Quantum Bianchi identities via DG categories

JOURNAL OF GEOMETRY AND PHYSICS

卷:124

Article

Beggs, Edwin J.¹  Majid, Shahn² 

[1] Swansea Univ, Dept Math, Swansea SA2 8PP, W Glam, Wales

[2] Queen Mary Univ London, Sch Math Sci, Mile End Rd, London E1 4NS, England

关键词: Quantum Riemannian geometry;    Noncommutative geometry;    Chern-Connes pairing;    Bianchi identity;    q-Sphere;    Quantum gravity;   

DOI  :  10.1016/j.geomphys.2017.11.005

来源: Elsevier

【 摘 要 】

We use DG categories to derive analogues of the Bianchi identities for the curvature of a connection in noncommutative differential geometry. We also revisit the Chern-Connes pairing but following the line of Chern's original derivation. We show that a related DG category of extendable bimodule connections is a monoidal tensor category and in the metric compatible case obtain an analogue of a classical antisymmetry of the Riemann tensor. The monoidal structure implies the existence of a cup product on noncommutative sheaf cohomology. Another application shows that the curvature of a line module reduces to a 2-form on the base algebra. We illustrate the theory on the q-sphere, the permutation group S-3 and the bicrossproduct quantum spacetime [r, t] = lambda r. (C) 2017 Elsevier B.V. All rights reserved.

【 授权许可】

Free