[Paper Review] The notion of dimension in geometry and algebra

Yuri I. Manin|ArXiv.org|Feb 1, 2005

Advanced Operator Algebra Research25 references4 citations

TL;DR

This paper explores the concept of dimension across geometry, algebra, and theoretical physics, unifying ideas from fractal geometry, noncommutative geometry, and elliptic curves. It establishes an equivalence between holomorphic vector bundles on noncommutative tori and hearts of t-structures in coherent sheaf categories, providing a categorical framework for fractional and noncommutative dimensions.

ABSTRACT

This talk reviews some mathematical and physical ideas related to the notion of dimension. After a brief historical introduction, various modern constructions from fractal geometry, noncommutative geometry, and theoretical physics are invoked and compared.

Motivation & Objective

• To unify diverse mathematical and physical conceptions of dimension—fractal, noncommutative, algebraic—through a categorical framework.
• To understand how fractional and non-integer dimensions arise naturally in geometric and algebraic structures, particularly via s-densities and holomorphic bundles.
• To establish a bridge between noncommutative geometry (via noncommutative tori) and algebraic geometry (via coherent sheaf categories and t-structures).
• To provide a categorical and algebraic foundation for arithmetical applications in noncommutative geometry, especially for real quadratic irrationals.
• To generalize Kronecker’s idea of combining $√{d}$ and $√{-d}$ using elliptic functions to noncommutative rings via autoequivalences and graded algebras.

Proposed method

• Uses s-densities on differentiable manifolds to define fractional-dimensional objects via integration of $f|dx|^s$ for complex $s$.
• Constructs holomorphic connections $\overline{\nabla}_z$ on modules over noncommutative tori $A_\theta$, compatible with complex structures $\tau$.
• Applies torsion pairs and t-structures to the derived category of coherent sheaves on elliptic curves to define new abelian categories $\mathrm{Coh}^\theta(X)$.
• Establishes an equivalence between the category $\mathcal{C}_{\theta,\tau}$ of holomorphic bundles on noncommutative tori and $\mathrm{Coh}^{-\theta^{-1}}(T_{0,\tau})$.
• Constructs noncommutative graded rings $B = \oplus_{n\geq 0} H^0(E_g(\theta)^{\otimes n})$ from bimodules $E_g(\theta)$, which are autoequivalences when $\theta$ is a real quadratic irrational.
• Applies a general construction $A_{F,O} = \oplus_{n\geq 0} \mathrm{Hom}(O, F^n(O))$ to define noncommutative projective spectra for arithmetical applications.

Experimental results

Research questions

• RQ1 How can fractional dimensions be rigorously defined and integrated into geometric and algebraic frameworks?
• RQ2 What is the categorical structure underlying holomorphic bundles on noncommutative tori, and how does it relate to coherent sheaves on elliptic curves?
• RQ3 Can noncommutative analogues of algebraic geometry, such as noncommutative projective spectra, be constructed from autoequivalences of module categories?
• RQ4 How do real quadratic irrationals $\theta$ and complex structures $\tau$ interact to produce arithmetically meaningful noncommutative rings?
• RQ5 In what way does the equivalence $\mathcal{C}_{\theta,\tau} \simeq \mathrm{Coh}^{-\theta^{-1}}(T_{0,\tau})$ generalize classical results like Kronecker’s solution to Pell’s equation?

Key findings

• The category $\mathcal{C}_{\theta,\tau}$ of holomorphic bundles on a noncommutative torus $T_{\theta,\tau}$ forms an abelian category with a finite filtration whose quotients are standard bundles.
• There is a canonical equivalence $\mathcal{C}_{\theta,\tau} \simeq \mathrm{Coh}^{-\theta^{-1}}(T_{0,\tau})$, linking noncommutative geometry to coherent sheaf theory on elliptic curves.
• For real quadratic irrational $\theta$, the bimodule $E_g(\theta)$ induces a nontrivial autoequivalence on $A_\theta$-modules, enabling the construction of noncommutative graded rings.
• The ring $B = \oplus_{n\geq 0} H^0(E_g(\theta)^{\otimes n})$ is a noncommutative graded ring arising from a holomorphic bimodule, generalizing constructions in noncommutative algebraic geometry.
• The construction $A_{F,O} = \oplus_{n\geq 0} \mathrm{Hom}(O, F^n(O))$ yields a noncommutative projective spectrum whose Serre category is equivalent to $\mathrm{Coh}^\theta(T_{0,\tau})$ for real quadratic $\theta$.
• When $\theta \in \mathbb{Q}(\sqrt{d})$ and $\tau \in \mathbb{Q}(\sqrt{-d})$, the resulting noncommutative rings provide a refined, categorical realization of Kronecker’s idea to solve Pell’s equation using elliptic functions.