Cyclic Cohomology and the Local Index Formula¶

Job: develop cyclic cohomology $HC^\bullet$ / periodic cyclic $HP^\bullet$ as the NC replacement of de Rham cohomology, the Chern–Connes character pairing with $K$-theory, and the Connes–Moscovici local index formula — a density-level cocycle recovering the Atiyah–Singer integrand, with the NC torus area cocycle as the worked example.

Core¶

Hochschild and cyclic cohomology¶

For a unital algebra $A$ over $\mathbb{C}$, the Hochschild complex is $C^n(A)=\mathrm{Hom}(A^{\otimes(n+1)},\mathbb{C})$ with the Hochschild differential $$ (b\varphi)(a^0,\dots,a^{n+1})=\sum_{j=0}^{n}(-1)^j\varphi(\dots,a^ja^{j+1},\dots)+(-1)^{n+1}\varphi(a^{n+1}a^0,\dots,a^n). $$ Its cohomology is the Hochschild cohomology $HH^\bullet(A)$.

Connes' cyclic operator $B:C^n\to C^{n-1}$ is $B=(1-\lambda)sN$ where $\lambda$ is the cyclic permutation $\lambda\varphi(a^0,\dots,a^n)=(-1)^n\varphi(a^n,a^0,\dots,a^{n-1})$, $N=1+\lambda+\cdots+\lambda^n$, and $s$ is the extra-degeneracy map. One has $b^2=B^2=bB+Bb=0$, giving the $(B,b)$-bicomplex.

Cyclic cohomology $HC^\bullet(A)$ is the cohomology of the total complex of the $(B,b)$-bicomplex. Periodic cyclic cohomology $HP^\bullet(A)$ is the direct limit under Connes' $S$-operation $S:C^n\to C^{n+2}$ ($S=Bc+cb$ with the contracting homotopy $c$); it is $2$-periodic, $HP^\bullet=HP^{\bullet+2}$.

HKR and the de Rham ↔ cyclic dictionary¶

Theorem (Hochschild–Kostant–Rosenberg, Connes). For $A=C^\infty(M)$ (smooth functions on a compact manifold), there is a natural isomorphism $$ HP^\bullet(C^\infty(M))\cong \bigoplus_k H^{\bullet+2k}_{\mathrm{dR}}(M), $$ given on a closed de Rham form $\omega$ of degree $n$ by the cyclic cocycle $\varphi_\omega(f^0,\dots,f^n)=\int_M f^0\,df^1\wedge\dots\wedge df^n\,\omega$ (the Connes–Hochschild–Kostant–Rosenberg map). Hence periodic cyclic cohomology is the NC replacement of de Rham.

Classical (de Rham)	NC replacement (cyclic)
$H^\bullet_{\mathrm{dR}}(M)$	$HP^\bullet(C^\infty(M))\cong\oplus_k H^{\bullet+2k}_{\mathrm{dR}}$
wedge product	cup product of cyclic cocycles
exterior differential	Hochschild differential $b$
de Rham differential → periods	Connes $S$-operation (period 2)
integration current	cyclic cocycle $\varphi_\omega$ above
closed forms	cocycles modulo $B$-boundaries + Hochschild boundaries

The Chern–Connes character¶

Theorem (Chern–Connes character). There is a natural transformation $$ \mathrm{ch}:K_\bullet(A)\longrightarrow HP^\bullet(A) $$ compatible with the $S$-operation. For $\bullet=0$ and a projection $e\in M_N(A)$, $$ \mathrm{ch}{2k}(e)=\mathrm{Tr}\bigl((e-\tfrac12)\underbrace{\delta e\wedge\dots\wedge\delta e}(A) $$ (the universal differential }\bigr)\quad\text{in }HC^{2k$\delta:A\to\Omega^1_A$ with $\delta(ab)=\delta a\,b+a\,\delta b$). For $\bullet=1$ and a unitary $u\in M_N(A)$, $$ \mathrm{ch}{2k+1}(u)=\mathrm{Tr}\bigl((u^{-1}-1)\delta u\,\underbrace{(\delta u^{-1}\delta u)^k}{ }\bigr). $$

Pairing $\langle\mathrm{ch}(e),\varphi\rangle\in\mathbb{Z}$ for $e\in M_N(A)$, $\varphi\in HP^{2k}(A)$ (even) and $\langle\mathrm{ch}(u),\varphi\rangle$ for $u\in M_N(A)$, $\varphi\in HP^{2k+1}(A)$ (odd) gives the numerical invariants — the NC analog of Chern numbers.

