The one-loop potential along the abelian valley of the torus is periodic, reproduces C133’s linear term at the origin, and saturates at order 1/L
For \(SU(2)\) Yang–Mills on \(T^3_L\) with a constant abelian background \(a=a_i^{\,3}T^3\), the zero-point energy of all modes, relative to \(a=0\), is the exact periodic function \[U(a)=\frac{2\hbar c}{\pi^2L}\,\Phi(La),\qquad \Phi(b)=\sum_{m\in\mathbb Z^3\setminus\{0\}}\frac{1-\cos(m\cdot b)}{|m|^4}\ \ge0,\] with \(a\) of dimension inverse length (\([A]=L^{-1}\) in the convention of G07 Proposition 7), obtained by Poisson summation of the charged frequencies \(\hbar c|k\pm a|\) over \(k\in(2\pi/L)\mathbb Z^3\); the divergent part of the sum is \(a\)-independent under any periodic regularization and drops out. Near the origin \(\Phi(b)=\pi^2|b|+\tfrac16C_M|b|^2+O(|b|^3)\) with \(C_M=\lim_{b\to0}[\sum'_me^{im\cdot b}|m|^{-2}-2\pi^2/|b|]\), so \(U(a)=2\hbar c|a|+\tfrac{C_M}{3\pi^2}\hbar cL|a|^2+\cdots\). The linear term \(2\hbar c|a|\) is exactly the \(k=0\) contribution, four oscillators of angular frequency \(c|a|\) with zero-point energy \(\tfrac12\hbar c|a|\) each, which are the transverse oscillators of C133 along its abelian valley: the field-theory computation reproduces the mechanism of G07 and G08 term by term. The nonzero modes contribute the correction \(U_\perp=U-2\hbar c|a|=O(\hbar cL|a|^2)\), of order \(g^{4/3}\hbar c/L\) in the rescaled variables of the Feshbach note, which is the input (H3) needs, with the sign of \(C_M\) (negative for the cubic lattice) deciding that the correction lowers the valley potential. Globally \(U\) is bounded by \(2\hbar c\max\Phi/(\pi^2L)=O(\hbar c/L)\) and vanishes exactly at \(La\in2\pi\mathbb Z^3\), the holonomies that are central in the fundamental representation: the confinement of the constant modes along the valley is linear only for \(La\ll1\), saturates at order \(\hbar c/L\), and has degenerate minima at the centre elements, which is the small-volume origin of the electric-flux sectors. Since the zero-mode gap is \(g^{2/3}\hbar c/L\ll\hbar c/L\), the ground state is localized at \(La\sim g^{2/3}\) and never reaches the saturation; the crossover \(z\simeq2\) is where it does. Sources: this is the spatial-torus form of Weiss’s finite-temperature potential (Phys. Rev. D 24 (1981) 475, metadata), and it appears in the small-volume literature (Lüscher 1983, abstract); the derivation below is self-contained. Nothing here is promoted.
1. Frequencies in a constant abelian background
Let \(a_i=\alpha_iT^3\), \(i=1,2,3\), with \(\vec\alpha\in\mathbb R^3\), and write \(a\) for \(\vec\alpha\) when no confusion arises. Decompose the adjoint fluctuation \(\tilde A_i=\tilde A_i^3T^3+\tilde A_i^+T^++\tilde A_i^-T^-\). The covariant derivative \(D_j=\partial_j+[a_j,\cdot]\) acts on the neutral component as \(\partial_j\) and on the charged components as \(\partial_j\pm i\alpha_j\), so on the Fourier mode \(e^{ik\cdot x}\), \(k\in(2\pi/L)\mathbb Z^3\), the charged components have covariant momentum \(k\pm a\). The quadratic form \(\frac1{2g^2}\int|D_i\tilde A_j-D_j\tilde A_i|^2\) is, for a constant abelian background, the free transverse form with \(k\) replaced by \(k\pm a\) on the charged components, so after the usual transverse gauge choice the frequencies are \[\hbar\omega^{(0)}_k=\hbar c|k|\quad(\text{neutral, two polarizations}),\] \[\hbar\omega^{(\pm)}_k=\hbar c|k\pm a|\quad(\text{charged, two polarizations each}),\] for \(k\ne0\). At \(k=0\) the neutral components are the valley directions themselves and the charged components of \(\tilde A_j\) have frequency \(|a|\) for each of the two charges and, since there is no transversality at \(k=0\), for the components of \(\tilde A_j\) orthogonal to the valley in field space: with \(a=\alpha_1T^3\) along \(x_1\) only, these are the charged parts of \(\tilde A_2,\tilde A_3\), four real oscillators of angular frequency \(c|a|\), while the charged parts of \(\tilde A_1\) are gauge directions of frequency zero removed by Gauss’s law.
