Ran D and Ker D2 = Ker D. Functional calculus… view at source ↗
read the original abstract

We investigate the spectral and index-theoretic properties of the Hodge-Dirac operator $D = \mathrm{d} + \mathrm{d}^*$ acting on the Banach space $\mathrm{L}^p(\Omega^\bullet(M))$ of differential forms over a compact Riemannian manifold $M$. Relying on the compactness of $M$, we establish that this operator is bisectorial and admits a bounded $\mathrm{H}^\infty$ functional calculus, without curvature assumptions. This result enables us to prove that the triple $(\mathrm{C}(M), \mathrm{L}^p(\Omega^\bullet(M)), D)$ constitutes a compact Banach spectral triple. We then investigate consistent pairings between the Banach K-homology and the K-theory of the algebra $\mathrm{C}(M)$, identifying the resulting Fredholm indices with classical topological invariants, and hence showing that they are independent of $p$. We recover the classical Euler characteristic and the Hirzebruch signature as $\mathrm{L}^p$-indices, demonstrating the effectiveness of Banach noncommutative geometry for geometric analysis, beyond the Hilbertian setting.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper studies the Hodge-Dirac operator D = d + d* on the Banach space L^p(Ω^•(M)) for a compact Riemannian manifold M. Relying solely on compactness, it establishes that D is bisectorial and admits a bounded H^∞ functional calculus with no curvature hypotheses required. This is used to show that (C(M), L^p(Ω^•(M)), D) forms a compact Banach spectral triple. The authors then construct consistent pairings between Banach K-homology and the K-theory of C(M), identify the resulting Fredholm indices with classical topological invariants, and conclude that these indices are independent of p. In particular, the Euler characteristic and Hirzebruch signature are recovered as L^p-indices.

Significance. If the central claims hold, the work provides a concrete demonstration that classical topological invariants remain accessible in the Banach-space setting of noncommutative geometry. It shows that the passage from L^2 to L^p does not alter the index values once the functional calculus is available, thereby extending the reach of spectral triples beyond Hilbert spaces while preserving geometric content. The absence of curvature assumptions strengthens the result for general compact manifolds.

minor comments (3)
  1. The abstract and introduction assert bisectoriality and the bounded H^∞ calculus without curvature assumptions, but the manuscript should include an explicit citation to the precise elliptic regularity or symbol-estimate theorem (e.g., the version of the Calderón–Zygmund theory or resolvent bounds employed) that justifies the p-independent sectoriality constants.
  2. In the construction of the Banach spectral triple, clarify whether the compactness of the resolvent is proved directly from the discrete spectrum of the Hodge Laplacian or via the functional calculus; a short paragraph contrasting the L^2 and L^p cases would improve readability.
  3. The identification of the Fredholm indices with the Euler characteristic and signature is stated as a consequence of the pairings; a brief remark on how the Chern character or local index formula adapts to the Banach setting would make the recovery step more transparent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their supportive summary and recommendation of minor revision. The report accurately captures the main results: bisectoriality and bounded H^∞ functional calculus for the Hodge-Dirac operator on L^p(Ω^•(M)) relying only on compactness, the resulting compact Banach spectral triple, and the p-independence of the indices recovering the Euler characteristic and Hirzebruch signature. No specific major comments were raised.

Circularity Check

0 steps flagged

No circularity: derivation proceeds from compactness and standard elliptic estimates

full rationale

The central steps rely on the compactness of M to obtain bisectoriality and bounded H^∞ calculus for D = d + d* on L^p(Ω•(M)) via standard symbol estimates and elliptic regularity that hold independently of p and without curvature hypotheses. The subsequent construction of the compact Banach spectral triple and the identification of Fredholm indices with the Euler characteristic and Hirzebruch signature follow formally from the functional calculus and K-homology pairings; these are consequences rather than redefinitions or fitted inputs. No self-citation chain, ansatz smuggling, or reduction of a claimed prediction to its own fitted parameters appears in the argument outline.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard differential-geometric definition of the Hodge-Dirac operator and on compactness of the manifold to obtain spectral properties; no free parameters or new postulated entities are introduced.

axioms (2)
  • domain assumption M is a compact Riemannian manifold
    Invoked to guarantee bisectoriality and bounded H^∞ calculus without curvature assumptions.
  • standard math The Hodge-Dirac operator is defined as d + d* on the graded bundle of differential forms
    Standard construction from differential geometry used throughout.

pith-pipeline@v0.9.0 · 5491 in / 1472 out tokens · 55287 ms · 2026-05-16T19:38:03.436503+00:00 · methodology

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