Open problems in the Atiyah–Manton complex-geometry model of nuclei and atoms: metrics, energy functionals, quantization, neutrons, and dynamics

In 'Complex geometry of nuclei and atoms' (arXiv:1609.02816; Int. J. Mod. Phys. A 33 (2018) 1830022), M. F. Atiyah and N. S. Manton proposed modelling neutral atoms by compact, complex algebraic surfaces — real four-dimensional manifolds — with the proton number $P$ and neutron number $N$ of the nucleus read off from the surface's topological invariants: $P=(c_1^2+c_2)/12$, $N=(3c_2-c_1^2)/4$ in terms of the Chern numbers. Classification constraints on algebraic surfaces then bound $0\le N\le\min(9P,7P+6)$, encompassing all known isotopes, and identify the valley of stability $N=P$ with the line of zero signature. The model, however, is purely topological. The authors explicitly list the further work needed to make it a physical theory: (1) choosing a metric structure on the surfaces (self-dual metrics are too rigid; Kähler–Einstein metrics do not exist on all algebraic surfaces); (2) constructing an energy functional combining topological invariants and curvature integrals that reproduces ground-state binding energies, the neutron preference at large $P$, pairing, magic-number effects, and excitation spectra; (3) formulating quantum mechanics, via Laplace or Dirac operators on a fixed surface or by quantising the surface moduli, and deriving spins, isospin and energy gaps; (4) modelling single neutrons and neutron clusters (e.g. by class VII surfaces or $\overline{\mathbb{CP}^2}$-type geometry); and (5) modelling dynamical processes such as fission, alpha emission and molecular binding. All five problems remain open in their original generality.

Background

A long tradition in mathematical physics seeks to model matter by geometry rather than by point particles. In the 'geometric models of matter' program (Atiyah–Manton–Schroers 2012), particles are modelled by four-dimensional Riemannian manifolds: for example, the electron by the Taub–NUT space (a non-compact four-manifold with a non-contractible two-cycle, familiar as a gravitational instanton — a complete non-compact solution of the Euclidean-signature Einstein equations), the proton by the Atiyah–Hitchin manifold (the moduli space of two magnetic monopoles), and the neutron by the complex projective plane $\mathbb{CP}^2$ with its Fubini–Study metric. Conserved quantum numbers arise as topological invariants, and dynamics is carried by the manifold's moduli — the continuous parameters labelling its shape. A closely related inspiration is the Skyrme model, a nonlinear field theory whose three-dimensional solitons model nuclei with baryon number as a topological charge.

In Atiyah–Manton (2018), the authors extended this program to entire atoms. A complex algebraic surface is a two-dimensional complex manifold — a real four-dimensional space locally described by polynomial equations; examples include $\mathbb{CP}^2$ and products of algebraic curves. They proposed that a neutral atom (a nucleus of $P$ protons and $N$ neutrons, orbited by $P$ electrons) is modelled by a compact complex algebraic surface $X$ whose physical quantum numbers are determined by the Chern numbers $c_1^2$ (the self-intersection number of the canonical class, the cohomology class of the line bundle of holomorphic two-forms) and $c_2$ (equal to the Euler number $e$):

$P=\frac{1}{12}(c_1^2+c_2),\qquad N=\frac{1}{4}(3c_2-c_1^2),\qquad B=P+N.$

Equivalently $P=(e+\tau)/4$ and $N=(e-3\tau)/4$, where $\tau=b_2^+-b_2^-$ is the signature (the difference of the dimensions of the self-dual and anti-self-dual parts of second homology; the Betti numbers $b_2^\pm$ count independent two-cycles of positive and negative self-intersection). Thus hydrogen is $\mathbb{CP}^2$ ($P=1$, $N=0$, the proton being a two-cycle of self-intersection $+1$); the deuterium atom is minimally modelled by the quadric surface $\mathbb{CP}^1\times\mathbb{CP}^1$ ($P=N=1$), while the Hirzebruch surface obtained by blowing up one point of $\mathbb{CP}^2$ (a blow-up replaces a point by a $\mathbb{CP}^1$ of self-intersection $-1$; it adds one neutron, $N=1$) represents the same atom with its proton, neutron and electron separated rather than bound; and the K3 surface (the simply connected complex surface with trivial canonical class) sits at the neutron-rich extreme $(P,N)=(2,18)$.

