Accidental symmetries are wildly ramified

Accidental symmetries are wildly ramified

Standalone note. Claim: the ramification filtration on a Galois action and the EFT filtration on the symmetries of a Lagrangian are the same structure.

1. The two filtrations

Arithmetic. Let E/F be a Galois extension of local fields, G = Gal(E/F). The lower ramification groups are

G_i = { σ ∈ G : v_E(σ(x) − x) ≥ i + 1 for all x ∈ 𝔬_E },

i.e. the elements acting trivially to order i near the identity. G_0 is inertia, G_1 wild inertia. An outer action ψ: G → Out(Ĝ) on a dual group inherits this filtration; its depth is the largest i with ψ(G_i) ≠ 1. Tame = depth 0.

Physics. Fix a Lagrangian 𝓛 = Σ_d 𝓛_d, graded by operator dimension. A symmetry of the theory has a breaking dimension

d(S) = min { d : 𝓛_d is not S-invariant },

or ∞ if exact. Set depth(S) = d(S) − 4 (dimension of the marginal layer in 4d). Exact-at-marginal symmetries broken only by irrelevant operators are the accidental symmetries.

Identification. depth of the ramification filtration ↔ depth of the breaking. G_i ↔ symmetries exact through order i.

2. Standard Model table

Charge conjugation is the diagram automorphism of SU(N), N ≥ 3, so it literally is a rung-1 outer action; the table says it is a tame one.

3. What transfers

The depth-shift theorem. Arithmetic: under local Langlands for tori, depth is preserved iff the torus is tame (Yu); for wild tori the shift is governed by the Hasse–Herbrand function of the splitting extension (Mishra–Pattanayak). Mechanism: v_E(σu − u) ≥ v_E(u) + b, i.e. the action is trivial to depth b. Physics translation: an observable violating a symmetry of depth b is suppressed by (v/Λ)^b relative to naive scaling. Neutrino mass m_ν ~ v²/Λ is the depth-1 case; proton decay ~ 1/Λ⁴ the depth-2 case. The "shift" is the extra power of the cutoff.

Inner twists carry no depth. Arithmetic: inner forms are unramified twists (Lang + Steinberg), so Jacquet–Langlands preserves depth. Physics: inner automorphisms are gauge transformations, which break nothing at any order. Consistent.

Two sources of wildness. Arithmetic distinguishes (i) wild ramification of the form (the outer action) from (ii) irregular singularities of the parameter (Stokes data of the connection; slope = depth, Frenkel–Gross, Chen–Kamgarpour). Over ℂ only (ii) exists. Physics candidate for (ii): the asymptotic, Borel-resummable structure of the coupling expansion — resurgence, whose Stokes lines are literally Stokes phenomena of the irregular-singular Schwinger–Dyson/RG equations. (Bet, unverified.)

4. Open

• Herbrand's function converts lower numbering (how far the element moves a uniformizer) to upper numbering (well-behaved under quotients). Candidate physical meaning: UV operator dimension vs IR operator dimension across integrating-out (matching + anomalous dimensions). Not checked.
• Unramified symmetries "act only through Frobenius." Physical analogue: symmetries acting on topological sectors but on no local operator (e.g. a symmetry permuting vacua or superselection sectors). Not developed.

5. Status

The identification is a definition on the physics side and a theorem-backed structure on the arithmetic side; its content is that the same qualitative theorems (tame preserves, wild shifts by the break, inner twists are invisible) hold verbatim after translation. No new theorem. Value: it names the EFT filtration as a ramification filtration, which makes the Galois-side toolbox (Herbrand, Swan conductors, conductor-discriminant) available as questions about accidental symmetries.