The Mass Gap of F₂ Faddeev-Skyrme Theory
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2026
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| author | Novickis, Alexander |
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| contents | <p><strong>Title:</strong> The Mass Gap of F$_2$ Faddeev-Skyrme Theory: A Constructive Proof on the F$_2$-Direct Lattice</p> <p><strong>Author:</strong> Alexander Novickis (alex.novickis@gmail.com)</p> <p>We prove the existence of a positive mass gap for four-dimensional F$_2$ Faddeev-Skyrme theory on the F$_2$-direct lattice, where $F_2 = SU(3)/(U(1) \times U(1))$ is the complete flag manifold of $\mathbb{C}^3$. The proof uses a single-scale cluster expansion over the F$_2$-direct action -- programme-native and free of any underlying $SU(N)$ Wilson formulation -- combining a Brydges-Yau gradient-field bound, a Kotecky-Preiss polymer-counting threshold, and a Brydges-Kennedy tree-graph identity, with the Plücker-bound spectral chain (Lemma 4.5) controlling the F$_2$ Cartan-Killing inner product across links. <strong>Theorem 1 (lattice):</strong> for $\beta \geq \beta_{\rm cluster} = 138$ the branched 6-point correlator on $\mathbb{Z}^4$ has positive mass gap $\Delta_{\rm latt} \geq 0.0516$. <strong>Theorem 2 (continuum):</strong> for $\beta(a)$ following the F$_2$-direct asymptotic-freedom rate, the continuum branched correlator exists, is reflection-positive and sequence-independent, with $m_{\rm gap}^{\rm cont} \geq c_*^{\rm IR} \cdot \Lambda_{\rm phys}^{F_2} > 0$. A new framework-relating renormalisation factor $Z^{F_2 \leftrightarrow {\rm CFN}}$ (existence, finiteness, positivity, $a$-independence sub-lemmas) connects $\Lambda_{\rm phys}^{F_2}$ to the CIII v3 $SU(2)$ Yang-Mills mass gap (companion paper) without requiring its explicit value. The two papers together provide multi-path independent verification of the mass-gap claim. Forward work (§8): all-$n$-point extension; full Osterwalder-Schrader reconstruction with Euclidean $SO(4)$ covariance restoration; F$_2$-direct to standard $SU(N)$ Yang-Mills bridge.</p> <p><strong>Keywords:</strong> mathematical physics, mass gap, F2 flag manifold, Faddeev-Skyrme, Faddeev-Niemi, F2-direct lattice, cluster expansion, Brydges-Yau, Brydges-Kennedy, Kotecky-Preiss, Plücker bound, branched correlator, six-point correlator, lattice gauge theory, constructive QFT, reflection positivity, Osterwalder-Schrader reconstruction, Cho-Faddeev-Niemi decomposition, CFN decomposition, SU(2) Yang-Mills, Yang-Mills, asymptotic freedom, Callan-Symanzik equation, dimensional transmutation, single-scale cluster expansion, polymer activity, tree-graph inequality, Aizenman-Newman, Glimm-Jaffe, Hopf soliton, Hopf fibration, topological soliton, sigma model, flag manifold, SU(3)/(U(1)xU(1)), Hopf Soliton Programme, Bałaban programme, millennium prize, Clay Millennium Prize</p> <p><strong>Series:</strong> Paper CXL in the Hopf Soliton Programme (companion to Paper CIII v3, DOI 10.5281/zenodo.19802204)</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19816726 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Mass Gap of F₂ Faddeev-Skyrme Theory Novickis, Alexander topological soliton Hopf fibration <p><strong>Title:</strong> The Mass Gap of F$_2$ Faddeev-Skyrme Theory: A Constructive Proof on the F$_2$-Direct Lattice</p> <p><strong>Author:</strong> Alexander Novickis (alex.novickis@gmail.com)</p> <p>We prove the existence of a positive mass gap for four-dimensional F$_2$ Faddeev-Skyrme theory on the F$_2$-direct lattice, where $F_2 = SU(3)/(U(1) \times U(1))$ is the complete flag manifold of $\mathbb{C}^3$. The proof uses a single-scale cluster expansion over the F$_2$-direct action -- programme-native and free of any underlying $SU(N)$ Wilson formulation -- combining a Brydges-Yau gradient-field bound, a Kotecky-Preiss polymer-counting threshold, and a Brydges-Kennedy tree-graph identity, with the Plücker-bound spectral chain (Lemma 4.5) controlling the F$_2$ Cartan-Killing inner product across links. <strong>Theorem 1 (lattice):</strong> for $\beta \geq \beta_{\rm cluster} = 138$ the branched 6-point correlator on $\mathbb{Z}^4$ has positive mass gap $\Delta_{\rm latt} \geq 0.0516$. <strong>Theorem 2 (continuum):</strong> for $\beta(a)$ following the F$_2$-direct asymptotic-freedom rate, the continuum branched correlator exists, is reflection-positive and sequence-independent, with $m_{\rm gap}^{\rm cont} \geq c_*^{\rm IR} \cdot \Lambda_{\rm phys}^{F_2} > 0$. A new framework-relating renormalisation factor $Z^{F_2 \leftrightarrow {\rm CFN}}$ (existence, finiteness, positivity, $a$-independence sub-lemmas) connects $\Lambda_{\rm phys}^{F_2}$ to the CIII v3 $SU(2)$ Yang-Mills mass gap (companion paper) without requiring its explicit value. The two papers together provide multi-path independent verification of the mass-gap claim. Forward work (§8): all-$n$-point extension; full Osterwalder-Schrader reconstruction with Euclidean $SO(4)$ covariance restoration; F$_2$-direct to standard $SU(N)$ Yang-Mills bridge.</p> <p><strong>Keywords:</strong> mathematical physics, mass gap, F2 flag manifold, Faddeev-Skyrme, Faddeev-Niemi, F2-direct lattice, cluster expansion, Brydges-Yau, Brydges-Kennedy, Kotecky-Preiss, Plücker bound, branched correlator, six-point correlator, lattice gauge theory, constructive QFT, reflection positivity, Osterwalder-Schrader reconstruction, Cho-Faddeev-Niemi decomposition, CFN decomposition, SU(2) Yang-Mills, Yang-Mills, asymptotic freedom, Callan-Symanzik equation, dimensional transmutation, single-scale cluster expansion, polymer activity, tree-graph inequality, Aizenman-Newman, Glimm-Jaffe, Hopf soliton, Hopf fibration, topological soliton, sigma model, flag manifold, SU(3)/(U(1)xU(1)), Hopf Soliton Programme, Bałaban programme, millennium prize, Clay Millennium Prize</p> <p><strong>Series:</strong> Paper CXL in the Hopf Soliton Programme (companion to Paper CIII v3, DOI 10.5281/zenodo.19802204)</p> |
| title | The Mass Gap of F₂ Faddeev-Skyrme Theory |
| topic | topological soliton Hopf fibration |
| url | https://doi.org/10.5281/zenodo.19816726 |