Connes and Moscovici (2014) computed, in closed form, the local curvature invariants of the curved noncommutative two-torus, a genuinely type-III (non-tracial) geometry whose volume weight φ(a)=φ0(ae−h)\varphi(a)=\varphi_0(ae^{-h}) is deformed by a noncommutative Weyl factor e−he^{-h} and whose curvature is governed by the Tomita-Takesaki modular operator. The Gaussian curvature of the functions-Laplacian is built from two explicit transcendental functions K0K_0 and H0H_0 acting through the modular derivation, while the isospectral Laplacian on (0,1)(0,1)-forms acquires an additional operator-valued skew term S(∇1,∇2)S(\nabla_1,\nabla_2) applied to a skew quadratic form; the authors also identified the gradient of the Ray-Singer log-determinant as the scalar curvature and defined the modular Ricci flow ∂th=Kφ\partial_t h = K_\varphi, the formal analogue of Hamilton's Ricci flow. The problem is the conceptual and analytic completion of this theory: give an intrinsic interpretation of the curvature constituents - above all the skew term, currently known only as the output of computer-assisted symbolic computation - thereby explaining the geometric origin of the asymmetry between the curvatures of the two isospectral partial Laplacians; establish existence, uniqueness, and long-time behavior for the modular Ricci flow on Aθ∞A_\theta^\infty; fix the intrinsic normalization of the scalar-curvature functional in terms of spectral data alone and determine the relationship between the two competing noncommutative Ricci-flow proposals; and extend the modular curvature calculus beyond the conformally deformed two-torus. Later literature resolved substantial parts of the extension question within toric θ\theta-deformed, conformally flat classes (Yang Liu; Connes-Fathizadeh; van Nuland-Sukochev-Zanin; Dong-Ghorbanpour-Khalkhali) and supplied genuine conceptual machinery for the functional relations (Lesch's divided differences; Yang Liu's hypergeometric closed forms and cyclic-category structure), but the intrinsic interpretation of the skew term, the analytic theory of the flow, the comparison of the flow proposals, and curvature beyond the toric conformally flat class remain open.

Background

In noncommutative geometry, a 'space' is described by an algebra of coordinates rather than by points, and a Riemannian-type metric is encoded by a spectral triple (A,H,D)(\mathcal{A},\mathcal{H},D): a representation of a ∗*-algebra A\mathcal{A} on a Hilbert space H\mathcal{H} together with an unbounded self-adjoint operator DD having bounded commutators with the algebra; the prototype is the Dirac operator on a compact spin manifold. The simplest genuinely noncommutative test-bed is the noncommutative two-torus AθA_\theta, the universal C∗C^*-algebra generated by unitaries U,VU, V satisfying VU=e2πiθUVVU = e^{2\pi i\theta}UV with θ\theta irrational. Its smooth subalgebra Aθ∞A_\theta^\infty consists of the series ∑amnUmVn\sum a_{mn}U^mV^n with rapidly decreasing coefficients; it carries two canonical commuting derivations δ1,δ2\delta_1, \delta_2 (the analogues of 1i∂/∂xj\frac{1}{i}\partial/\partial x_j) and a unique trace φ0\varphi_0 (the coefficient a0,0a_{0,0}). A translation-invariant complex structure is fixed by a parameter τ∈C\tau \in \mathbb{C}, ℑ(τ)>0\Im(\tau) > 0; the flat metric is encoded by a Dolbeault-type operator built from the derivations, and the total Laplacian splits, as on an ordinary complex torus, into a Laplacian on functions and a Laplacian Δφ(0,1)\Delta_\varphi^{(0,1)} on (0,1)(0,1)-forms (formal anti-holomorphic one-forms), the two being isospectral outside zero.

Deforming the flat metric by a noncommutative Weyl factor - a positive invertible element k=eh/2k = e^{h/2}, h=h∗∈Aθ∞h = h^* \in A_\theta^\infty, plays the role of the conformal factor - changes the natural volume weight to φ(a)=φ0(ae−h)\varphi(a) = \varphi_0(ae^{-h}), which is no longer a trace. By Tomita-Takesaki modular theory (the structure theory of non-tracial weights on operator algebras), the weight φ\varphi determines a modular operator Δφ\Delta_\varphi on the GNS Hilbert space L2(Aθ,φ)L^2(A_\theta,\varphi); its logarithm ∇=log⁡Δφ\nabla = \log\Delta_\varphi acts on AθA_\theta as the inner derivation a↦−[h,a]a \mapsto -[h,a]. In this curved situation the relevant spectral data form a twisted (or modular) spectral triple: the bounded commutators [D,a][D,a] of the classical definition are replaced by twisted commutators Da−σ(a)DDa - \sigma(a)D with σ(a)=kak−1\sigma(a) = kak^{-1} an inner automorphism. This is the first and paradigmatic example of curvature on a genuinely non-tracial ('type-III') noncommutative space.

