The force on the Wilson line, drawn on 143a1's own torus
The one-loop potential of PA-01's relative U(1) holonomy θ = (θ₁, θ₂) around the two loops of 143a1, and the force
−G⁻¹∇V it exerts, for each of the three even (Scherk–Schwarz) spin structures. The page sums the series itself; its values agree
with mtft.matter.vacuum_atlas.wilson_line_torus, which tests/test_pg01_wilson_torus.py checks by two routes (PG-6, CC-66).
| Critical point | θ | V (one-loop units) |
|---|
What is drawn
V(θ) = −2 Σw odd |w|⁻⁸ cos 2π θ·w over windings w = (m, n), |w| = |m + nτ|/√Im τ with τ = ½ + 1.0232745927 i. "Odd" means s₁m + s₂n is odd for the spin structure s. The weight is bosons minus fermions: even windings cancel, and with no twist (supersymmetry unbroken) the potential vanishes identically — the force exists because of the Scherk–Schwarz twist.
The torus is drawn in its kinetic metric G = [[|τ|²/y, −x/y], [−x/y, 1/y]] (det 1), so the parallelogram has 143a1's shape and its four corners are one point. Flow lines follow −G⁻¹∇V: every generic line starts at the untwisting point and drains into the jet vacuum.
For a Morse function on a torus #min − #saddle + #max = 0, so one sink and one source force two saddles; the half-periods are critical by symmetry, two independent searches find no other critical points, and all four are nondegenerate. In the real spin structure both saddles sit at exactly V = 0, by the pairing (m, n) → (−m − n, n), which keeps |m + nτ| because Re τ = ½. The jet vacuum is the only minimum (PA1-1 unchanged); the saddles are the passes for winding the holonomy. Scope: one loop, PA-01's product approximation; the overall scale (MKK, sub-puzzle III.5) is unassigned.
Curvatures are metric quantities: at the jet vacuum the eigenvalues of G⁻¹H are 37.394 and 128.985 (real structure), not the θ-coordinate Hessian's 28.893 and 166.935; the product agrees because det G = 1. The drawing truncates the sum at |m|, |n| ≤ 7 (error at the half-periods ≤ 5 × 10⁻⁶).