Open issues in Connes' axiomatic foundation of noncommutative geometry (1996): finite-dimensional realization of the algebra-Dirac constraints, the quantum-group origin of the Standard Model algebra, hypoelliptic (quartic) extension, and the Planckian fading of commutativity

In 1996, Connes proposed that a noncommutative geometry is governed by simple algebraic relations between an involutive algebra of coordinates $\mathscr{A}$ and an infinitesimal line element $ds=D^{-1}$, where $D$ is a Dirac operator, and showed that the almost-commutative algebra $\mathscr{A}=C^\infty(M)\otimes(\mathbb{C}\oplus\mathbb{H}\oplus M_3(\mathbb{C}))$, equipped with a purely gravitational spectral action, encodes the Standard Model coupled to gravity. The derivation of the full Lagrangian from the spectral action, deferred in that paper, was completed shortly afterwards with Chamseddine. The paper's four remaining foundational issues are all still open. (1) The nonlinear constraints linking $\mathscr{A}$ and $D$ -- the order-one condition $[[D,a],Jb^*J^{-1}]=0$ and the orientability axiom requiring a Hochschild 4-cycle -- admit no finite-dimensional realization, because Hochschild homology in positive degree vanishes for the semisimple finite-dimensional algebras available, and existing finite or 'fuzzy' geometries drop or replace the obstructed axiom rather than satisfy it. (2) The suggested derivation of the finite internal algebra $\mathscr{A}_F$ and its bimodule from quantum groups at roots of unity, including the extension from $SU(2)$ to $Spin(4)$, has been worked out only in fragments. (3) The axiom system is tailored to elliptic Dirac operators, and the quartic-type relations needed to cover hypoelliptic index-theoretic situations have not been formulated. (4) The predicted fading of near-commutativity at Planckian energies, with inner automorphisms exhausting the automorphism group, remains purely qualitative. The problem is to resolve these four linked components and thereby complete the proposed axiomatic foundation of geometry coupled to matter.

Background

A spectral triple $(\mathscr{A},\mathscr{H},D)$ is the basic object of Alain Connes' noncommutative geometry: $\mathscr{A}$ is an involutive algebra of 'coordinates' represented on a Hilbert space $\mathscr{H}$, and $D$ is a self-adjoint operator generalizing the Dirac operator of a spin manifold. In the commutative case $\mathscr{A}=C^\infty(M)$, $\mathscr{H}=L^2(M,S)$ (square-integrable spinors) and $D$ the Dirac operator, all metric data of the compact Riemannian spin manifold $M$ can be recovered from the pair $(\mathscr{A},D)$: the geodesic distance is given by Connes' formula $d(p,q)=\sup\{|f(p)-f(q)|:\|[D,f]\|\le 1\}$, the volume form and the Einstein-Hilbert action are recovered from the spectrum of $D$, and the inverse $ds=D^{-1}$ plays the role of an infinitesimal line element -- an operator whose singular values are the infinitesimals of order $1/n$ in dimension $n$.

In the source paper (Commun. Math. Phys. 182 (1996) 155-176), Connes distilled the relation between $\mathscr{A}$ and $ds$ into axioms from which ordinary Riemannian geometry can be reconstructed. Two of them matter here. Axiom 2, the order-one condition, requires $[[D,a],b]=0$ for all $a,b\in\mathscr{A}$: the commutators $[D,a]$ behave like first-order differential operators. Axiom 4, orientability, requires a Hochschild cycle $c\in Z_n(\mathscr{A},\mathscr{A})$: Hochschild homology $H_n(\mathscr{A},\mathscr{A})$ is the homology of the complex of $(n+1)$-fold tensor powers of $\mathscr{A}$ with the Hochschild boundary $b$ (a signed alternation of the products of adjacent tensor factors); a cycle satisfies $bc=0$, and its representation $\pi(c)=\sum_j a_{0j}[D,a_{1j}]\cdots[D,a_{nj}]$ is required to equal the chirality operator $\gamma$ (the abstract $\gamma_5$ matrix) in even dimension $n$. The cycle is the algebraic substitute for the volume form. In the noncommutative case the axioms are twisted by Tomita's involution $J$, an antilinear isometry on $\mathscr{H}$ abstracting charge conjugation: axiom 2 becomes $[[D,a],Jb^*J^{-1}]=0$, and axiom 4 asks for a cycle with coefficients in $\mathscr{A}\otimes\mathscr{A}^\circ$ (the tensor product of $\mathscr{A}$ with its opposite algebra). A real noncommutative $n$-dimensional geometry is a triple $(\mathscr{A},\mathscr{H},D)$ with real structure $J$ satisfying these.

