For the 2×22\times2 fused Forrester–Baxter RSOS(m,m′)(m,m') lattice models in the regime m′>2mm'>2m, where the shaded 2-bands that would support level-2 coset ground states are absent, Fehér, Pearce and Vittorini-Orgeas (J. Stat. Mech. 1807 (2018) 073104) argued a level-reduction phenomenon: the continuum scaling limit is the level-1 minimal model M(m,m′)\mathcal{M}(m,m'), and the corner-transfer-matrix one-dimensional configuration sums Xabc(N)(q)X_{abc}^{(N)}(q) match conjectured bosonic finitized characters built from qq-trinomial coefficients, verified numerically up to N=12N=12 for the ground-state boundary data b=cb=c. The source leaves this match as an open conjecture. The problem is to prove, or refute, the conjectured identity between the one-dimensional sums and the bosonic qq-trinomial expressions for all NN, all coprime m′>2mm'>2m and all ground-state boundary data, thereby putting the level-reduction argument on rigorous combinatorial ground; the source further raises, as its own secondary open direction, obtaining the bosonic expressions for the remaining boundary data c=bc=b and c=b±2c=b\pm2 and proving that these expressions satisfy Baxter's corner-transfer-matrix recursion.

Background

A restricted solid-on-solid (RSOS) lattice model assigns integer "heights" to the sites of a square lattice, with adjacent heights differing by exactly one, as if walking on the Dynkin diagram Am′−1A_{m'-1} (a path graph with m′−1m'-1 nodes). The integrable (Yang–Baxter solvable) critical RSOS(m,m′)(m,m') models of Forrester and Baxter depend on a pair of coprime integers (m,m′)(m,m') through the crossing parameter λ=(m′−m)π/m′\lambda=(m'-m)\pi/m', and their continuum scaling limits are the Virasoro minimal models M(m,m′)\mathcal{M}(m,m') — the rational conformal field theories with central charge c=1−6(m′−m)2/(mm′)c=1-6(m'-m)^2/(mm') and finitely many primary fields labelled by Kac labels (r,s)(r,s). The n×nn\times n fused models generalize this by "fusion" (the nn-fold tensor-product construction of Andrews–Baxter–Forrester): heights step by 00 or ±n\pm n instead of ±1\pm1, so a 2×22\times2 fused path moves up, down, or stays flat in steps of 2 and has definite parity. Tartaglia and Pearce argued that the n×nn\times n fused RSOS(m,m′)(m,m') models have as scaling limits the higher-level coset minimal models M(M,M′,n)=(A1(1))k⊗(A1(1))n/(A1(1))k+n\mathcal{M}(M,M',n)=(A_1^{(1)})_k\otimes(A_1^{(1)})_n/(A_1^{(1)})_{k+n} — conformal field theories built as cosets of the affine (Wess–Zumino–Witten) su^(2)\widehat{\mathfrak{su}}(2) current algebra at levels kk and nn, with fractional level k=nM/(M′−M)−2k=nM/(M'-M)-2 and (M,M′)=(nm−(n−1)m′,m′)(M,M')=(nm-(n-1)m',m') — which requires nm>(n−1)m′nm>(n-1)m'; when nm<(n−1)m′nm<(n-1)m' the shaded nn-bands (contiguous runs of nn shaded 1-bands, located between heights ρ(r)=⌊rm′/m⌋\rho(r)=\lfloor rm'/m\rfloor and ρ(r)+1\rho(r)+1, that support the level-nn ground states) do not exist, and the scaling limit is expected to "default" to a lower effective fusion level n′<nn'<n.

