Open problems of the geometric models of matter: energy functional, spin-$1/2$ states, refined particle models, and geometric particle processes

The 2011 proposal of Atiyah, Manton and Schroers models the stable particles — electron, proton, neutron, neutrino — by self-dual Einstein 4-manifolds (Taub-NUT, Atiyah-Hitchin, the Fubini-Study $\mathbb{CP}^2$, and $S^4$), with electric charge read off the asymptotic Kaluza-Klein circle fibration and baryon number from the signature. The paper deliberately left the framework static, topological and qualitative. The task is to complete the agenda the paper itself states: fixing an energy or action functional that makes quantitative contact with particle masses, binding energies and forces; constructing spin-$1/2$ states on the model manifolds; refining the tentative neutrino model, which does not distinguish neutrino from antineutrino, and the neutron model within the Hitchin family $M(k)$; giving a geometric account of beta decay via gluing and deformation of self-dual manifolds, obstructed by the mismatch of asymptotic fundamental groups between the proton and electron models; and quantifying the models' length scales against experiment. Follow-up work addresses only the leptonic components of this agenda; the central issues remain open.

Background

The Skyrme model, proposed by T. H. R. Skyrme in 1961, models atomic nuclei as solitons — stable, spatially localized finite-energy solutions of a nonlinear field theory — with the topological winding number of the configuration identified with the baryon number. Atiyah, Manton and Schroers (2011) proposed a more radical, purely Riemannian version of this idea: a static elementary particle is modelled by a complete Riemannian 4-manifold that is self-dual, meaning that the anti-self-dual part $W^-$ of its Weyl curvature tensor $W$ (the conformally invariant piece of the Riemann curvature, which in four dimensions splits as $W = W^+ \oplus W^-$) vanishes, making it a 4-dimensional analogue of a Riemann surface; the models are typically Einstein metrics. Equivalently, such manifolds carry twistor spaces in the sense of Penrose — auxiliary complex 3-manifolds fibred by 2-spheres over the underlying 4-manifold, which encode the self-duality of the metric. The main non-compact examples are gravitational instantons: complete hyperkähler or Einstein 4-manifolds, asymptotically locally flat (ALF), fibred at infinity by circles of bounded length over a 3-dimensional base. Two central examples are the Taub-NUT manifold (TN), the moduli space (the space of solutions modulo symmetry) of a single centred $SU(2)$ monopole (a static soliton of Yang-Mills-Higgs theory), asymptotic to $\mathbb{R}^3\times S^1$, and the Atiyah-Hitchin manifold (AH), the moduli space of two centred $SU(2)$ monopoles. Mimicking Kaluza-Klein theory (in which electromagnetism appears geometrically from a circle fibration added to a theory of gravity) but exchanging the roles of electric and magnetic fields, the authors interpret the asymptotic circle fibration electromagnetically: minus the first Chern class of the circle bundle at infinity is the electric charge, and the signature of the 4-manifold (the difference of the dimensions of self-dual and anti-self-dual harmonic 2-forms) is the baryon number. The programme assigns TN to the electron, AH to the proton, $\mathbb{CP}^2$ with its Fubini-Study metric (a compact self-dual Einstein manifold of positive scalar curvature) to the neutron, and the round 4-sphere to the neutrino. Harmonic 2-forms representing the non-trivial cohomology (for example the bounded self-dual form on AH associated with the core 2-sphere) play the role of electric fields; their exponential fall-off is reminiscent of the Yukawa pion field of the proton (the exponentially decaying meson field that mediates the residual nuclear force in Yukawa's description). The construction is fermionic in the sense that the rotation action lifts from $SO(3)$ to $SU(2)$ on spin bundles, but it stops short of producing quantum states. The paper's concluding section lists what is missing for the framework to become a physical theory. First, no action functional or definite energy measure is chosen — geometrically natural candidates such as $L^2$-norms of curvature or of harmonic forms exist, but none is fixed — so there is no quantitative contact with particle masses, forces or binding energies. Second, the spin-$1/2$ states have not been constructed; they are expected to arise from eigenspaces of the Dirac operator (the first-order operator acting on spinor fields, whose square is essentially the Laplacian), possibly twisted by a $U(1)$ connection whose curvature is proportional to one of the harmonic 2-forms, with $\mathrm{Spin}^c$ structures (spin structures relaxed by twisting with a line bundle, well suited to such Dirac twisting) and Seiberg-Witten theory (whose monopole equations, built from such twisted Dirac operators, give differential-topological invariants of 4-manifolds) possibly relevant. Third, the neutrino model $S^4$ is tentative and does not distinguish neutrino from antineutrino. Fourth, the $\mathbb{CP}^2$ neutron may be too naive: Hitchin's one-parameter family of self-dual Einstein metrics $M(k)$ — with $M(4)$ the Fubini-Study metric on $\mathbb{CP}^2\setminus\mathbb{RP}^2$, $M(k)$ for even $k\geq4$ completing to metrics on $\mathbb{CP}^2$ with a conical singularity of angle $4\pi/(k-2)$ along $\mathbb{RP}^2$, the family extending to real $k>0$ and converging to AH as $k\to\infty$ — offers more flexible possibilities, but the appropriate value of $k$ is not fixed. Fifth, particle processes such as beta decay ($n \to p + e^- + \bar\nu$) should be described by gluing and deformation of self-dual 4-manifolds, but the asymptotic fundamental groups of the proton model AH (base $\mathbb{RP}^2$ with $\pi_1 = \mathbb{Z}_2$) and the electron model TN (boundary $S^3$) differ, which obstructs a naive gluing picture and argues for a more sophisticated neutrino model. The paper also wonders whether the global $SU(3)$ symmetry of the Fubini-Study $\mathbb{CP}^2$ might give a link to quarks, but leaves this unexplored. Finally, the model length scales — the core radius of order $\pi$ in geometric units, the asymptotic circle radius 2, and the scale of the $\mathrm{e}^{-r}$ asymptotic corrections — are linked only qualitatively to the physical proton radius and the pion Compton wavelength.

