Non-commutative Geometry

Spectral Triples: The Heart of NCG

Definition: Spectral Triple

A spectral triple $({\mathcal A}, {\mathcal H}, D)$ consists of:

• ${\mathcal A}$ - a *-algebra represented on
• ${\mathcal H}$ - a Hilbert space with
• $D$ - a self-adjoint operator (Dirac operator)

Such that $[D, a]$ is bounded for all $a \in {\mathcal A}$

The Non-commutative Torus

Deformation Parameter θ: 0.3 Animate Deformation

Moyal Product

The non-commutative multiplication on $C^∞(T^2_θ)$:

$$f \star_θ g = \sum_{n \in \mathbb{Z}^2} e^{-πi n^T θ m} \hat{f}(n)\hat{g}(m) e^{2πi(n+m) \cdot x}$$

where $UV = e^{2πiθ}VU$ generates the algebra

Connes' Distance Formula

The geodesic distance between pure states:

$$d(φ, ψ) = \sup\{|φ(a) - ψ(a)| : a \in {\mathcal A}, \|[D, a]\| \leq 1\}$$

This recovers the Riemannian distance for commutative manifolds!

Distance on Quantum Spaces

The Fuzzy Sphere

A finite-dimensional approximation to $S^2$ using matrix algebras:

Matrix Size N: 5

Algebra Structure

The fuzzy sphere $S^2_N$ is described by:

• Algebra: $M_N(\mathbb{C})$ (N×N matrices)
• Coordinates: $x_i = \frac{2}{N}J_i$ where $J_i$ are spin-j generators
• Constraint: $\sum x_i^2 = 1 + O(1/N)$

Quantum Groups & Hopf Algebras

SUq(2) - Quantum SU(2)

Generators $a, b, c, d$ with relations:

$$ab = qba, \quad ac = qca, \quad bd = qdb, \quad cd = qdc$$ $$bc = cb, \quad ad - da = (q - q^{-1})bc$$ $$ad - q^{-1}bc = 1, \quad det_q = 1$$

The Spectral Action Principle

Action Functional

The spectral action for a spectral triple:

$$S = \text{Tr}(f(D/Λ)) + \langle ψ, Dψ \rangle$$

where $f$ is a cutoff function and $Λ$ is the energy scale

🌟 The Non-commutative Standard Model

Connes and Chamseddine showed that the Standard Model of particle physics emerges naturally from the spectral action on:

$M × F$ where $M$ is spacetime and $F$ is a finite non-commutative space

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Matrix Geometries

Fuzzy Spaces

Finite matrix approximations to continuous geometries

Matrix Models

IKKT, BFSS models describing M-theory

Emergent Geometry

Spacetime emerging from matrix eigenvalues

Cyclic Cohomology

The Dixmier Trace

For compact operators with eigenvalues $μ_n$:

$$\text{Tr}_ω(T) = \lim_{n→ω} \frac{1}{\log n} \sum_{k=1}^n μ_k$$

Essential for defining integration in NCG

K-Theory & Index Theory

Connes-Kasparov Isomorphism

The analytic and topological K-theory are related by:

$$K_*(C^*_r(Γ)) \cong K^*(BΓ)$$

for discrete groups $Γ$ satisfying the Baum-Connes conjecture

Revolutionary Applications

Quantum Field Theory

Renormalization via Hopf algebras and motivic Galois theory

Number Theory

Approach to the Riemann Hypothesis via spectral interpretation

Quantum Gravity

Spectral triple approach to quantum spacetime

Condensed Matter

Quantum Hall effect and topological insulators

Foundational Papers

Noncommutative Geometry

Alain Connes (1994)

Academic Press

The foundational textbook establishing the field.

The Spectral Action Principle

Ali Chamseddine, Alain Connes (1997)

Comm. Math. Phys.

Deriving physics from spectral data.

Gravity and the Standard Model with Neutrino Mixing

Chamseddine, Connes, Marcolli (2007)

Adv. Theor. Math. Phys.

The non-commutative geometric Standard Model.

From Physics to Number Theory via NCG

Alain Connes, Matilde Marcolli (2008)

AMS

Connections to the Riemann zeta function.

Leading Research Centers

• IHES - Alain Connes' group
• Penn State - NCG and mathematical physics
• Copenhagen - Operator algebras and K-theory
• Vanderbilt - NCG and number theory
• ANU Canberra - Spectral triples and index theory