Matrix Regularizations in Noncommutative Geometry

• Matrix regularizations are techniques that represent smooth manifolds as sequences of finite-dimensional matrix algebras, ensuring the emergence of classical geometry in the large‐N limit.
• They utilize properties such as multiplicative consistency, Poisson–commutator correspondence, and trace–integral relations to rigorously encode geometric structures.
• Canonical examples like the fuzzy sphere and torus demonstrate the method’s power in modeling emergent gravity and quantized geometries within matrix models.

Matrix regularization is a foundational technique in noncommutative geometry for realizing smooth manifolds and their associated geometric structures as sequences of finite-dimensional matrix algebras, such that the classical, commutative geometry emerges in the large matrix-size limit. This correspondence establishes a bridge between differentiable manifolds (equipped with symplectic or Poisson structures) and algebras of matrices, underpinning much of the modern theory of emergent geometry and gravity in Yang–Mills matrix models, and enabling the study of noncommutative spaces with rigorously controlled classical limits (Steinacker, 2011).

1. Definition and Axiomatic Structure

Let $M$ denote a compact, $2n$-dimensional symplectic manifold with symplectic form $\omega$. A matrix regularization is a sequence of linear maps

$$\mathcal{I}_N: C^\infty(M) \longrightarrow \operatorname{Mat}(N, \mathbb{C})$$

such that as $N \to \infty$, and for all $f, g \in C^\infty(M)$, the following hold (Steinacker, 2011, Steinacker, 2011):

• Multiplicative property:

$$\mathcal{I}_N(f)\, \mathcal{I}_N(g) = \mathcal{I}_N(fg) + O(N^{-1})$$

• Poisson–commutator correspondence:

$$\frac{1}{i}[\mathcal{I}_N(f), \mathcal{I}_N(g)] = \mathcal{I}_N(\{f,g\}) + O(N^{-2})$$

where $\{f,g\}$ is the Poisson bracket associated to $\omega$.

• Trace–integral property:

$$(2\pi)^n\, \operatorname{Tr}(\mathcal{I}_N(f)) = \int_M \frac{\omega^n}{n!} f + O(N^{-1})$$

If $x^a: M \hookrightarrow \mathbb{R}^D$ is an embedding, the "matrix embedding coordinates" are $$X^a_N := \mathcal{I}_N(x^a) \in \operatorname{Mat}(N, \mathbb{C})$$, with commutators

$$[X^a_N, X^b_N] = i\, \mathcal{I}_N(\{x^a, x^b\}) + O(N^{-2}).$$

These encode the underlying semi-classical Poisson structure of $M$ (Steinacker, 2011).

2. Canonical Examples: Fuzzy Spaces

The archetypes of matrix regularization are the fuzzy sphere $S^2_N$ and the fuzzy torus $T^2_N$ (Steinacker, 2011, Steinacker, 2011, Ydri, 2016).

Fuzzy Sphere:

• The coordinates are quantized as

$$[X^a, X^b] = \frac{i}{\sqrt{C_N}}\, \varepsilon^{abc} X^c,\qquad \sum_{a=1}^3 X^a X^a = 1,\qquad C_N = \frac{1}{4}(N^2 - 1)$$

• Functions on $S^2$ are mapped to matrices via the correspondence between spherical harmonics $Y^l_m$ (with $l < N$) and "fuzzy harmonics" $\widehat{Y}^l_m$ (Ydri, 2016).
• The matrix Laplacian

$\Box = [X^a, [X^a,\,\cdot\,]]$

reproduces the eigenvalues $l(l+1)$ of the classical Laplacian up to $l = N-1$.

Fuzzy Torus:

• Defined using clock–shift operators $U, V$ satisfying $UV = q\,VU$, $q^N = 1$;
• The embedding and commutation relations quantize the classical torus $T^2 \hookrightarrow \mathbb{R}^4$ (Steinacker, 2011, Trzetrzelewski, 2012).

Moyal–Weyl Plane:

• Regularized by infinite-dimensional operators with commutation relations $[X^\mu, X^\nu] = i\,\theta^{\mu\nu} \,1$ (Steinacker, 2011), providing a matrix model for the quantization of flat symplectic spaces.

3. Emergent Geometry and Matrix Models

Central to Yang–Mills-type matrix models is the interpretation of certain matrix configurations as noncommutative (NC) branes—matrix analogues of embedded Poisson manifolds. The action

$$S = -\,\operatorname{Tr}\bigl[X^a, X^b\bigr]\bigl[X^{a'},X^{b'}\bigr]\delta_{a\,a'}\,\delta_{b\,b'} + \text{(fermions)}$$

admits background solutions $X^a \sim x^a: M \hookrightarrow \mathbb{R}^D$, and under small deformations $\widetilde{X}^a = X^a + A^a$, the emergent metric structure on $M$ is encoded in the matrix commutators (Steinacker, 2011).