Problem Statement

Within the geometric-models-of-matter framework of Atiyah, Manton and Schroers (2011), in which the electron, proton, neutron and neutrino are modelled respectively by the self-dual Einstein 4-manifolds Taub-NUT (TN), Atiyah-Hitchin (AH), the Fubini-Study $\mathbb{CP}^2$ and $S^4$, with electric charge given by minus the first Chern class of the asymptotic circle fibration and baryon number by the signature of the 4-manifold, complete the framework into a quantitative and dynamical theory of these particles. The source-faithful open question, as stated in the paper's own outlook, comprises the following objectives, each of which may be advanced independently: (1) choose a definite energy (or action) functional on the model 4-manifolds and show that it makes quantitative contact with particle masses, binding energies, and forces between particles; (2) construct explicitly the spin-$1/2$ states of the particles on the model manifolds; (3) refine the particle models where they are tentative or too simple: a neutrino model that distinguishes neutrino from antineutrino and can bridge the difference between the asymptotic fundamental groups of the proton model AH (base $\mathbb{RP}^2$, the real projective plane, with $\pi_1=\mathbb{Z}_2$) and the electron model TN (boundary $S^3$), and a neutron model based on the Hitchin family $M(k)$ with an appropriate choice of the (integer or real) parameter $k$; (4) give a geometric description of particle processes — in particular beta decay, and more generally fusion, fission and decays — in terms of gluing and deformation of self-dual 4-manifolds, including the required deformation theory and moduli spaces of self-dual manifolds; and (5) connect the geometric length scales of the models quantitatively to physical scales such as the proton radius and the pion Compton wavelength. These components share one scientific objective: to convert the static topological correspondence between self-dual 4-manifolds and stable particles into a model with quantitative, testable contact with particle properties.

