The $\mathrm{L}^p$-index of the Hodge-Dirac operator on compact Riemannian manifolds
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911714942910464 |
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| author | Arhancet, Cédric |
| author_facet | Arhancet, Cédric |
| contents | We investigate the spectral and index-theoretic properties of the Hodge-Dirac operator $D = \mathrm{d} + \mathrm{d}^*$ acting on the Banach space $\mathrm{L}^p(Ω^\bullet(M))$ 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 $(\mathrm{C}(M), \mathrm{L}^p(Ω^\bullet(M)), D)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22517 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $\mathrm{L}^p$-index of the Hodge-Dirac operator on compact Riemannian manifolds Arhancet, Cédric Functional Analysis Differential Geometry K-Theory and Homology We investigate the spectral and index-theoretic properties of the Hodge-Dirac operator $D = \mathrm{d} + \mathrm{d}^*$ acting on the Banach space $\mathrm{L}^p(Ω^\bullet(M))$ 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 $(\mathrm{C}(M), \mathrm{L}^p(Ω^\bullet(M)), D)$ 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. |
| title | The $\mathrm{L}^p$-index of the Hodge-Dirac operator on compact Riemannian manifolds |
| topic | Functional Analysis Differential Geometry K-Theory and Homology |
| url | https://arxiv.org/abs/2512.22517 |