Scalar curvature in conformal geometry of Connes-Landi noncommutative manifolds

TL;DR

This paper develops a conformal geometry framework for Connes-Landi noncommutative manifolds, defining a scalar curvature that generalizes the classical case and reveals deep relations with index theory and characteristic classes.

Contribution

It introduces a new scalar curvature concept for noncommutative manifolds, with explicit formulas and connections to index theory, extending previous results on noncommutative tori.

Findings

Scalar curvature reduces to classical Riemannian curvature in the commutative limit.

Explicit formulas for curvature functions in all even dimensions, especially four.

Deep relations between local curvature functions confirmed through variational analysis.

Abstract

We first propose a conformal geometry for Connes-Landi noncommutative manifolds and study the associated scalar curvature. The new scalar curvature contains its Riemannian counterpart as the commutative limit. Similar to the results on noncommutative two tori, the quantum part of the curvature consists of actions of the modular derivation through two local curvature functions. Explicit expressions for those functions are obtained for all even dimensions (greater than two). In dimension four, the one variable function shows striking similarity to the analytic functions of the characteristic classes appeared in the Atiyah-Singer local index formula, namely, it is roughly a product of the $j$-function (which defines the $\hat A$-class of a manifold) and an exponential function (which defines the Chern character of a bundle). By performing two different computations for the variation of the Einstein-Hilbert action, we obtain a deep internal relations between two local curvature functions. Straightforward verification for those relations gives a strong conceptual confirmation for the whole computational machinery we have developed so far, especially the Mathematica code hidden behind the paper.