References

1. M. F. Atiyah, N. S. Manton and B. J. Schroers, Geometric models of matter, Proceedings of the Royal Society A 468 (2012) 1252-1279, DOI 10.1098/rspa.2011.0616, arXiv:1108.5151, https://arxiv.org/abs/1108.5151
2. G. Franchetti and N. S. Manton, Gravitational instantons as models for charged particle systems, Journal of High Energy Physics 03 (2013) 072, DOI 10.1007/JHEP03(2013)072, arXiv:1301.1624, https://arxiv.org/abs/1301.1624
3. R. Jante and B. J. Schroers, Dirac operators on the Taub-NUT space, monopoles and SU(2) representations, Journal of High Energy Physics 01 (2014) 114, DOI 10.1007/JHEP01(2014)114, arXiv:1312.4879, https://arxiv.org/abs/1312.4879
4. G. Franchetti, Harmonic forms on ALF gravitational instantons, Journal of High Energy Physics 12 (2014) 075, DOI 10.1007/JHEP12(2014)075, arXiv:1410.2864, https://arxiv.org/abs/1410.2864
5. M. F. Atiyah, G. Franchetti and B. J. Schroers, Time evolution in a geometric model of a particle, Journal of High Energy Physics 02 (2015) 062, DOI 10.1007/JHEP02(2015)062, arXiv:1412.5915, https://arxiv.org/abs/1412.5915
6. R. Jante and B. J. Schroers, Taub-NUT dynamics with a magnetic field, Journal of Geometry and Physics 104 (2016) 305-328, DOI 10.1016/j.geomphys.2016.02.016, arXiv:1507.08165, https://arxiv.org/abs/1507.08165
7. R. Jante and B. J. Schroers, Spectral properties of Schwarzschild instantons, Classical and Quantum Gravity 33 (2016) 205008, DOI 10.1088/0264-9381/33/20/205008, arXiv:1604.06080, https://arxiv.org/abs/1604.06080
8. M. F. Atiyah and N. S. Manton, Complex geometry of nuclei and atoms, International Journal of Modern Physics A 33 (2018) 1830022, DOI 10.1142/s0217751x18300223, arXiv:1609.02816, https://arxiv.org/abs/1609.02816
9. M. F. Atiyah and M. Marcolli, Anyons in geometric models of matter, Journal of High Energy Physics 07 (2017) 076, DOI 10.1007/JHEP07(2017)076, arXiv:1611.04047, https://arxiv.org/abs/1611.04047
10. M. F. Atiyah, Geometric models of Helium, Modern Physics Letters A 32 (2017) 1750079, DOI 10.1142/s0217732317500791, arXiv:1703.02532, https://arxiv.org/abs/1703.02532
11. G. Franchetti, Harmonic spinors on a family of Einstein manifolds, Nonlinearity 31 (2018) 2419-2441, DOI 10.1088/1361-6544/aab0bd, arXiv:1705.02666, https://arxiv.org/abs/1705.02666
12. G. Franchetti, Harmonic forms and spinors on the Taub-bolt space, Journal of Geometry and Physics 141 (2019) 11-28, DOI 10.1016/j.geomphys.2019.03.005, arXiv:1812.07512, https://arxiv.org/abs/1812.07512
13. B. J. Schroers, Michael Atiyah and Physics: the Later Years, arXiv:1910.10630 (2019), https://arxiv.org/abs/1910.10630
14. B. J. Schroers and M. A. Singer, D_k gravitational instantons as superpositions of Atiyah-Hitchin and Taub-NUT geometries, The Quarterly Journal of Mathematics 72 (2021) 277-337, DOI 10.1093/qmath/haab002, arXiv:2004.02759, https://arxiv.org/abs/2004.02759
15. L. Boulton, B. J. Schroers and K. Smedley-Williams, Spectral geometry of nuts and bolts, Journal of Physics A: Mathematical and Theoretical 55 (2022) 235202, DOI 10.1088/1751-8121/ac6996, arXiv:2112.11411, https://arxiv.org/abs/2112.11411
16. G. Franchetti and K. Krasnov, Eguchi-Hanson harmonic spinors revisited, Journal of Geometry and Physics 217 (2025) 105640, DOI 10.1016/j.geomphys.2025.105640, arXiv:2309.08453, https://arxiv.org/abs/2309.08453
17. A. Garnier, Particle dynamics in spherically symmetric electro-vacuum instantons, The European Physical Journal C 84 (2024) 374, DOI 10.1140/epjc/s10052-024-12719-4, arXiv:2401.15809, https://arxiv.org/abs/2401.15809
18. X. Zhu, A gluing construction of D_k ALF gravitational instantons and existence of non-holomorphic minimal spheres, Proceedings of the American Mathematical Society 154 (2026) 2175-2187, DOI 10.1090/proc/17577, https://doi.org/10.1090/proc/17577