The Enriques–Kodaira classification of compact complex surfaces (the analogue, for surfaces, of organising curves by genus) imposes constraints on the possible pairs $(c_1^2,c_2)$ of minimal surfaces (surfaces that are not themselves blow-ups); the most relevant class is that of surfaces of general type — the analogues, for surfaces, of algebraic curves of genus at least two — for which the Noether inequality and the Bogomolov–Miyaoka–Yau inequality $c_1^2\le 3c_2$ hold. In the physical dictionary these become the bounds $0\le N\le\min(9P,7P+6)$, which encompass all known isotopes, and the observation that the line $N=P$ — the centre of the valley of nuclear stability for light and medium nuclei — coincides with the line $\tau=0$ of zero signature. Further structure comes from the intersection form (the integral bilinear form on two-cycles): a form is even when every two-cycle has even self-intersection, and even forms can occur only when the neutron excess $N-P$ is a multiple of 8, a possible geometric account of certain anomalously stable neutron-rich isotopes. The nuclear landscape itself is organised by the valley of stability, by pairing (favouring even proton and neutron counts), and by magic numbers ($2, 8, 20, 28, 50,\dots$ — proton or neutron counts at which nuclei are especially stable, conventionally explained by the nuclear shell model).

The model, however, uses only the classification data of surfaces; it is purely topological. The authors close the paper by stating that 'much further work is needed to develop these ideas into a physical model of nuclei and atoms'. No metric has been chosen on the surfaces: self-dual metrics (metrics in four dimensions whose Weyl curvature tensor has vanishing anti-self-dual part; their complete non-compact examples are the gravitational instantons) are too rigid, since compact examples are rare, while Kähler–Einstein metrics — Hermitian metrics whose Ricci tensor is proportional to the metric — do not exist on all algebraic surfaces. No energy function exists: the authors suggest combining the topological term $\tau^2$ with the $L^2$ norms of the scalar and Ricci curvature (the latter constrained by LeBrun's inequalities relating curvature norms to $c_1^2$), calibrated against nuclear data. There is no formulation of quantum mechanics: candidates are spectral theory of the Laplacian or Dirac operator on a fixed surface, or quantization of the continuous moduli of surfaces by analogy with collective-coordinate quantization of solitons. There is no model of the single free neutron or of neutron clusters: candidates are the class VII surfaces (compact complex, non-Kähler surfaces with $b_1=1$, whose classification is itself an open problem in complex geometry) or surfaces built from two-cycles of self-intersection $-1$, such as the orientation-reversed projective plane $\overline{\mathbb{CP}^2}$. And there is no dynamics for processes such as nuclear fission, alpha emission, or the binding of atoms into molecules.

Problem Statement

The task is to promote the topological Atiyah–Manton model — neutral atoms as compact complex algebraic surfaces with $P=(c_1^2+c_2)/12$ and $N=(3c_2-c_1^2)/4$ — into a predictive physical theory of nuclei and atoms. The paper poses five linked problems that jointly serve this single objective:

(1) Metric. Decide what metric structure the surfaces need. Self-dual metrics (gravitational instantons) are too rigid, as compact examples are rare; Kähler–Einstein metrics do not exist for all algebraic surfaces. The metric structure the surfaces physically need is not fixed by the mere existence of some metric class — since every compact complex algebraic surface can be embedded in a complex projective space and thereby acquires a Kähler metric from the ambient Fubini–Study metric — and such a metric structure must exist on surfaces realizing the topological types in the physically allowed range $P\ge 1$, $0\le N\le\min(9P,7P+6)$.

(2) Energy. Construct a concrete energy functional on the space (moduli) of surfaces — combining topological invariants with non-topological curvature integrals, candidates being combinations of $\tau^2$ (with $\tau$ the signature of the surface) and the $L^2$ norms of scalar or Ricci curvature (constrained by LeBrun-type inequalities relating curvature norms to $c_1^2$) — whose minimization reproduces the detailed ground-state binding energies of nuclei and atoms, the preference for neutrons over protons in larger nuclei, pairing, magic-number effects, and discrete excitation spectra, and which accounts for the geometry of the valley of stability ($\tau=0$).

