The logarithmic minimal models LM(p,p′)\mathcal{LM}(p,p') are Yang-Baxter integrable loop models whose scaling limits are logarithmic conformal field theories, and their integrable boundary conditions labelled by Kac labels (r,s)(r,s) are built from an rr-type seam of width ρ−1\rho-1 carrying a boundary field ξ\xi and an ss-type seam of width s−1s-1. Pearce, Tartaglia and Couvreur (2015) confirmed numerically that these boundary conditions realize the infinitely extended Kac weights Δr,s\Delta_{r,s} and computed the boundary free energies analytically by inversion relations, but the analytic derivation of the full conformal spectra was left open: the logarithmic TT- and YY-systems of the transfer matrices were known only for the vacuum boundary (r,s)=(1,1)(r,s)=(1,1), and the inversion-relation solutions rest on analyticity and string-zero-removal assumptions justified only numerically. The problem is to derive and solve the logarithmic TT- and YY-systems (fusion hierarchies) for general (r,s)(r,s) Kac boundary conditions, thereby obtaining the conformal weights and finitized characters analytically from the lattice.

Background

The logarithmic minimal models LM(p,p′)\mathcal{LM}(p,p'), with 1≤p<p′1\le p<p' coprime, form a family of Yang-Baxter integrable lattice models (exactly solvable models whose commuting transfer matrices follow from the Yang-Baxter consistency equation) whose continuum scaling limits are logarithmic conformal field theories: nonunitary, nonrational theories in which the Virasoro dilatation generator L0L_0 acts reducibly but indecomposably, so that correlation functions can contain logarithms. On the lattice, the models are loop models built on the planar Temperley-Lieb algebra (the diagram algebra generated by arc operators eje_j with ej2=βeje_j^2=\beta e_j, where β=2cos⁡λ\beta=2\cos\lambda is the loop fugacity and the crossing parameter λ=(p′−p)π/p′\lambda=(p'-p)\pi/p' is a rational fraction of π\pi). Their central charges are c=1−6(p′−p)2/(pp′)c=1-6(p'-p)^2/(pp') and their conformal weights fill an infinitely extended Kac table labelled by r,s=1,2,3,…r,s=1,2,3,\ldots, with Δr,s=((p′r−sp)2−(p′−p)2)/(4pp′)\Delta_{r,s}=((p'r-sp)^2-(p'-p)^2)/(4pp').

On a strip of width NN the model is described by commuting double-row transfer matrices (the strip analogue of a periodic transfer matrix, obtained by reflecting the lattice so that both edges enter one operator), and conformal data are extracted from the finite-size corrections to the eigenvalues of the associated quantum Hamiltonians. Integrable boundary conditions are solutions of the boundary Yang-Baxter equation. The (r,s)(r,s) Kac boundary conditions, conjugate to the Virasoro Kac representations, are constructed by acting on the vacuum (1,1)(1,1) boundary with an ss-type seam of s−1s-1 columns and an rr-type seam of ρ−1\rho-1 columns carrying a boundary field ξ\xi (a seam is a column of modified faces inserted next to the boundary that implements fusion with a Kac representation). The usual fusion construction relies on Wenzl-Jones projectors (Temperley-Lieb idempotents implementing the Jones-Wenzl fusion projection), which become singular at the ranks needed for large rr; the diagrammatic fusion construction removes this restriction. Pearce, Tartaglia and Couvreur confirmed numerically, by exact diagonalization for N≤32−ρ−sN\le 32-\rho-s together with analytic boundary free energies, that the Kac weights Δr,s\Delta_{r,s} are obtained when ρ=⌊rp′/p⌋\rho=\lfloor rp'/p\rfloor and ξ\xi takes specialized values, and they gave empirical finitized characters (finite-NN truncations of the Kac characters) as qq-binomial expressions.

The analytic route to such conformal data is the functional-equation method of Klümper and Pearce: the commuting family of fused transfer matrices satisfies a fusion hierarchy of functional relations that yields a TT-system and, through the ratios of adjacent TTs, a YY-system of bilinear relations, solvable by nonlinear integral equations (integral reformulations whose unknowns are auxiliary counting functions) or dilogarithm techniques. For the logarithmic minimal models these relations were derived by Morin-Duchesne, Pearce and Rasmussen only for the vacuum boundary (r,s)=(1,1)(r,s)=(1,1) and for the periodic cylinder. The inversion-relation solutions for the boundary free energies also rest on analyticity assumptions: eigenvalues must be analytic and nonzero (ANZ) in specified strips of the spectral parameter, after the controlled removal of finitely many zeros lying in 1-string patterns (on the vertical line through the crossing centre) or short 2-string patterns (pairs straddling it) — assumptions supported by numerics and partial arguments, not proved. For Robin boundary conditions, integrable linear combinations of Neumann-type (loop-reflecting) and Dirichlet-type (loop-absorbing) boundary behaviour built on a twist solution of the boundary Yang-Baxter equation, the member LM(1,2)\mathcal{LM}(1,2) was solved exactly by Pearce, Rasmussen and Tipunin and exhibits half-integer Kac labels Δr,s−1/2\Delta_{r,s-1/2}.

Problem Statement

Derive the logarithmic TT-system and YY-system — the fusion hierarchy of functional relations — satisfied by the commuting double-row transfer matrices of the logarithmic minimal models LM(p,p′)\mathcal{LM}(p,p') (1≤p<p′1\le p<p' coprime, p′>2p'>2) on the strip with the vacuum (1,1)(1,1) boundary condition on one edge and the general (r,s)(r,s) Kac boundary condition on the other edge (an rr-type seam of width ρ−1=⌊rp′/p⌋−1\rho-1=\lfloor rp'/p\rfloor-1 carrying the specialized boundary field ξ\xi — with [x][x] denoting the fractional part, ξ=π/2\xi=\pi/2 if [ρ/p′]=0[\rho/p']=0 and ξ=[ρp/p′] π/2\xi=[\rho p/p']\,\pi/2 otherwise — and an ss-type seam of width s−1s-1), for arbitrary Kac labels r,s∈Nr,s\in\mathbb{N}; and solve these functional relations analytically — for instance by nonlinear integral equations or dilogarithm techniques — to obtain from the lattice the boundary conformal weights Δr,s\Delta_{r,s} of the infinitely extended Kac formula and the finitized characters in the scaling limit.

The parallel analytic derivation of the conformal spectra for the (r,s)(r,s) Robin boundary conditions — for which only numerical spectra, analytic boundary free energies and conjectural characters are known — is the corresponding open extension of the same program; it requires the same analytic machinery and would extend the result to the half-integer Kac labels Δr,s−1/2\Delta_{r,s-1/2}.

This Robin programme is not part of the required deliverable; it is described for context.

The problem is complete when the functional relations hold for the general (r,s)(r,s) Kac table and are solved analytically so as to yield the boundary conformal weights Δr,s\Delta_{r,s} and the finitized characters from the lattice; treatments restricted to special subfamilies (particular p,p′p,p' or finitely many small labels), or stopping at numerical extrapolation, do not complete it.

References

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