BPS Structure and Fixed-Point Theorem for the Density-Feedback Faddeev--Niemi Hopfion

The companion paper establishes numerically that the Euler--Lagrange (EL) minimum of the density-feedback Faddeev--Niemi energy $E_{\mathrm{fb}} = K_{\mathrm{fb}}\cdot J_4$ satisfies $K_{\mathrm{fb}}/J_4 = \lambda/2^{1/3}$ to within $O(h^2)$ discretisation error ($+0.020\%$ on a $256\times256$ lattice), where $\lambda=\varphi^6$ and $\varphi$ is the golden ratio. Here we address this analytically in three steps.

We introduce the generalised functional $\mathcal{G}[f;\lambda_{\mathrm{eff}}] = K_{\mathrm{fb}}[f] + \lambda_{\mathrm{eff}}\,J_4[f]$ and define the ratio map
$H(\lambda_{\mathrm{eff}}) := K_{\mathrm{fb}}[f_{\lambda_{\mathrm{eff}}}]/J_4[f_{\lambda_{\mathrm{eff}}}]$ where $f_{\lambda_{\mathrm{eff}}}$ is the critical point of $\mathcal{G}[\,\cdot\,;\lambda_{\mathrm{eff}}]$ in the $Q=2$ topological sector.

Main results
1.(Existence) For each $\lambda_{\mathrm{eff}}>0$, $\mathcal{G}[\,\cdot\,;\lambda_{\mathrm{eff}}]$ has a critical   point $f_{\lambda_{\mathrm{eff}}}$ in the $Q=2$ sector, via a mountain-pass argument in   the energy space $X_2$, subject to   a concentration-compactness condition.

2.(Monotonicity and uniqueness) $H$ is strictly decreasing  with $H(0^+)>0$ and $H(\lambda_{\mathrm{eff}})-\lambda_{\mathrm{eff}}\to-\infty$, so the fixed-point equation $H(\lambda_{\mathrm{eff}}^*)=\lambda_{\mathrm{eff}}^*$ has exactly one solution, subject to a non-degeneracy condition.

3.(Fixed-point value --- thin-torus) In the thin-torus  approximation, restricting to the Battye--Sutcliffe ansatz family  $f=2\arctan(C/d)$, we prove exactly by Beta-function integration  that $I_4/I_{2a} = 3/(4C^2)$, from which the self-consistency  gives $I_4/I_{2a}\big|_{C=C^*} = 2^{4/3}/\varphi^5$. For the full 2D EL equation, the value $\lambda_{\mathrm{eff}}^*=\lambda/2^{1/3}$ is the Linking Scale Conjecture :  $s_{\mathrm{opt}}^{2k} = N_{\mathrm{fus}} = 2$, where  $N_{\mathrm{fus}}$ is the $\widehat{\mathfrak{su}(2)}_k$ WZW fusion multiplicity. Three WZW identities at $k=3$ motivate the conjecture,  whose full proof requires a modular Ward identity  connecting the PDE virial condition to the WZW $S$-matrix.

Steps~(i) and~(ii) are proved subject to standard functional-analytic conditions. Step~(iii) is proved in the thin-torus model and conjectured in full generality; it is  confirmed numerically to $0.020\%$ at $R_0=3$ and consistent with a three-point $R_0\in\{3,4,5\}$ dataset.