Noncommutative Geometry: Foundations¶

Job: Connes spectral triples as NC Riemannian manifolds, a pointer to cyclic cohomology and the Chern–Connes pairing, noncommutative tori as the basic example, and the philosophy of the local index formula.

Core¶

From spaces to algebras¶

Gelfand–Naimark: compact Hausdorff $X$ $\leftrightarrow$ unital commutative $C^*$-algebra $C(X)$. Noncommutative geometry studies noncommutative $C^*$- / dense $*$-subalgebras $A$ as “function algebras on NC spaces,” with additional analytic structure for metric geometry.

Typical sources of noncommutativity.

Source	Algebra
Quotients $X/G$ bad orbit spaces	reduced crossed products $C(X)\rtimes_r G$
Foliations	Connes foliation $C^*$-algebra
Quantum tori / deformations	$A_\theta$, strict deformation quantization
Group duals	reduced group $C^*_r\Gamma$

Spectral triples (Connes)¶

A (unital, even/odd) spectral triple is data $(A,H,D)$ where:

$A$ is a dense $*$-subalgebra of a $C^*$-algebra acting on Hilbert space $H$;
$D$ is an unbounded self-adjoint operator on $H$ (the Dirac operator);
$[D,a]$ extends to a bounded operator for all $a\in A$;
$(1+D^2)^{-1/2}$ is compact (discrete spectrum / finite volume).

Even: $\mathbb{Z}/2$-grading $\gamma$ on $H$ with $\gamma a=a\gamma$, $\gamma D=-D\gamma$. Real structure: antiunitary $J$ implementing KO-dimension axioms (Connes).

Metric (Connes distance). On states / points of the spectrum when commutative: $$ d(\varphi,\psi)=\sup\bigl{|\varphi(a)-\psi(a)|:\ a\in A,\ |[D,a]|\le 1\bigr}. $$ For $(C^\infty(M),L^2(S),D\!\!\!/\,)$ on a compact spin manifold, this recovers geodesic distance.

Dimension. $D$ has spectral dimension $n$ if eigenvalues of $|D|$ grow like Weyl’s law $\lambda_k\sim c\,k^{1/n}$; more precisely via dimension spectrum of zeta functions $\zeta_b(s)=\operatorname{Tr}(b|D|^{-s})$.

Reconstruction (pointer). Connes’ reconstruction theorem: under additional axioms (regularity, orientability, Poincaré duality in $K$-theory, etc.), commutative spectral triples arise from spin$^c$ Riemannian manifolds.

Cyclic cohomology (pointer)¶

Cyclic cohomology $HC^*(A)$ is computed from the cyclic complex of multilinear forms $\varphi:A^{n+1}\to\mathbb{C}$ with cyclic symmetry and Hochschild coboundary constraints. Periodic cyclic cohomology $HP^*$ is the target of the Chern–Connes character $$ \operatorname{ch}:K_(A)\to HP^(A) $$ pairing $K$-theory classes with cyclic cocycles to give numerical indices.

Classical	NC replacement
de Rham $H^*_{\mathrm{dR}}(M)$	periodic cyclic $HP^*(C^\infty(M))$
Chern character $K\to H^{\mathrm{even}}$	$\operatorname{ch}:K_(A)\to HP^$
Fundamental class $[M]$	Fredholm module / $K$-homology class of $D$

Pairing. $\langle\operatorname{ch}(e),[D]\rangle=\operatorname{Index}(eDe)$ for projections $e$ — NC index formula at the cohomological level.

Noncommutative tori¶

Fix $\theta\in\mathbb{R}$ (or skew matrix $\Theta$ in higher rank). The noncommutative torus $A_\theta$ is the universal $C^*$-algebra generated by unitaries $U,V$ with $$ UV=e^{2\pi i\theta}VU. $$ Smooth subalgebra $A_\theta^\infty$: rapid-decay Fourier series $\sum a_{mn}U^m V^n$.

Feature	Fact
Trace	unique normalized trace $\tau(\sum a_{mn}U^mV^n)=a_{00}$ (irrational $\theta$)
$K_0$	$K_0(A_\theta)\cong\mathbb{Z}^2$; range of $\tau_*$ is $\mathbb{Z}+\theta\mathbb{Z}$
Morita	$A_\theta\sim_M A_{\theta'}$ under $\mathrm{SL}_2(\mathbb{Z})$ action on $\theta$
Spectral triple	Connes–Rieffel Dirac-type operators from derivations $\delta_\tau,\delta_\sigma$

NC tori are the test bed for NC Yang–Mills, connections on projective modules, and Chern numbers pairing with cyclic cocycles.

