Mass Gap of the F₂ Faddeev-Skyrme Model
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2026
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| _version_ | 1866901502011899904 |
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| author | Novickis, Alexander |
| author_facet | Novickis, Alexander |
| contents | We prove that the Faddeev-Skyrme model with target the flag manifold \text{F}_2 = \mathrm{SU}(3)/[U(1) \times U(1)], specified by the Lagrangian \mathcal{L} = \kappa_2\,(\partial_\mu \hat n)^2 + \frac{1}{4}\kappa_4\,F_{\mu\nu} F^{\mu\nu} with F_{\mu\nu} = \hat n^* \omega_K the pull-back of the Kähler form, defines a well-posed Euclidean quantum field theory on \mathbb{R}^4 satisfying the Osterwalder-Schrader axioms, with a unique translation-invariant vacuum and a positive mass gap m_{\text{gap}}(\kappa_2, \kappa_4) > 0. The proof uses compact-target Bogomolny coercivity on a hypercubic lattice, a character-expansion / cluster-expansion case split on the \mathrm{SU}(3)-irreducible decomposition of L^2(\text{F}_2), and Glimm-Jaffe Osterwalder-Schrader reconstruction. The compact target obviates the gauge-fixing, infrared-regularisation, and rotational-symmetry-restoration obstructions that have historically blocked the analogous construction for non-Abelian Yang-Mills. As a corollary, via the Cho-Faddeev-Niemi decomposition (§6), the \mathrm{SU}(2) Yang-Mills mass-gap statement of the Clay Millennium Prize is the physics-level shadow of the present theorem: \text{F}_2-Faddeev-Skyrme is the programme-native field, and its mass gap is the programme-native formulation of the Clay question. Keywords: Faddeev-Skyrme model, flag manifold, \text{F}_2 = \mathrm{SU}(3)/[U(1)\times U(1)], Hopf soliton, Bogomolny bound, Berger metric, Laplace-Beltrami spectrum, Osterwalder-Schrader reconstruction, Cho-Faddeev-Niemi decomposition, mass gap, lattice gauge theory. MSC (2020): 81T13 (quantum field theory on lattices), 58J50 (spectral problems; spectral geometry), 81T25 (gauge theory on a lattice), 35Q75 (other equations from mathematical physics), 53C07 (special Riemannian geometry). |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19718769 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Mass Gap of the F₂ Faddeev-Skyrme Model Novickis, Alexander topological soliton Hopf fibration We prove that the Faddeev-Skyrme model with target the flag manifold \text{F}_2 = \mathrm{SU}(3)/[U(1) \times U(1)], specified by the Lagrangian \mathcal{L} = \kappa_2\,(\partial_\mu \hat n)^2 + \frac{1}{4}\kappa_4\,F_{\mu\nu} F^{\mu\nu} with F_{\mu\nu} = \hat n^* \omega_K the pull-back of the Kähler form, defines a well-posed Euclidean quantum field theory on \mathbb{R}^4 satisfying the Osterwalder-Schrader axioms, with a unique translation-invariant vacuum and a positive mass gap m_{\text{gap}}(\kappa_2, \kappa_4) > 0. The proof uses compact-target Bogomolny coercivity on a hypercubic lattice, a character-expansion / cluster-expansion case split on the \mathrm{SU}(3)-irreducible decomposition of L^2(\text{F}_2), and Glimm-Jaffe Osterwalder-Schrader reconstruction. The compact target obviates the gauge-fixing, infrared-regularisation, and rotational-symmetry-restoration obstructions that have historically blocked the analogous construction for non-Abelian Yang-Mills. As a corollary, via the Cho-Faddeev-Niemi decomposition (§6), the \mathrm{SU}(2) Yang-Mills mass-gap statement of the Clay Millennium Prize is the physics-level shadow of the present theorem: \text{F}_2-Faddeev-Skyrme is the programme-native field, and its mass gap is the programme-native formulation of the Clay question. Keywords: Faddeev-Skyrme model, flag manifold, \text{F}_2 = \mathrm{SU}(3)/[U(1)\times U(1)], Hopf soliton, Bogomolny bound, Berger metric, Laplace-Beltrami spectrum, Osterwalder-Schrader reconstruction, Cho-Faddeev-Niemi decomposition, mass gap, lattice gauge theory. MSC (2020): 81T13 (quantum field theory on lattices), 58J50 (spectral problems; spectral geometry), 81T25 (gauge theory on a lattice), 35Q75 (other equations from mathematical physics), 53C07 (special Riemannian geometry). |
| title | Mass Gap of the F₂ Faddeev-Skyrme Model |
| topic | topological soliton Hopf fibration |
| url | https://doi.org/10.5281/zenodo.19718769 |