The effective Riemannian metric seen by matter fields is determined by

$$G^{\mu\nu}(x) = e^{-\sigma(x)}\, \theta^{\mu\mu'} \theta^{\nu\nu'} g_{\mu'\nu'}$$

where $g_{\mu\nu}$ is the pullback of the ambient metric and $\theta^{\mu\nu}$ is the Poisson tensor. The conformal factor $e^{-\sigma}$ is fixed by the condition

$$e^{-(n-1)\sigma} = \frac{1}{\theta^n\, \sqrt{|g|}}.$$

This provides an explicit realization of gravity as an emergent, low-energy phenomenon in matrix models, with the spectrum of the matrix Laplacian $\Box$ approximating classical Laplace spectra (Steinacker, 2011, Steinacker, 2011).

4. Stability, Spectral Triples, and Spectral Geometry

Embedded NC branes are stable under sufficiently small perturbations, a fact ensured by the PBW-type quantization map. Any small deformation $A^a = f^a(X)$ again defines a quantized embedding of the same type (Steinacker, 2011).

Matrix geometry admits a natural spectral triple structure: $$(\mathcal{A}, \mathcal{H}, D) = (\operatorname{Mat}(N,\mathbb{C}),\,\,\mathbb{C}^N \text{ or spinors},\,\,D = \Gamma_a [X^a,\,\cdot\,])$$ with the square $D^2$ governed by the matrix Laplacian. For low eigenvalues, the spectrum of $\Box$ matches the classical Laplace–Beltrami operator on $(M, g)$ up to a cutoff of order $O(N)$, in accord with Weyl's law (Steinacker, 2011).

5. Matrix Regularization of Higher-Dimensional and Generic Geometries

Classical embedding theorems allow any 4-dimensional Riemannian manifold $(M^4, g)$ to be isometrically immersed in $\mathbb{R}^{10}$. By equipping $M$ with a closed (anti-)selfdual two-form $\omega$, yielding an (almost-)Kähler manifold structure, one can construct matrix regularizations such that

$$X^a = \mathcal{I}(x^a),\qquad x^a: M \hookrightarrow \mathbb{R}^{10}$$

with $[X^\mu, X^\nu] = i\,\theta^{\mu\nu}(x)$ ensuring the correct emergent metric. The corresponding matrix model action reproduces the low-energy dynamics of fields on $(M, g)$ (Steinacker, 2011).

6. Algebraic Formulation of Geometry in Matrix Language

All classical geometric constructs—second fundamental form, Weingarten operator, Ricci tensor, Codazzi–Mainardi equations, and the Gauss–Bonnet theorem—can be recast in algebraic form using higher-order Nambu or Poisson brackets on $C^\infty(M)$. These algebraic identities literally carry over to matrix regularizations, replacing functions by matrices and brackets by commutators. For example, the discrete (matrix) Gauss curvature and Euler characteristic are given by explicit matrix formulas: $$K_N = -\frac{1}{2\hbar^2} \sum_{i,j=1}^3 [X^i, X^j]^2, \qquad \chi_N = 2\pi\hbar \operatorname{Tr}(\gamma_N K_N)$$ with the convergence $K_N \to K$ and $\chi_N \to \chi(M)$ as $N \to \infty$, including a noncommutative Gauss–Bonnet theorem (Arnlind et al., 2010).

The spectrum of the discrete Laplacian on matrices also satisfies geometric bounds analogous to the classical setting: a positive lower bound on the discrete curvature yields a lower bound on the Laplacian eigenvalues, mirroring classical comparison theorems (Arnlind et al., 2010).

7. Broader Significance and Applications

Matrix regularization provides a rigorous framework for realizing noncommutative manifolds as sequences of finite-dimensional algebras that approximate smooth commutative spaces in the large-$N$ limit. This construction:

• Enables explicit realization of field theory, gravity, and emergent geometry within matrix models.
• Connects to spectral geometry via natural spectral triples, with matrix Laplacians encoding geometric information.
• Admits stable deformation classes, making it appropriate for dynamical models of geometry such as the IKKT/IIB matrix model (Steinacker, 2011, Steinacker, 2011).
• Admits explicit algebraic descriptions of curvature, topology, and their discrete analogues, providing noncommutative generalizations of classical theorems (Arnlind et al., 2010).

In conclusion, matrix regularizations constitute a foundational bridge between noncommutative algebraic structures and differential geometry, supporting extensive research on fuzzy spaces, emergent gravity, and quantized geometries using purely finite-dimensional matrix models (Steinacker, 2011, Arnlind et al., 2010, Steinacker, 2011).