Index theorem philosophy¶

Classical Atiyah–Singer: analytic index of elliptic $D$ equals topological index ($\hat A(M)\operatorname{ch}(E)$ integrated).

NC upgrade (Connes–Moscovici local index formula). For a spectral triple of finite dimension spectrum, $$ \operatorname{Index}(eDe)=\sum_k \int!!!!!!-\, \varphi_k\bigl(e, [D,e],\ldots\bigr), $$ a finite sum of residues of zeta functions involving nested commutators with $D$ — a local formula (density-level) in the NC sense. Recovers AS when $A=C^\infty(M)$ and $D$ is Dirac-type.

Baum–Connes / assembly (pointer). For group $C^*$-algebras, assembly maps relate $K$-homology of classifying spaces to $K_*(C^*_r\Gamma)$ — index theory for Galois coverings and higher signatures in NCG language.

Spectral action (physics pointer). Chamseddine–Connes: $S=\operatorname{Tr}f(D/\Lambda)$ expands in Seeley–DeWitt coefficients; applied to almost-commutative geometries (manifold $\times$ finite NC space) to package Einstein–Yang–Mills–Higgs Lagrangians.

See it¶

Foliation index. For a foliated manifold $(M,\mathcal{F})$, longitudinal elliptic operators define classes in $K$-theory of the foliation $C^*$-algebra; Connes’ index theorem pairs them with cyclic cocycles from holonomy-invariant transverse measures — analytic index lives in $\mathbb{R}$ (or more refined $K$-groups), not necessarily $\mathbb{Z}$. Leafwise Dirac on the Kronecker foliation of $T^2$ (slope $\theta$) is closely related to the NC torus picture.

Distance on the NC torus. Derivations and a Dirac operator built from them yield a metric geometry on the state space; classical $T^2$ is recovered as $\theta\to 0$ in an appropriate topology of spectral triples.

Check¶

Show that $\|[D,a]\|\le 1$ controls Lipschitz constants when $A=C^\infty(M)$ and $D=D\!\!\!/$.

For spin Dirac, $[D,f]$ acts as Clifford multiplication by $df$, so $\|[D,f]\|=\|df\|_\infty=\operatorname{Lip}(f)$. Thus $\sup\{|f(p)-f(q)|:\operatorname{Lip}(f)\le 1\}=d_g(p,q)$.

Why does irrational rotation algebra $A_\theta$ have no nontrivial projections that are ‘characteristic functions of open sets’?

For irrational $\theta$, $A_\theta$ is simple and has a unique trace; projections have traces in $(\mathbb{Z}+\theta\mathbb{Z})\cap[0,1]$, so there are projections (Rieffel), but they do not come from an underlying classical open set — there is no Gelfand spectrum realizing them as clopen sets of a compact Hausdorff space of dimension $2$ in the commutative way ($A_\theta$ is noncommutative).

Pitfalls¶

$K$-homology class of $D$ needs Fredholm module data (bounded transform $F=D|D|^{-1}$); unbounded triples require regular representations.
Summability: $p$-summable triples ($|D|^{-1}$ in $\mathcal{L}^{p+}$) determine which cyclic degree receives the Chern character.
Different NCG schools: analytic (Connes), algebraic (Artin, van den Bergg, Kontsevich–Soibelman motives), deformation quantization (Bayen–Flato–Frønsdal–Lichnerowicz–Sternheimer) — do not mix theorems freely.
Compact resolvent fails for noncompact manifolds without care (proper cocompact actions, Mooerdijk–Mrčun, etc.).

Refs¶

Connes, Noncommutative Geometry (Academic Press)
Connes–Moscovici, “The local index formula in noncommutative geometry”
Gracia-Bondía–Várilly–Figueroa, Elements of Noncommutative Geometry
Rieffel, “$C^*$-algebras associated with irrational rotations”

Next¶

NCG index · Cyclic Cohomology & Local Index · Spectral Triples & Reconstruction · Baum–Connes Assembly · Spectral Action & Physics · Functional analysis · K-theory · Differential geometry · Mathematical physics · Representation theory · Algebraic geometry · Operator algebras — C*/von Neumann algebras as the foundation of NCG

Classical	NC replacement
de Rham \(H^*_{\mathrm{dR}}(M)\)	periodic cyclic \(HP^*(C^\infty(M))\)
Chern character \(K\to H^{\mathrm{even}}\)	\(\operatorname{ch}:K_(A)\to HP^\)
Fundamental class \([M]\)	Fredholm module / \(K\)-homology class of \(D\)