The local index formula (Connes–Moscovici)¶

Let $(A,H,D)$ be a $p$-summable spectral triple with dimension spectrum $\Sigma\subset\mathbb{C}$ — a discrete set such that for every $b$ in the algebra generated by $A$, $[D,A]$, and $|D|^{-1}$, the zeta function $\zeta_b(s)=\mathrm{Tr}(b|D|^{-s})$ extends meromorphically to $\mathbb{C}$ with poles only in $\Sigma$ (assume simple poles for clarity). Define the Wodzicki-type residue $\mathrm{Res}_s(b|D|^{-s})$ at $s=n\in\Sigma$.

Theorem (Connes–Moscovici local index formula). The Chern character $\mathrm{ch}(e)\in HP^\bullet(A)$ (resp. $\mathrm{ch}(u)$) is represented by the local cocycle $$ \varphi_n(a^0,\dots,a^n)=\sum_{k_1,\dots,k_n\ge 0}c_{n,k}\,\mathrm{res}\Bigl(\mathrm{Tr}\bigl(\gamma\, a^0[D,a^1]^{(k_1)}\cdots[D,a^n]^{(k_n)}|D|^{-n-2|k|}\bigr)\Bigr) $$ (even case, $\gamma$ the grading), with $|k|=k_1+\cdots+k_n$, universal constants $c_{n,k}=\frac{(-1)^{|k|}}{2}\binom{n+2|k|}{n}\Gamma(|k|+n/2)$, the residues taken at $s=n+2|k|$, and the iterated commutator $[D,a]^{(k)}=[\,[D,a],\dots]\ (k\text{ folds})$. The odd case is analogous without $\gamma$ and with unitaries.

The cocycle is local — it depends only on the "density level" (residues of zeta functions of algebraic combinations of $a$ and $D$), not on global traces of powers of $D$.

Properties. - Recovery of Atiyah–Singer. When $A=C^\infty(M)$ and $D$ is a Dirac-type operator on a compact spin manifold, the local cocycle evaluates to the classical integrand $\hat A(M)\mathrm{ch}(E)$ (the $\hat A$-genus times the Chern character of the twisting bundle), recovering the Atiyah–Singer index theorem. - Periodicity. The cocycle is closed under the $S$-operation: $S\varphi=0$ in $HP$, hence defines a periodic class. - Transgression. There is a transgression formula relating $\mathrm{ch}$ and $\varphi$ via the gauge action / universal connection: the local cocycle is a representative of the periodic class of $\mathrm{ch}$, and the difference between representatives in consecutive degrees is a $B$-boundary.

Pairings and the index theorem¶

Theorem (Connes' index pairing). For an even spectral triple $(A,H,D)$ with grading $\gamma$: - Even case. For a projection $e\in M_N(A)$, $$ \langle\mathrm{ch}(e),[D]\rangle=\mathrm{Index}(eD^+e)\in\mathbb{Z}, $$ where $D^+=\frac{1+\gamma}{2}D\frac{1-\gamma}{2}$ is the positive chirality part. - Odd case. For a unitary $u\in M_N(A)$, $$ \langle\mathrm{ch}(u),[D]\rangle=\mathrm{Index}(P_u D P_u)\in\mathbb{Z}, $$ with $P_u=\frac{1}{2}(1+u)$ acting on $H\otimes\mathbb{C}^N$. The local index formula gives a cohomological representative $\varphi$ with $\langle\mathrm{ch}(e),\varphi\rangle=\mathrm{Index}(eD^+e)$ — a finite, computable residue formula.

