[Paper Review] On Gravity, Torsion and the Spectral Action Principle

Frank Pfaeffle, Christoph A. Stephan|arXiv (Cornell University)|Jan 7, 2011

Advanced Operator Algebra Research20 references3 citations

TL;DR

This paper investigates the spectral action principle in four-dimensional Riemannian spin manifolds equipped with orthogonal connections, focusing on totally anti-symmetric torsion. It derives the gravitational part of the Chamseddine-Connes spectral action, shows that certain terms vanish compared to prior work, and proves that the equations of motion include a Proca-type equation for the torsion 3-form, suggesting torsion behaves as a massive vector boson. The critical points include Einstein manifolds with zero torsion, and non-trivial critical points are ruled out for warped product geometries.

ABSTRACT

We consider compact Riemannian spin manifolds without boundary equipped with orthogonal connections. We investigate the induced Dirac operators and the associated commutative spectral triples. In case of dimension four and totally anti-symmetric torsion we compute the Chamseddine-Connes spectral action, deduce the equations of motions and discuss critical points.

Motivation & Objective

• To extend the Chamseddine-Connes spectral action principle to Riemannian spin manifolds with orthogonal connections, particularly those with totally anti-symmetric torsion.
• To compute the gravitational part of the spectral action in four dimensions and verify consistency with previous results.
• To derive the equations of motion from the spectral action and analyze their critical points, especially regarding non-zero torsion configurations.
• To investigate whether torsion can be reconstructed from spectral data in even dimensions, using Connes' reconstruction theorem.
• To rule out non-trivial critical points in warped product geometries with non-zero torsion, ensuring consistency with Einstein gravity in the zero-torsion limit.

Proposed method

• Uses Cartan's classification of orthogonal connections and decomposes torsion into vectorial, anti-symmetric, and Cartan-type components.
• Applies Friedrich and Sulanke's result to show that only anti-symmetric torsion yields symmetric Dirac operators, eliminating vectorial torsion.
• Computes curvature invariants—Ricci and Riemann tensors—under totally anti-symmetric torsion using a synchronous frame near a point.
• Derives the spectral action via the trace of a function of the Dirac operator, focusing on the purely gravitational part in four dimensions.
• Performs explicit computation of the action using differential forms and Hodge star duality, leading to a curvature-squared expression in terms of symmetric and anti-symmetric parts of Ricci and Riemann tensors.
• Applies the reconstruction theorem of Connes to show that anti-symmetric torsion is spectrally reconstructible in even dimensions.

Experimental results

Research questions

• RQ1 How does the spectral action principle behave when applied to Riemannian manifolds with orthogonal connections and totally anti-symmetric torsion?
• RQ2 What are the equations of motion derived from the spectral action in four dimensions, and what physical interpretation do they admit?
• RQ3 Which critical points of the action exist, and do they include Einstein manifolds with zero torsion?
• RQ4 Can non-trivial torsion configurations survive as critical points, particularly in warped product geometries?
• RQ5 Are the terms in the gravitational spectral action that were previously reported actually non-zero, or do they vanish under proper computation?

Key findings

• The purely gravitational part of the Chamseddine-Connes spectral action in four dimensions confirms the vanishing of certain terms previously reported in [HPS10], aligning with [ILV10].
• The equations of motion derived from the action include a Proca equation for the torsion 3-form, indicating that torsion behaves as a massive vector field.
• All Einstein manifolds (with zero torsion) are critical points of the action, consistent with general relativity in the absence of torsion.
• Critical points that are warped products with non-zero torsion are excluded, as shown in Lemmas 5.10 and 5.13, ruling out such configurations as solutions.
• The anti-symmetric part of the torsion is spectrally reconstructible from the spectral triple data in even dimensions, confirming its physical relevance.
• The norm of the anti-symmetric part of the Riemann curvature, ‖Riem^A‖², appears explicitly in the action and contributes to the spectral action, while the Cartan-type torsion has no effect on the Dirac operator.