Classification of spherical noncommutative manifolds beyond the $\theta$-deformation class and construction of their smooth and metric structures

Connes and Dubois-Violette defined an $n$-dimensional spherical noncommutative manifold by K-theoretic equations: its function algebra is generated by the matrix entries of a hermitian projection (for $n$ even) or a unitary (for $n$ odd) whose Connes-Chern character components vanish below the top degree, the top component being a nonzero Hochschild cycle that plays the role of a volume form. For $n=3$ this program is complete: the solutions form the three-parameter family $S^3_{\mathbf{u}}$ of deformed 3-spheres whose generic members are governed by 4-dimensional Sklyanin (elliptic) algebras, and the authors' Part II determined the moduli spaces, the classification with dualities and degenerations, the smooth algebra, the differential calculus via a ramified covering by a noncommutative nilmanifold, and the Jacobian. Outside this case almost nothing is known beyond the $\theta$-deformations, which exist in every dimension but all arise from non-formal torus actions. The open problem is to classify spherical noncommutative manifolds in dimensions $n\ge 4$ and beyond the $\theta$-class in every dimension, and to equip the non-toric classes -- including the generic, non-$\theta$ members already known in dimension 3 -- with the full structure of noncommutative geometry: smooth completion, differential calculus, and a spectral triple with Dirac operator, for which so far only an undeveloped one-sentence indication exists. Partial beyond-$\theta$ progress comes from R-matrix products of Euclidean spaces, which yield noncommutative 7-spheres, but with no classification and no metric structure.

Background

In noncommutative geometry a 'space' is studied through an algebra of functions: a compact smooth manifold $M$ is encoded by the involutive algebra $C^\infty(M)$ of smooth complex-valued functions, completed for topology to a $C^*$-algebra (a norm-closed algebra of bounded operators on a Hilbert space, closed under adjoints), and equipped with metric data by a spectral triple $(\mathcal{A},\mathcal{H},D)$: a dense involutive subalgebra $\mathcal{A}$ of a $C^*$-algebra, represented on a Hilbert space $\mathcal{H}$, together with a self-adjoint unbounded operator $D$ with compact resolvent and bounded commutators $[D,a]$, the noncommutative analogue of the Dirac operator of a spin manifold. The natural dimension notion for such algebras is Hochschild homology $HH_k(\mathcal{A})$, a homology theory for algebras built from chains $a_0\otimes\cdots\otimes a_k$; the Hochschild dimension of $\mathcal{A}$ is the largest $k$ with nonzero homology (with arbitrary coefficient bimodules). Complex K-theory, the generalized cohomology theory built from isomorphism classes of vector bundles, enters through the following construction.

Motivated by the polynomial equation of degree $n$ satisfied by the Dirac operator of an $n$-dimensional spin manifold, Connes and Dubois-Violette introduced spherical noncommutative manifolds by K-theoretic equations: the algebra $\mathcal{C}_{alg}(S)$ of 'polynomial functions' on an $n$-dimensional spherical noncommutative manifold $S$ is generated by the entries of a hermitian projection $e\in M_q(\mathcal{A})$ (a square matrix over the algebra with $e=e^*=e^2$; the case $n=2m$ even) or of a unitary $U\in M_q(\mathcal{A})$ (the case $n=2m+1$ odd), such that the components $ch_k$ of the Connes-Chern character -- explicit universal polynomials in the matrix entries with values in the cyclic homology of the algebra, a homology theory that in the commutative case recovers de Rham cohomology and plays the role of characteristic classes -- vanish, $ch_k(e)=0$ (resp. $ch_{k+1/2}(U)=0$) for $k=0,\ldots,m-1$, while the top component is a nonzero Hochschild cycle playing the role of the volume form. The classical Bott generators -- the explicit generators supplied by Bott periodicity, the fundamental periodicity phenomenon of topological K-theory -- of the K-theory of the ordinary sphere $S^n$ give commutative solutions, and genuinely noncommutative solutions first appear for $n=3$.

Part I of the authors' series obtained a complete description for $n=3$: a three-parameter family $S^3_{\mathbf{u}}$ of deformations of $S^3$, indexed by a parameter space $\Sigma$, with a compatible deformation $\mathbb{R}^4_{\mathbf{u}}$ of the polynomial algebra of $\mathbb{R}^4$. For generic parameter values the algebra of $\mathbb{R}^4_{\mathbf{u}}$ is isomorphic to a 4-dimensional Sklyanin algebra: a quadratic algebra (generated in degree one with only quadratic relations) attached to an elliptic curve (a nonsingular complex curve of genus one) through a solution of the Yang-Baxter equation (the consistency relation for the scattering matrix of a solvable lattice model), introduced by Sklyanin and developed by Odesskii and Feigin, with homological properties analyzed by Tate and Van den Bergh. Distinct spheres $S^3_{\mathbf{u}}$ can span the same $\mathbb{R}^4_{\mathbf{u}}$; the corresponding equivalence classes are the leaves of a scaling foliation of $\Sigma$, and the critical leaves are two one-parameter families $C_\pm$ where the Sklyanin isomorphism breaks down. These critical families are $\theta$-deformations: deformations of the type of the noncommutative torus (the algebra generated by two unitaries with $UV=e^{2\pi i\theta}VU$), produced by Rieffel's deformation quantization -- a construction that deforms the product of two functions by combining their Fourier decompositions under the group action -- applied to a non-formal, i.e. genuinely topological rather than infinitesimal, action of the torus $\mathbb{T}^n$; applied to spheres this gives noncommutative spheres $S^m_\theta$ in every dimension which are spherical manifolds, closely related to the isospectral deformations of Connes and Landi (deformations preserving the spectrum of the Dirac operator). The authors stress that the method essentially requires torus actions: their results 'would be generically wrong for actions of $\mathbb{R}^n$'.