The physics enters through almost-commutative geometries $\mathscr{A}=C^\infty(M)\otimes\mathscr{A}_F$, where $\mathscr{A}_F=\mathbb{C}\oplus\mathbb{H}\oplus M_3(\mathbb{C})$ ($\mathbb{H}$ the quaternions) is a finite-dimensional algebra acting on a finite-dimensional Hilbert space $\mathscr{H}_F$ that carries the fermions of one Standard Model generation with their hypercharges; $D$ is the sum of the spacetime Dirac operator and a finite internal operator whose off-diagonal part is the Higgs field. Automorphisms of $\mathscr{A}$ generalize diffeomorphisms, and inner automorphisms $a\mapsto uau^*$ (conjugation by unitary elements) form a normal subgroup acting as the internal gauge transformations; because $\mathscr{A}$ is only slightly noncommutative, this inner subgroup is small. Connes proposed the spectral action $I=\mathrm{Trace}\,\varphi(ds/\ell_p)+\langle D\psi,\psi\rangle$ -- the trace of a cutoff function of the line element in Planck units, plus the fermionic bilinear -- as the pure gravitational action of such a geometry, and the companion papers with Chamseddine (Phys. Rev. Lett. 77 (1996) 4868; Commun. Math. Phys. 186 (1997) 731-750) carried out the deferred derivation: this action yields the Standard Model Lagrangian coupled to Einstein and Weyl gravity, with $SU(5)$-type relations among the gauge couplings at a fixed high energy scale. In the commutative case the axiom system was later proven to characterize compact spin Riemannian manifolds (the reconstruction theorem of Connes, J. Noncommut. Geom. 7 (2013) 1-82), so the axioms are rigid.

The paper's concluding section (pp. 27-29) leaves four issues open. First, finite-dimensional realization: the constraints from axioms 2 and 4 'do not have finite dimensional realizations since the Hochschild homology $H_4(\mathscr{A},\mathscr{A})$ vanishes for any finite dimensional algebra $\mathscr{A}$' -- a statement that is a theorem for the semisimple (separable) finite-dimensional algebras, such as $\mathscr{A}_F$, that are the natural candidates, the paper noting in the same connection that it is tempting to follow the approximation scheme of Grosse, Klimcik and Presnajder but that this vanishing of $H_4$ obstructs a finite-dimensional realization -- so constructing finite-dimensional approximations or a controlled regularization scheme for these constraints remains unresolved. Second, the quantum-group route suggested in the same passage: at a root of unity $q$ (that is, $q^m=1$), the quantized enveloping algebra of $SL(2)$ admits a finite-dimensional Hopf algebra quotient with a Frobenius-type map to the classical group, giving nontrivial finite-order quantum extensions $1\to H\to SU_q(2)\to SU(2)\to 1$ (for $q$ a root of unity of odd order), where the kernel $H$ is a finite quantum group (the source's notation, not to be confused with the quaternion algebra above) whose finite-dimensional Hopf algebra is closely related to $\mathscr{A}_F$ and whose spin representation defines a bimodule closely resembling $\mathscr{H}_F$; 'the details of the adaptation of these ideas to $Spin(4)=SU(2)\times SU(2)$ still remain to be elucidated'. Third, the hypoelliptic extension: an operator is hypoelliptic if $Pu=f$ with smooth $f$ forces $u$ to be smooth; on contact or Carnot manifolds (manifolds carrying a bracket-generating subbundle of the tangent bundle, such as the horizontal directions of the Heisenberg group) the natural differential operators are hypoelliptic but not elliptic, and the transverse index theory of foliations where this arises (the local index formula of Connes and Moscovici, Geom. Funct. Anal. 5 (1995) 174-243) exhibits higher-order commutator structures; the paper states that 'more involved relations of a quartic type are necessary to cover the hypoelliptic situation', but provides none. Fourth, the Planckian dynamics: the near-commutativity of $\mathscr{A}$ is expected to fade at energies $\Lambda\sim m_P$ (the Planck mass), so that the inner automorphisms come to exhaust more and more of the automorphism group and internal symmetries merge with spacetime symmetries; this discussion is qualitative only.