Classical (Atiyah–Singer)	NC (Connes–Moscovici)
elliptic $D$ on compact $M$	spectral triple $(A,H,D)$
$\mathrm{Index}(D)=\int_M\hat A(M)\mathrm{ch}(E)$	$\langle\mathrm{ch}(e),\varphi\rangle=\mathrm{Index}(eD^+e)$
integrand = Chern–Weil form	integrand = local cocycle (Wodzicki residues of $\zeta_b$)
Chern character $K^0\to H^{\mathrm{even}}_{\mathrm{dR}}$	Chern–Connes $\mathrm{ch}:K_0(A)\to HP^{\mathrm{even}}(A)$
Thom isomorphism / Bott	suspension in cyclic cohomology via $S$-operation
families index	NC families; leafwise index for foliations

See it¶

The area cocycle on the NC torus and the Rieffel projection. Let $A_\theta$ be the NC 2-torus generated by unitaries $U,V$ with $UV=e^{2\pi i\theta}VU$, smooth subalgebra $A_\theta^\infty$, canonical trace $\tau(\sum a_{mn}U^mV^n)=a_{00}$. The derivations $\delta_1(U)=2\pi iU$, $\delta_1(V)=0$, $\delta_2(U)=0$, $\delta_2(V)=2\pi iV$ give the Connes–Rieffel Dirac $D=\delta_1+\delta_2\gamma$ on $H=A_\theta^\infty\otimes\mathbb{C}^2$.

The area cyclic 2-cocycle is $$ \varphi_2(a^0,a^1,a^2)=\frac{1}{2\pi i}\,\tau\bigl(a^0(\delta_1 a^1\,\delta_2 a^2-\delta_2 a^1\,\delta_1 a^2)\bigr). $$ Evaluating on generators: $\varphi_2(U,V,U^{-1})$ vanishes (only $UV$-cyclic patterns with both derivations nonzero contribute), while $$ \varphi_2(U,V,U^{-1},V^{-1})\ \text{(as a 3-cocycle, odd, for the odd index)}\quad\text{or}\quad\varphi_2(U,V,U^{-1}V^{-1})\propto\theta $$ in the even formulation; more precisely the fundamental cyclic 2-cocycle gives $\varphi_2(U,V,U^{-1})=2\pi i\,\theta$ up to the chosen normalization, encoding the symplectic area form on the underlying classical $T^2$ at $\theta=0$.

The Rieffel projection $p\in A_\theta^\infty$ (for $0<\theta<1$, built from a Powers–Rieffel Gaussian-type function in $A_\theta$) satisfies $$ \tau(p)=\theta,\qquad p=p^*=p^2. $$ The even index pairing with the Dirac and $\varphi_2$: $$ \langle\mathrm{ch}(p),\varphi_2\rangle=\mathrm{Index}(pDp)=\theta-\lfloor\theta\rfloor\quad(\text{for }0<\theta<1,\ \lfloor\theta\rfloor=0,\ =\theta). $$ This is the NC analog of the $\hat A$-genus / Chern-number pairing on $T^2$: the "Chern number" of the Rieffel projective module is the fractional part of $\theta$, recovered as a residue of $\zeta_p(s)=\tau(p|D|^{-s})$ via the local index formula. The integrality of $\mathrm{Index}$ forces the range of $\tau$ on $K_0(A_\theta)$ to be $\mathbb{Z}+\theta\mathbb{Z}$ (Pimsner–Voiculescu).

Check¶

Show that for $A=C^\infty(M)$ the local cocycle $\varphi_n(f^0,\dots,f^n)$ reduces to $\int f^0\,df^1\wedge\dots\wedge df^n\,\omega$ for a Dirac-type $D$.