Part I deferred the generic spheres to Part II: 'our goal in part II will be to analyse general spherical 3-manifolds including their smooth structure, their differential calculus and metric aspect.' Part II, announced in 2003 and published in 2008, delivered most of this in dimension 3: it identified the real moduli space (through the space of unitary symmetric matrices and the alcove structure of the root system $D_3$, a root system being a finite configuration of vectors in Euclidean space whose associated hyperplane reflections generate a finite reflection group) and the complex moduli space (a net -- a two-parameter linear system -- of elliptic curves in complex projective three-space, described through the orbits of a birational cubic automorphism), proved that the isomorphism classes of the algebras $\mathbb{R}^4_{\mathbf{u}}$ are exactly the leaves of the scaling foliation, classified the algebras together with their degenerations and dualities, constructed the smooth algebra $C^\infty(S^3_{\mathbf{u}})$ as the closure of the polynomial algebra under holomorphic functional calculus (the operation $a\mapsto f(a)$ for holomorphic $f$, taken inside the $C^*$-completion), and built the differential calculus through a ramified covering of the generic spheres by a noncommutative 3-dimensional nilmanifold (a quotient of a nilpotent Lie group by a lattice, here in noncommutative form), computing the Jacobian -- the determinant of the tangent map -- of the covering as an elliptic period times a rational function. The metric aspect, however, is only indicated: a single remark states that combining the ramified cover with the natural spectral triple on the nilmanifold 'yields a natural spectral triple on $S^3_{\mathbf{u}}$ in the generic case', without developing the construction or verifying any axioms. Part II's introduction is explicit about what remains: 'we believe that a lot remains to be discovered for the abstract construction of the calculus as well as for the general computation of the Jacobian, let alone in the case of higher dimensional spheres which we do not adress here.' Thus, after the authors' own two-part work, the classification of spherical noncommutative manifolds is open outside dimension 3 and outside the $\theta$-class, and the full noncommutative-geometric structure of the non-toric classes is open even in dimension 3.

Problem Statement

Determine the spherical noncommutative manifolds beyond the classes already constructed, and endow them with the full structure of noncommutative geometry. The defining equations are those of Connes-Dubois-Violette: for each dimension $n$, the algebra $\mathcal{A}$ is generated by the entries of a hermitian projection $e\in M_q(\mathcal{A})$ when $n=2m$ is even, or of a unitary $U\in M_q(\mathcal{A})$ when $n=2m+1$ is odd, satisfying $ch_k(e)=0$ (resp. $ch_{k+1/2}(U)=0$) for $k=0,\ldots,m-1$, the top component $ch_m(e)$ (resp. $ch_{m+1/2}(U)$) being a nonzero Hochschild cycle playing the role of the volume form, and the Hochschild dimension being $n$. The classes already known are the commutative spheres, the $\theta$-deformations $S^m_\theta$ available in all dimensions via non-formal torus actions, and the three-parameter family $S^3_{\mathbf{u}}$ with its degenerations in dimension 3. The problem asks:

(1) Classification: for dimensions $n\ge 4$, decide whether spherical noncommutative manifolds exist beyond the commutative sphere and the $\theta$-deformations, and classify them -- enumerate the isomorphism and deformation classes, describe the moduli space, and prove completeness of the enumeration up to the natural equivalences -- change of basis in $M_q(\mathcal{A})$ and rescaling of generators (the identification of spheres spanning the same deformed ambient space $\mathbb{R}^4_{\mathbf{u}}$, i.e. the leaves of the scaling foliation, was the operative equivalence in the $n=3$ classification) -- or prove that no further classes exist.

(2) Geometric completion: for the non-toric classes -- the generic (Sklyanin-type) members of the known families, including the generic $S^3_{\mathbf{u}}$ for which only a one-sentence indication exists, and any newly classified manifolds -- construct the smooth algebra stable under holomorphic functional calculus in the $C^*$-completion, the differential calculus, and a spectral triple $(\mathcal{A},\mathcal{H},D)$ whose Dirac operator realizes the metric geometry, verifying the axioms of noncommutative geometry and the compatibility of the fundamental class (the volume-form class in cyclic cohomology carried by the spectral triple) with the K-theoretic volume cycle.

(3) General theory: develop the abstract construction of the differential calculus and the computation of the Jacobian for involutive quadratic algebras via central quadratic forms -- a central quadratic form being a central element of degree two in the algebra, on which Part II builds the calculus (Sections 11--13) -- the theory that Part II explicitly declares incomplete and on which any systematic higher-dimensional classification depends.

The scientific target comprises (1)--(3): it extends the $n=3$ classification theorem to higher dimensions, completes, for the non-$\theta$ examples, the path 'from the algebra of polynomials to the full-fledged structure of noncommutative geometry' that Part I completed only for the $\theta$-class, and develops the general theory of the differential calculus and the Jacobian for involutive quadratic algebras.

References

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