Local curvature invariants are read off the small-time heat expansion: for a positive elliptic operator Δ\Delta of Laplace type, Tr(ae−tΔ)∼t→0∑qa2q(a,Δ)tq−1\mathrm{Tr}(ae^{-t\Delta}) \sim_{t\to 0} \sum_q a_{2q}(a,\Delta)t^{q-1}, and in dimension two the coefficient a2(a,Δ)a_2(a,\Delta) - equivalently the value at s=0s=0 of the zeta function ζΔ(a,s)=Tr(aΔ−s)\zeta_\Delta(a,s) = \mathrm{Tr}(a\Delta^{-s}) - is, classically, up to normalization the pairing of aa with the Gaussian curvature. After work initiated in the early 1990s (Cohen and Connes' conformal geometry of the irrational rotation algebra) showing that a non-tracial conformal weight gives the irrational rotation algebra a non-flat geometry, culminating in the Gauss-Bonnet theorem of Connes-Tretkoff and the independent scalar-curvature computation of Fathizadeh-Khalkhali, Connes and Moscovici computed this coefficient in full for the modular spectral triples of the curved noncommutative two-torus, by a symbolic pseudo-differential calculus governed by a 'Rearrangement Lemma' for functions of ∇\nabla, with computer assistance.

Their result for the Laplacian on functions has the form a2(a,Δφ)=c(τ) φ0(a K0(∇)(□(h))+H0(∇1,∇2)(Ωℜ(h)))a_2(a,\Delta_\varphi) = c(\tau)\,\varphi_0\big(a\,K_0(\nabla)(\Box(h)) + H_0(\nabla_1,\nabla_2)(\Omega_\Re(h))\big), with c(τ)c(\tau) an explicit elementary constant, □(h)=δ12(h)+2ℜ(τ)δ1δ2(h)+∣τ∣2δ22(h)\Box(h) = \delta_1^2(h) + 2\Re(\tau)\delta_1\delta_2(h) + |\tau|^2\delta_2^2(h) the noncommutative Laplacian of the dilaton, Ωℜ(ℓ)=(δ1ℓ)2+ℜ(τ)(δ1ℓ δ2ℓ+δ2ℓ δ1ℓ)+∣τ∣2(δ2ℓ)2\Omega_\Re(\ell) = (\delta_1\ell)^2 + \Re(\tau)(\delta_1\ell\,\delta_2\ell + \delta_2\ell\,\delta_1\ell) + |\tau|^2(\delta_2\ell)^2 a Dirichlet-type quadratic form, and ∇i\nabla_i signifying that ∇\nabla acts on the ii-th factor. The functions K0K_0 and H0H_0 are explicit transcendental functions of one and two variables respectively; up to a constant factor, the associated function K~0\tilde K_0 is the generating function of the Bernoulli numbers, K~0(u)∝∑n≥1B2n(2n)!u2n−2\tilde K_0(u) \propto \sum_{n\ge1}\frac{B_{2n}}{(2n)!}u^{2n-2}. For the Laplacian on (0,1)(0,1)-forms, the two-variable function is replaced by a simply related one, and a genuinely new term appears: the operator S(∇1,∇2)S(\nabla_1,\nabla_2) applied to the skew quadratic form Ω=(ℓ)=iℑ(τ)(δ1(ℓ)δ2(ℓ)−δ2(ℓ)δ1(ℓ))\Omega_=(\ell) = i\Im(\tau)(\delta_1(\ell)\delta_2(\ell) - \delta_2(\ell)\delta_1(\ell)), ℓ=2h\ell = 2h. The paper proves, variationally and a priori, a functional relation expressing H~0\tilde H_0 through divided differences of K~0\tilde K_0; a variational proof of Gauss-Bonnet; a closed formula for the Ray-Singer log-determinant of Δφ\Delta_\varphi (the modular analogue of Polyakov's conformal anomaly formula, involving the Dedekind eta function of τ\tau); and, by operator positivity of K~0\tilde K_0, the analogue of the Osgood-Phillips-Sarnak theorem: the scale-invariant determinant functional attains its extreme value only at the flat metric. Lesch-Moscovici later proved that the modular Gaussian curvature is invariant under the Morita equivalences of AθA_\theta implemented by Heisenberg bimodules - a Theorema-Egregium-type property.