$[D,f]=c(df)$ is Clifford multiplication by $df$, so nested commutators $[D,f]^{(k)}$ vanish for $k\ge 2$ (Clifford algebra relations: $[c(df),c(dg)]=0$ at the level of Clifford action on the bundle up to curvature). Hence only the $|k|=0$ term survives, and the residue $\mathrm{res}(\mathrm{Tr}(\gamma f^0 c(df^1)\cdots c(df^n)|D|^{-n}))$ computes, by the heat-kernel / Wodzicki-residue on the Dirac, the integral $\int f^0\,df^1\wedge\dots\wedge df^n\,\hat A(M)\mathrm{ch}(S)$ — the classical integrand.

Why must the dimension spectrum be simple for the local index formula as stated?

With multiple (higher-order) poles, the residue is replaced by coefficients of Laurent expansions (Taylor residues) and the universal constants $c_{n,k}$ must be augmented with the order-of-pole data; the cocycle then includes extra "logarithmic" terms. Poles of order $>1$ correspond to modular-type anomalies (e.g. modular forms in the Connes–Moscovici transverse geometry of foliations) and the formula generalizes but the simple-pole statement above is no longer valid verbatim.

Pitfalls¶

$HP^\bullet$ is 2-periodic, not $\mathbb{Z}$-graded: $HC^\bullet$ is the finer invariant; the local index formula naturally lives in $HP$ but its representatives are explicit in $HC$.
Dimension spectrum must be simple/discrete for the formula as written; higher-order poles require the refined (residue-with-log) version (Connes–Moscovici, Cyclic cohomology and the transverse index).
The $(B,b)$-bicomplex is not a first-quadrant bicomplex in the naive sense: the $S$-operation shifts by 2, and periodicity kills information in each fixed degree.
Chern–Connes vs. classical Chern character: the classical one factors through de Rham; the NC one lands in $HP$ directly and the HKR map is the bridge (only valid for smooth commutative $A$).
Pairing integrality $\langle\mathrm{ch}(e),\varphi\rangle\in\mathbb{Z}$ requires $\varphi$ to be a cyclic cocycle (not just Hochschild) — the cyclic condition is the integrality condition.
The local formula is local in $D$ and $a$, not in the state: it is a residue formula, not a heat-trace asymptotic. The two coincide for commutative $A$ but differ in general.

Refs¶

Connes, "Noncommutative differential geometry" (Publ. Math. IHÉS 1985); Noncommutative Geometry, Ch. III, IV
Connes–Moscovici, "The local index formula in noncommutative geometry" (GAFA 1995)
Connes–Moscovici, "Cyclic cohomology and the transverse index theorem" (Annals 1998)
Loday, Cyclic Homology; Khalkhali, Basic Noncommutative Geometry, Ch. 3–4
Rieffel, "$C^*$-algebras associated with irrational rotations" (Pac. J. Math. 1981); Pimsner–Voiculescu, "Exact sequences for $K$-groups and Ext-groups"

Next¶

NCG index · Foundations · Spectral Triples & Reconstruction · Baum–Connes Assembly · Spectral Action & Physics · DAG / derived moduli & perfect — the Hochschild as the $\mathrm{QCoh}$ of the loop space (the Ben-Zvi–Nadler) · K-theory · Operator algebras

Classical (Atiyah–Singer)	NC (Connes–Moscovici)
elliptic \(D\) on compact \(M\)	spectral triple \((A,H,D)\)
\(\mathrm{Index}(D)=\int_M\hat A(M)\mathrm{ch}(E)\)	\(\langle\mathrm{ch}(e),\varphi\rangle=\mathrm{Index}(eD^+e)\)
integrand = Chern–Weil form	integrand = local cocycle (Wodzicki residues of \(\zeta_b\))
Chern character \(K^0\to H^{\mathrm{even}}_{\mathrm{dR}}\)	Chern–Connes \(\mathrm{ch}:K_0(A)\to HP^{\mathrm{even}}(A)\)
Thom isomorphism / Bott	suspension in cyclic cohomology via \(S\)-operation
families index	NC families; leafwise index for foliations