(3) Quantum mechanics. Formulate a quantum version of this geometric setting and derive from it the energies and spins of ground and excited states, isospin assignments (isospin being the internal nuclear symmetry under which the proton and the neutron are two states of one particle), and discrete energy gaps (expected to arise from discrete geometry changes such as replacing a blown-up surface by a minimal one, changing $b_1$ (the first Betti number) at fixed $(P,N)$, comparing different projective embeddings, or the even/odd choice of intersection form).

(4) Neutrons. Identify which complex surfaces model a single free neutron and clusters of neutrons — candidates include class VII surfaces (compact, complex, non-Kähler) and surfaces containing two-cycles of self-intersection $-1$ such as $\overline{\mathbb{CP}^2}$ (the complex projective plane with its orientation reversed) — including highly neutron-rich systems near the geometric boundary of the isotope chart, and relate these models to observed near-threshold resonances (dineutron, tetraneutron).

(5) Dynamics. Extend the static model to physical processes: the fission of larger nuclei, alpha emission, and the binding of atoms into molecules.

The five problems jointly serve a single objective: a consistent framework whose resulting theory predicts, rather than merely accommodates, the nuclear data enumerated above. Each of the five problems is posed independently above, and the stated objective is their joint one.

References

1. M. F. Atiyah and N. S. Manton, 'Complex geometry of nuclei and atoms', Int. J. Mod. Phys. A 33 (2018) 1830022. DOI: 10.1142/S0217751X18300223, https://doi.org/10.1142/S0217751X18300223 (preprint arXiv:1609.02816, https://arxiv.org/abs/1609.02816; republished in 'Topology and Physics', World Scientific, 2019, DOI: 10.1142/9789813278677_0001, https://doi.org/10.1142/9789813278677_0001)
2. M. Atiyah, N. S. Manton and B. J. Schroers, 'Geometric models of matter', Proc. R. Soc. Lond. A 468 (2012) 1252. DOI: 10.1098/rspa.2011.0616, https://doi.org/10.1098/rspa.2011.0616 (preprint arXiv:1108.5151, https://arxiv.org/abs/1108.5151)
3. M. F. Atiyah, 'Geometric models of Helium', Mod. Phys. Lett. A 32 (2017) 1750079. DOI: 10.1142/S0217732317500791, https://doi.org/10.1142/S0217732317500791 (preprint arXiv:1703.02532, https://arxiv.org/abs/1703.02532)
4. M. F. Atiyah and M. Marcolli, 'Anyons in Geometric Models of Matter', JHEP 07 (2017) 076. DOI: 10.1007/JHEP07(2017)076, https://doi.org/10.1007/JHEP07(2017)076 (preprint arXiv:1611.04047, https://arxiv.org/abs/1611.04047)
5. Michael Atiyah and Matilde Marcolli, 'Anyon Networks from Geometric Models of Matter', Q. J. Math. 72 (2021) 717-733. DOI: 10.1093/qmath/haab004, https://doi.org/10.1093/qmath/haab004
6. Nicholas S Manton, 'Skyrmions, Tetrahedra and Magic Numbers', Q. J. Math. 72 (2021) 735-753. DOI: 10.1093/qmathj/haaa025, https://doi.org/10.1093/qmathj/haaa025
7. B. J. Schroers, 'Michael Atiyah and Physics: the Later Years', arXiv:1910.10630 (2019), https://arxiv.org/abs/1910.10630
8. M. F. Atiyah and J. Malkoun, 'The Relativistic Geometry and Dynamics of Electrons', Found. Phys. 48 (2018) 199-208. DOI: 10.1007/s10701-018-0139-2, https://doi.org/10.1007/s10701-018-0139-2
9. Claude LeBrun, 'Ricci curvature, minimal volumes, and Seiberg-Witten theory', Invent. Math. 145 (2001) 279. DOI: 10.1007/s002220100148, https://doi.org/10.1007/s002220100148 (preprint arXiv:math/0003068, https://arxiv.org/abs/math/0003068)
10. H. M. Maridi, 'Global Trends of Nuclear Radii and Binding Energies from the NUCLEI-PACK Framework', arXiv:2509.13750 (2025). DOI: 10.48550/arXiv.2509.13750, https://arxiv.org/abs/2509.13750