Two classical facts motivate the flow. First, on a closed surface, Hamilton's Ricci flow ∂tg=−Ric\partial_t g = -\mathrm{Ric} reduces within a conformal class to an evolution of the log-conformal factor by (half) the Gaussian curvature; solutions exist for all time and every metric on the commutative two-torus flows to the flat metric. Second, the determinant of the Laplacian (analytic torsion, introduced by Ray and Singer) is extremal in a conformal class exactly at the constant-curvature metric, and its gradient is the curvature. Connes and Moscovici define the scalar curvature KφK_\varphi as the gradient of their log-determinant functional on the space of Weyl factors and derive the evolution equations ∂th=Kφ\partial_t h = K_\varphi for the conformal factor and ∂tgφ=−Rφ\partial_t g_\varphi = -R_\varphi for the associated Hermitian metric, observing that this is 'precisely the analogue of Hamilton's Ricci flow'. A different, spectral version of Ricci flow for the noncommutative two-torus - defined through the evolution of the Laplace eigenvalues - was proposed by Bhuyain and Marcolli.

The theory is thus computationally complete but conceptually and analytically incomplete: the curvature constituents, above all the skew term S(∇1,∇2)S(\nabla_1,\nabla_2), are known only as the output of the symbolic computation; the flow is defined but has no analytic theory; the normalization of the scalar curvature functional by spectral data alone is only partially carried out; and the whole calculus is specific to the conformally deformed two-torus. These are the gaps the problem addresses.

Problem Statement

The problem is the conceptual and analytic completion of the modular curvature theory of the noncommutative two-torus, as founded by Connes-Moscovici (JAMS 2014). It comprises four linked open directions, sharing one common objective: to replace computationally derived objects of the modular geometry of AθA_\theta - a curvature term, a formally defined flow, an ad hoc normalization, a two-torus-specific calculus - by intrinsically characterized and analytically controlled ones.

(i) Intrinsic interpretation of the modular curvature constituents, above all the skew term. The curvature of the (0,1)(0,1)-Laplacian Δφ(0,1)\Delta_\varphi^{(0,1)} contains, besides the expected analogue of the functions-Laplacian curvature, the operator S(∇1,∇2)S(\nabla_1,\nabla_2) applied to the skew quadratic form Ω=(ℓ)=iℑ(τ)(δ1(ℓ)δ2(ℓ)−δ2(ℓ)δ1(ℓ))\Omega_=(\ell) = i\Im(\tau)(\delta_1(\ell)\delta_2(\ell) - \delta_2(\ell)\delta_1(\ell)), ℓ=2h\ell = 2h, where ∇=log⁡Δφ\nabla = \log\Delta_\varphi is the modular derivation. Give a fully conceptual (intrinsic) account of this term: characterize it from structural principles of the modular spectral triple rather than from the symbolic computation, and thereby explain the geometric origin of the asymmetry between the curvatures of the two isospectral (outside zero) partial Laplacians. What is sought is a structural principle or invariant of the modular spectral triple that determines S(∇1,∇2)S(\nabla_1,\nabla_2) and accounts for that asymmetry, rather than a re-derivation of the computed expression. (In the source's words: 'It could be useful to find a fully conceptual understanding of the meaning of this term.')

(ii) Extension of the modular curvature calculus beyond the conformally deformed noncommutative two-torus, that is, to at least one of: a genuinely non-toric modular spectral triple; a metric that is not conformally flat; or a higher-dimensional modular spectral triple outside the toric conformally flat class. The toric θ\theta-deformed, conformally flat cases have been substantially treated in subsequent work (Yang Liu; Connes-Fathizadeh; van Nuland-Sukochev-Zanin; Dong-Ghorbanpour-Khalkhali), so the surviving targets are the genuinely non-toric or non-conformally-flat settings; the class of functional metrics is the class of metrics whose coefficients are smooth functions gij(h)g_{ij}(h) of a self-adjoint element hh (Ghorbanpour-Khalkhali [28]).

(iii) Analytic theory of the modular Ricci flow. For the flow equations ∂th=Kφ\partial_t h = K_\varphi on the conformal factor and ∂tgφ=−Rφ\partial_t g_\varphi = -R_\varphi on the associated Hermitian metric (Eqs. (122)-(124) of the source), where KφK_\varphi is the gradient of the scale-invariant Ray-Singer log-determinant functional: establish (or refute) well-posedness of the Cauchy problem for initial data h0=h0∗h_0 = h_0^* in Aθ∞A_\theta^\infty - existence, uniqueness, continuous dependence - and determine the long-time behavior, e.g. whether the scale-normalized flow converges to the flat metric for arbitrary initial data, as it does for Hamilton's Ricci flow on the commutative two-torus.

(iv) Intrinsic normalization and comparison of Ricci-flow proposals. Complete the normalization of the scalar-curvature functional in terms of the spectral data of the twisted spectral triple alone (the chiral normalization of the source's Section 5.2 is partial), and systematically compare the Connes-Moscovici gradient flow with the spectral (eigenvalue-based) Ricci flow of Bhuyain-Marcolli for the same geometry: determine under what conditions the two flows agree or differ.

Progress on these directions is expected to be linked: the intrinsic interpretation (i) should guide the flow analysis (iii) and the extensions (ii), and conversely. The four directions have independent standing.

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