Two-chirality law for a general 2D CFT (rests on an unproved chiral factorisation)

Status: conjectural — believed, with partial evidence; not proved.

Area: Theorem A: exact recovery error of the free fermion · Card type: conjecture · Knowledge-base id: two-chirality-law-general-cft

Statement

Conjecture: for a two-dimensional CFT with chiral central charges c = c-bar, the rotated Petz maps of the adjacent-interval configuration give -log F^(lambda) = Phi_c(z(1+e^{-pi lambda})) + Phi_c-bar(z(1+e^{pi lambda})), z = eta_V/(1 - eta_V), with Phi_c the single-chirality zero-collar recovery error of the chiral half (Note 5 eq:collapse-vwz; paper 1 prop:twobranch). The argument uses, without proof, (i) that the net, the vacuum and the rotated Petz map factorise into the two chiral halves, the Bisognano-Wichmann boost acting as t on one light ray and as -t on the other (Note 5 states this in the VWZ paragraph, l. 432-434; paper 1 assumes a tensor-product net A_chiral x A-bar_chiral), and (ii) a per-chirality function Phi_c: the zero-collar collapse (Note 5 cor:collapse) and the normal extension of the glued map behind it are proved only for the free fermion, and for a general chiral net only the second-order coefficient f2 = c/(12 pi^2) is known (Theorem B). Proved special case: the 1+1 massless Dirac fermion (card two-chirality-law).

How to verify

fidelity_recovered_quasifree.tex sec:rotated, VWZ paragraph and eq:collapse-vwz (l. 426-440); paper1/sec_2d_lattice.tex prop:twobranch and its proof (factorisation over the light-ray coordinates assumed for A_chiral x A-bar_chiral); the proved Dirac case is check_note5_theory.tex prop:chirality <1>1-<1>7.

Evidence

• fidelity_recovered_quasifree.tex — document, at eq:collapse-vwz
• fidelity_recovered_quasifree.tex — document, at sec:rotated
• paper1/sec_2d_lattice.tex — document, at prop:twobranch

Related claims

• Two-chirality law for the massless Dirac fermion: -log F^(lambda) = Phi(z(1+e^{-pi lambda})) + Phi(z(1+e^{pi lambda})), even in lambda — proved
• Theorem B: f2 = c/(12 pi^2) for every diffeomorphism-covariant net — refereed

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-B5.

The referee's report and the card's event log stay in the private knowledge base; only the verdict, the date and the referee are published here.

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