The \(k=0\) term is C133. The SU(2) matrix model of C133 with \(\vec x_1=\lambda\hat n\), \(\vec x_2=\vec x_3=0\) has potential \(\tfrac12g^2(|\vec x_1\times\vec x_2|^2+|\vec x_1\times\vec x_3|^2+\cdots)\), whose transverse Hessian at that point gives the components of \(\vec x_2,\vec x_3\) orthogonal to \(\hat n\), four oscillators of frequency \(g|\lambda|/\sqrt m\); in the field-theory variables of G07 Proposition 7 (\(m=\hbar L^3/(g^2c)\), \(g_B^2=\hbar cL^3/g^2\)) this frequency is \(c|a|\). The zero-point energy \(4\cdot\tfrac12\hbar c|a|=2\hbar c|a|\) is the confining term of G07 Theorem 6(1) and the floor of G08 Theorem 2 along the valley.
2. Poisson summation
The zero-point energy of the fluctuations relative to \(a=0\), summing two polarizations and both charges, is \[U(a)=\frac{\hbar c}2\sum_{k\in(2\pi/L)\mathbb Z^3}\sum_{\rm pol=1,2}\big(|k+a|+|k-a|-2|k|\big) =\hbar c\sum_{k\in(2\pi/L)\mathbb Z^3}\big(|k+a|+|k-a|-2|k|\big),\] where the \(k=0\) term is \(2\hbar c|a|\) and the neutral modes cancel. Each term is nonnegative by convexity of \(|\cdot|\) and of order \(|a|^2/|k|\), so the sum diverges in three dimensions; under a periodic regularization (lattice dispersion \(\omega_{\rm lat}\) with the exact momentum grid) the divergent part is \(a\)-independent, because \(\sum_{k\in\text{grid}}\omega_{\rm lat}(k+a)\) is a periodic function of \(a\) with period \(2\pi/L\) whose mean over a period is the \(a\)-independent Brillouin-zone integral. The \(a\)-dependent part is obtained by Poisson summation. For \(f(k)=|k|\) the distributional Fourier transform is \(\hat f(x)=\int|k|e^{-ik\cdot x}d^3k=-8\pi/|x|^4\) for \(x\ne0\) (the formula \(\widehat{|k|^\alpha}=2^{\alpha+3}\pi^{3/2}\frac{\Gamma((3+\alpha)/2)}{\Gamma(-\alpha/2)}|x|^{-3-\alpha}\) at \(\alpha=1\), with \(\Gamma(-\tfrac12)=-2\sqrt\pi\)), and \[\sum_{k\in(2\pi/L)\mathbb Z^3}f(k+a)=\Big(\frac L{2\pi}\Big)^3\sum_{m\in\mathbb Z^3}\hat f(mL)\,e^{im\cdot La}\] with the \(m=0\) term the divergent, \(a\)-independent one. Hence \[U(a)=\hbar c\Big(\frac L{2\pi}\Big)^3\sum_{m\ne0}\Big(-\frac{8\pi}{L^4|m|^4}\Big)\big(e^{im\cdot La}+e^{-im\cdot La}-2\big) =\frac{2\hbar c}{\pi^2L}\sum_{m\ne0}\frac{1-\cos(m\cdot La)}{|m|^4}=\frac{2\hbar c}{\pi^2L}\Phi(La).\] Lattice corrections change \(\hat f\) only at \(|x|\lesssim a_{\rm lat}\), while the \(m\ne0\) terms live at \(|x|=|m|L\ge L\), so they are \(O(a_{\rm lat}^2/L^2)\) relative.