Problem Statement

Complete the axiomatic foundation of noncommutative geometry proposed in Connes (1996) -- the simple relations between the coordinate algebra $\mathscr{A}$ and the infinitesimal line element $ds=D^{-1}$, $D$ the Dirac operator -- along the four directions the paper's concluding section leaves open. The four requested results jointly serve this single objective: components 1 and 2 secure the finite, microscopic underpinning of the almost-commutative geometry of the Standard Model, component 3 secures the reach of the axioms beyond the elliptic case, and component 4 secures their dynamical behaviour at Planck scale.

1. (Finite-dimensional realization or controlled regularization of the constraint system.) The nonlinear constraints linking $\mathscr{A}$ and $D$ coming from axioms 2 and 4 -- the order-one condition $[[D,a],Jb^*J^{-1}]=0$ for all $a,b\in\mathscr{A}$, and the orientability condition requiring a Hochschild 4-cycle $c$ with $bc=0$ and $\pi(c)=\gamma$, $\gamma^2=1$, $\gamma D=-D\gamma$ -- do not have finite-dimensional realizations, since Hochschild homology $H_4(\mathscr{A},\mathscr{A})$ vanishes for finite-dimensional $\mathscr{A}$ (the source's statement; a theorem for the semisimple algebras that are the natural candidates, as explained in the background). Construct finite-dimensional approximations to this constraint system, or a controlled regularization scheme for it, in which the constraints are satisfied in a precise approximate or asymptotic sense suitable as an ultraviolet regularization of the spectral action; the solver must fix and justify the sense of approximation, stating which identities hold exactly, in which norm the asymptotic statement is made, and at what rate the constraints are approached.

2. (Quantum-group origin of the internal algebra.) Elucidate in detail the explanation of the finite algebra $\mathscr{A}_F=\mathbb{C}\oplus\mathbb{H}\oplus M_3(\mathbb{C})$ and its bimodule $\mathscr{H}_F$ from the theory of quantum groups at roots of unity: starting from the nontrivial finite-order quantum extensions $1\to H\to SU_q(2)\to SU(2)\to 1$ with $q$ a root of unity of odd order, where the kernel $H$ is a finite quantum group whose finite-dimensional Hopf algebra is closely related to $\mathscr{A}_F$ and whose spin representation defines a bimodule very similar in structure to $\mathscr{H}_F$, realize $\mathscr{A}_F$ and $\mathscr{H}_F$ (including the representation pattern of one Standard Model generation with its hypercharges) and carry out the adaptation of these ideas to $Spin(4)=SU(2)\times SU(2)$, whose details 'still remain to be elucidated'.

3. (Hypoelliptic extension of the axioms.) The relations of axioms 2 and 4 are tailored to elliptic (Dirac-type) situations. Formulate the more involved relations of a quartic type between $\mathscr{A}$ and $ds$ that are necessary to cover the hypoelliptic situations encountered in transverse index theory, and establish a corresponding axiom system: the elliptic case must be recovered as a special case, and genuinely hypoelliptic examples (on contact or Carnot manifolds, or in foliation theory) must satisfy the new relations with the analytic properties a spectral geometry requires.

4. (Quantitative Planckian dynamics.) The algebra $\mathscr{A}=C^\infty(M)\otimes\mathscr{A}_F$ is only slightly noncommutative, and this commutativity is expected to fade away at energies $\Lambda\sim m_P$ (the Planck mass), so that the inner automorphisms Int$(\mathscr{A})$ exhaust more and more of the automorphism group Aut$(\mathscr{A})$ and internal symmetries merge with spacetime symmetries. Give a quantitative description of this transition -- defining the mathematical object whose energy dependence is tracked and the dynamics governing it -- consistent with the almost-commutative Standard Model geometry and its spectral action at low energies, and derive its implications for the unification of internal and spacetime symmetries.

Scope note: the source paper also deferred the proof that the spectral action $\mathrm{Trace}\,\varphi(ds/\ell_p)+\langle D\psi,\psi\rangle$ reproduces the Standard Model Lagrangian coupled to gravity; that component was completed by Chamseddine and Connes (1996, 1997) and is therefore not part of this problem.

References

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