Curvature and Weitzenbock formula for the Podleś quantum sphere

2024/06/26 by Bram Mesland, Mesland, Bram, Adam Rennie +1
Mathematics · #46L87 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.2406.18483

openalex publication_date 2024/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that there is a unique Levi-Civita connection on the one-forms of the Dabrowski-Sitarz spectral triple for the Podleś sphere S²_q. We compute the full curvature tensor, as well as the Ricci and scalar curvature of the Podleś sphere using the framework of \citeMRLC. The scalar curvature is a constant, and as the parameter q→ 1, the scalar curvature converges to the classical value 2. We prove a generalised Weitzenbock formula for the spinor bundle, which differs from the classical Lichnerowicz formula for q≠ 1, yet recovers it for q→ 1.

Related

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