Properties of \(\Phi\): it is nonnegative, periodic with period \(2\pi\) in each component, invariant under the cubic group, bounded by \(2\sum'|m|^{-4}\), and zero exactly on \(2\pi\mathbb Z^3\).
3. Expansion at the origin and the sign of the correction
\(\Phi\) is not smooth at \(0\). Its Laplacian is the periodic Green’s function \(G(b)=\sum'_me^{im\cdot b}|m|^{-2}\), which has the Coulomb singularity \(2\pi^2/|b|\) at the origin (from \(\int d^3m\,e^{im\cdot b}|m|^{-2}\)) and a finite remainder: \(G(b)=2\pi^2/|b|+C_M+O(|b|^2)\), where \(C_M\) is the constant term of the periodic Coulomb Green’s function of the cubic lattice, the analytically continued Epstein sum \(\sum'_m|m|^{-2}\). Since \(\Delta(\pi^2|b|)=2\pi^2/|b|\), the function \(\Phi-\pi^2|b|\) has Laplacian \(C_M+O(|b|^2)\) near \(0\), and the only quadratic form with cubic symmetry and Laplacian \(C_M\) is \(\tfrac16C_M|b|^2\); so \[\Phi(b)=\pi^2|b|+\tfrac16C_M|b|^2+O(|b|^3),\qquad U(a)=2\hbar c|a|+\frac{C_M}{3\pi^2}\,\hbar cL|a|^2+O(\hbar cL^2|a|^3).\] The linear term \(2\hbar c|a|\) is the \(k=0\) contribution of Section 1, so the nonzero modes contribute \[U_\perp(a)=U(a)-2\hbar c|a|=\frac{C_M}{3\pi^2}\hbar cL|a|^2+O(\hbar cL^2|a|^3),\] with \(C_M<0\) for the cubic lattice (lattice-sum tables; the sign is not re-derived here and affects only the direction of a subleading correction). In the rescaled variables \(a=g^{2/3}\xi/L\) this is \(\frac{C_M}{3\pi^2}g^{4/3}|\xi|^2\,\hbar c/L\), of relative order \(g^{2/3}\) to the zero-mode Hamiltonian \(g^{2/3}h_3\,\hbar c/L\): the size that (H3) of the Feshbach reduction assumes, now with its coefficient identified.
3a. General constant backgrounds at quadratic order
The abelian computation fixes the quadratic term of \(U_\perp\) for every constant background. For \(|a|\) small against \(2\pi/L\) each nonzero-mode quadratic form is positive definite and depends smoothly on \((a_1,a_2,a_3)\in(\mathfrak{su}(2))^3\), and the trace of the square root of a smooth positive-definite family is smooth, so with a lattice regularization \(U_\perp\) is a smooth function near \(a=0\) (the non-smooth \(|a|\) behaviour of \(U\) comes only from the \(k=0\) term). It is invariant under the constant gauge transformations, which rotate the three colour vectors \(\vec a_i\) simultaneously, under the cubic group acting on the spatial index, and under \(a\to-a\). The quadratic invariants of \((\vec a_1,\vec a_2,\vec a_3)\) under \(SO(3)_{\rm colour}\) are the bilinears \(\vec a_i\cdot\vec a_j\), and the only cubic-invariant symmetric form in the spatial indices is \(\delta_{ij}\), so \[U_\perp(a)=c\,\hbar cL\sum_{i=1}^3|\vec a_i|^2+O(\hbar cL^2|a|^3),\] with a single pure number \(c\), which the abelian valley (\(\vec a_i=\alpha_i\hat e_3\), \(\sum_i|\vec a_i|^2=|\alpha|^2\)) determines: \(c=C_M/(3\pi^2)\). The Landau-level structure of non-commuting backgrounds enters only at the cubic order \(O(\hbar cL^2|a|^3)\), which is of relative order \(g^{4/3}\) to the zero-mode Hamiltonian; (H3) at its leading correction therefore needs no non-abelian spectral input beyond the symmetry argument.
