[Paper Review] Quantum Gravity and Riemannian Geometry on the Fuzzy Sphere

Evelyn Lira Torres, Shahn Majid|arXiv (Cornell University)|Apr 29, 2020

Noncommutative and Quantum Gravity Theories22 references4 citations

TL;DR

This paper develops quantum Riemannian geometry on the fuzzy sphere using a novel 3D rotationally invariant differential calculus. It proves the existence and uniqueness of a quantum Levi-Civita connection with constant coefficients for any symmetric 3×3 metric tensor, deriving a closed-form expression for scalar curvature; this enables construction of Euclidean quantum gravity on the fuzzy sphere, revealing UV and IR divergences in the functional integral that cancel in ratios, yielding finite expectation values with power-law scaling.

ABSTRACT

We study the quantum geometry of the fuzzy sphere defined as the angular momentum algebra $[x_i,x_j]=2\imathλ_p ε_{ijk}x_k$ modulo setting $\sum_i x_i^2$ to a constant, using a recently introduced 3D rotationally invariant differential structure. Metrics are given by symmetric $3 imes 3$ matrices $g$ and we show that for each metric there is a unique quantum Levi-Civita connection with constant coefficients, with scalar curvature $ \frac{1}{2}({ m Tr}(g^2)-\frac{1}{2}{ m Tr}(g)^2)/\det(g)$. As an application, we construct Euclidean quantum gravity on the fuzzy unit sphere. We also calculate the charge 1 monopole for the 3D differential structure.

Motivation & Objective

• To develop a consistent quantum Riemannian geometry on the fuzzy sphere using a recently introduced 3D rotationally invariant differential calculus.
• To resolve the failure of standard differential structures to descend to the fuzzy sphere by constructing a new calculus that preserves rotational symmetry.
• To establish the existence and uniqueness of a quantum Levi-Civita connection for arbitrary symmetric 3×3 quantum metrics on the fuzzy sphere.
• To construct a model of Euclidean quantum gravity on the fuzzy sphere by integrating over all quantum geometries.
• To explore the implications of quantum geometry rigidity, including metric quantization and divergences in the functional integral.

Proposed method

• The paper uses a 3D differential calculus on the fuzzy sphere derived from the angular momentum algebra, with differential forms closed under the exterior derivative and compatible with rotational symmetry.
• Metrics are defined as symmetric 3×3 matrices $ g_{ij} $, representing quantum Riemannian structures on the noncommutative sphere.
• The quantum Levi-Civita connection is constructed as a bimodule connection with constant coefficients, uniquely determined by metric compatibility and torsion-freeness in the noncommutative setting.
• The scalar curvature is computed as $ \frac{1}{2} \left( \mathrm{Tr}(g^2) - \frac{1}{2} \mathrm{Tr}(g)^2 \right) / \det(g) $, providing a quantum Ricci scalar for the action.
• Euclidean quantum gravity is formulated by integrating over all such quantum metrics using a functional integral with a scalar curvature action.
• The charge 1 monopole is computed as a natural geometric object in the 3D calculus, relevant for spinor and spectral triple constructions.

Experimental results

Research questions

• RQ1 Does a quantum Levi-Civita connection exist and is it unique for any symmetric quantum metric on the fuzzy sphere?
• RQ2 Can a consistent quantum Riemannian geometry be defined on the fuzzy sphere using a rotationally invariant differential calculus that does not descend from flat space?
• RQ3 How does the scalar curvature of the fuzzy sphere depend on the choice of quantum metric, and what is its closed-form expression?
• RQ4 What are the quantum gravity path integral properties on the fuzzy sphere, particularly regarding UV and IR divergences?
• RQ5 Can a geometric spectral triple be constructed on the fuzzy sphere using the quantum differential structure and monopole bundle?

Key findings

• For every symmetric 3×3 matrix metric $ g $, there exists a unique quantum Levi-Civita connection with constant coefficients on the fuzzy sphere.
• The scalar curvature is given by the closed-form expression $ \frac{1}{2} \left( \mathrm{Tr}(g^2) - \frac{1}{2} \mathrm{Tr}(g)^2 \right) / \det(g) $, which depends nontrivially on the metric.
• The quantum geometry of the fuzzy sphere is rigid: all quantum metrics and connections are effectively constant over the algebra, resembling a single-point geometry.
• Euclidean quantum gravity on the fuzzy sphere exhibits UV and IR divergences at $ \lambda_i = 0 $ and $ \lambda_i = \infty $, but their ratio remains finite in expectation values.
• The expectation value of an $ n $-th power of the metric eigenvalues diverges as $ L^n $, where $ L $ is a cutoff, but ratios of such expectations remain well-defined in the $ L \to \infty $ limit.
• A natural charge 1 monopole is constructed as a line bundle in the 3D calculus, suggesting a path toward geometric spectral triples and spinor geometry on the fuzzy sphere.