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Asymmetric noncommutative torus has vanishing Einstein tensor

Deeponjit Bose, Andrzej Sitarz

TL;DR

The paper studies a two-dimensional asymmetric noncommutative torus endowed with a partially conformally rescaled Dirac operator $D_k$ and develops its spectral-geometry via Connes' spectral-triple framework. Using the noncommutative pseudodifferential calculus, it explicitly computes the symbols of $D_k$, $D_k^{-1}$, and $D_k^{-2}$, and then evaluates the metric, torsion, and Einstein spectral functionals through the Wodzicki residue. It proves that the torsion functional vanishes identically and that the Einstein functional also vanishes, supporting conjectures about curvature quantities in 2D spectral geometry and aligning with Gauss–Bonnet results in this NC setting. The results underscore the robustness of the spectral-functional approach for diagnosing geometric properties from Dirac operators alone and illustrate the NC torus as a fruitful testbed for NC differential geometry.

Abstract

We explicitly compute the spectral metric, torsion and Einstein tensors for a nontrivial spectral triple on a noncommutative torus, with the Dirac operator related to the fully equivariant Dirac by a partial conformal rescaling (as introduced in [1]). The results show that the spectral triple has vanishing torsion and the Einstein tensor also identically vanishes.

Asymmetric noncommutative torus has vanishing Einstein tensor

TL;DR

The paper studies a two-dimensional asymmetric noncommutative torus endowed with a partially conformally rescaled Dirac operator and develops its spectral-geometry via Connes' spectral-triple framework. Using the noncommutative pseudodifferential calculus, it explicitly computes the symbols of , , and , and then evaluates the metric, torsion, and Einstein spectral functionals through the Wodzicki residue. It proves that the torsion functional vanishes identically and that the Einstein functional also vanishes, supporting conjectures about curvature quantities in 2D spectral geometry and aligning with Gauss–Bonnet results in this NC setting. The results underscore the robustness of the spectral-functional approach for diagnosing geometric properties from Dirac operators alone and illustrate the NC torus as a fruitful testbed for NC differential geometry.

Abstract

We explicitly compute the spectral metric, torsion and Einstein tensors for a nontrivial spectral triple on a noncommutative torus, with the Dirac operator related to the fully equivariant Dirac by a partial conformal rescaling (as introduced in [1]). The results show that the spectral triple has vanishing torsion and the Einstein tensor also identically vanishes.

Paper Structure

This paper contains 11 sections, 2 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. The spectral triple of the asymmetric torus.
  3. Symbols of the Dirac
  4. Spectral functionals
  5. The metric functional
  6. The torsion functional
  7. The spectral Einstein functional
  8. Evaluating the Wodzicki residue.
  9. Conclusions
  10. Appendix: the symbol.
  11. Appendix: the integrals

Key Result

Proposition 4.1

Let $u_i,v_j \in A(\mathbb{T}^2_\theta)$, $i,j=1,2$ and $u = u_1 \sigma_1 + k u_2 \sigma_2$, $v = v_1 \sigma_1 + k v_2 \sigma_2$ be the one-forms in $\Omega^1(A(\mathbb{T}^2_\theta))$. Then the metric functional for the spectral triple of the asymmetric noncommutative torus is:

Theorems & Definitions (3)

  • Definition 2.1
  • Proposition 4.1
  • Proposition 4.2