3b. The abelian theory in the same expansion
For compact \(U(1)\) the constant modes commute, the quartic potential vanishes, and the photons are neutral, so the one-loop potential of the holonomy is identically zero: the constant-mode Hamiltonian is the free Laplacian on the torus of holonomies, \(H_0=\frac{e^2c}{2\hbar L^3}|p|^2\) with \(a\) periodic of period \(2\pi/L\), spectrum \(e^2\hbar c|n|^2/(2L)\), \(n\in\mathbb Z^3\), gap \(e^2\hbar c/(2L)\). The abelian small-volume gap is linear in \(e^2\), not \(e^{2/3}\), because there is no commutator to confine the holonomy and no charged mode to give it a potential; with \(e\) fixed by the absence of running, the \(1/L\) decay is uncompensated and the infinite-volume theory is gapless, in agreement with the Coulomb phase of Guth and Fröhlich–Spencer. In the non-abelian theory both the \(g^{2/3}\) and the running of \(g(L)\) come from the charged modes, the former through the \(k=0\) commutator term and the latter through the logarithm in the quartic coefficient of \(U\).
4. Saturation and the centre-degenerate minima
\(U(a)\le2\hbar c\max\Phi/(\pi^2L)=O(\hbar c/L)\) for all \(a\), while the \(k=0\) term \(2\hbar c|a|\) alone grows without bound. The linear confinement of the constant modes along the abelian valley, which in C133 holds for all \(|a|\), is in the torus theory the small-\(La\) regime of a bounded periodic function. The minima of \(U\) are at \(La\in2\pi\mathbb Z^3\): for \(SU(2)\) with the charged modes of charge \(\pm1\), \(La=2\pi e_i\) corresponds to the fundamental holonomy \(\exp(2\pi iT^3)=-1\), a centre element, trivial on all adjoint fields but a distinct state of the theory; the degenerate minima are connected by tunnelling through the valley barrier of height \(O(\hbar c/L)\), which produces the splittings between the ’t Hooft electric-flux sectors of the small-volume spectrum (van Baal’s programme; not derived here).
Consequence for the small-volume expansion: the zero-mode ground state has width \(La\sim g^{2/3}\) and energy \(g^{2/3}\hbar c/L\), far below the valley saturation scale \(\hbar c/L\) and the tunnelling barrier, so the C133 picture is exact at leading order and the corrections enter at relative order \(g^{2/3}\), as in the Feshbach reduction. The regime where the zero-mode energy reaches \(\hbar c/L\), \(g(L)^{2/3}\sim1\), is where the holonomy delocalizes over the whole compact valley, the flux sectors become degenerate and the description changes character: the crossover \(z\simeq2\) of Lüscher–Münster in this language.
5. Consequence for STATE
(H3)’s one-loop input along the abelian valley is now explicit: \(U_\perp=\frac{C_M}{3\pi^2}\hbar cL|a|^2+O(\hbar cL^2|a|^3)\), bounded by \(O(\hbar c/L)\) globally, periodic with centre-degenerate minima. By the symmetry argument of Section 3a the same quadratic term holds for every constant background, so (H3) at its leading correction is a relatively bounded perturbation of \(h_3\) by \(-|C_M|g^{4/3}\sum_i|\xi_i|^2\,\hbar c/(3\pi^2L)\) inside the region \(La\ll1\) and by a bounded potential outside it, which the confining bound of C133 controls; the Landau-level input enters only at the next order. The abelian theory has no valley potential at all, which is the small-volume form of the abelian/non-abelian distinction.