Baxter's corner transfer matrix (CTM) formalism — the off-critical counterpart of radial quantization — expresses local state probabilities of a solvable model as one-dimensional configuration sums. For the 2×22\times2 fused RSOS models these are Xabc(N)(q)=∑σqE(σ)X_{abc}^{(N)}(q)=\sum_{\sigma} q^{E(\sigma)}, finite sums over (N+1)(N{+}1)-step fused paths σ=(σ0,…,σN+1)\sigma=(\sigma_0,\ldots,\sigma_{N+1}) with boundary heights (σ0,σN,σN+1)=(a,b,c)(\sigma_0,\sigma_N,\sigma_{N+1})=(a,b,c), weighted by the energy statistic E(σ)=∑j=1Nj H(σj−1,σj,σj+1)E(\sigma)=\sum_{j=1}^N j\,H(\sigma_{j-1},\sigma_j,\sigma_{j+1}), where the local energy functions H(d,a,b)H(d,a,b) are extracted from the elliptic Boltzmann face weights in the conjugate-modulus (low-temperature) limit in which the corner transfer matrix acts and, in the nonnegative gauge of the source, take only the values 0,12,10,\tfrac12,1. The sums obey the CTM recursion Xabc(N)(q)=∑d∼bqNH(d,b,c)Xadb(N−1)(q)X_{abc}^{(N)}(q)=\sum_{d\sim b} q^{N H(d,b,c)} X_{adb}^{(N-1)}(q), where d∼bd\sim b runs over heights adjacent to bb at fusion level 2, with boundary conditions Xabc(0)(q)=δa,bX_{abc}^{(0)}(q)=\delta_{a,b} and Xa0c(N)(q)=Xam′c(N)(q)=0X_{a0c}^{(N)}(q)=X_{am'c}^{(N)}(q)=0. In the N→∞N\to\infty limit these polynomials converge to Virasoro characters; at finite NN they are finitized characters — polynomial, finite-N truncations of the characters that count conformal states below a cutoff.

For the unfused (1×11\times1) Andrews–Baxter–Forrester model, the identification of one-dimensional sums with finitized characters is the content of the finitized polynomial identities conjectured by Melzer and proved by Warnaar, a landmark of "physical combinatorics". At fusion level 2, Jacob and Mathieu had earlier studied spin-1 (half-integer height) path sums that reproduce the characters of the nonunitary models M(m,2m+1)\mathcal{M}(m,2m+1) without an underlying integrable structure; Fehér, Pearce and Vittorini-Orgeas supplied that structure by exhibiting an energy-preserving bijection between these half-integer paths and the 2×22\times2 fused RSOS(m,2m+1)(m,2m+1) paths. For general m′>2mm'>2m (crossing parameter π/2<λ<π\pi/2<\lambda<\pi), they conjectured an explicit bosonic form for the finitized characters using the qq-trinomial coefficients Tk(N)(q)=∑j=0Nqj(j+k)(Nj,j+k)qT_k^{(N)}(q)=\sum_{j=0}^{N} q^{j(j+k)}\binom{N}{j,j+k}_q, where (nℓ,m)q\binom{n}{\ell,m}_q denotes the qq-multinomial coefficient, and verified the conjecture by computer out to system sizes N=12N=12 for the ground-state boundary data b=cb=c and several (m,m′)(m,m'). The conjecture is the lynchpin of their level-reduction argument: if the one-dimensional sums are exactly the finitized characters of M(m,m′)\mathcal{M}(m,m'), then the 2×22\times2 fused model lies in the M(m,m′)\mathcal{M}(m,m') universality class despite being a genuinely different lattice model from the 1×11\times1 one.

Problem Statement

Prove, or refute, the conjecture of Fehér, Pearce and Vittorini-Orgeas for the 2×22\times2 fused RSOS(m,m′)(m,m') lattice models in the regime m′>2mm'>2m:

(i) Show that for the ground-state boundary data a=sa=s, b=c=ρμ(r)b=c=\rho_\mu(r) (with μ=s mod 2\mu=s \bmod 2 and Kac labels 1≤r≤m−11\le r\le m-1, 1≤s≤m′−11\le s\le m'-1 of M(m,m′)\mathcal{M}(m,m'); here ρ(r)=⌊rm′/m⌋\rho(r)=\lfloor rm'/m\rfloor and ρμ(r)\rho_\mu(r), μ∈{0,1}\mu\in\{0,1\}, denotes the subsequence of parity μ\mu drawn from the union of the two height sequences ρ(r)\rho(r) and ρ(r)+1\rho(r)+1), the one-dimensional configuration sums equal the conjectured bosonic finitized characters, i.e. for every N≥0N\ge0 and every coprime pair (m,m′)(m,m') with m′>2mm'>2m, Xabcm,m′;(N)(q)=∑k=−∞∞[qk(kmm′+m′r−ms)Tkm′+b−a2(N)(q)−q(km+r)(km′+s)Tkm′+b+a2(N)(q)],X_{abc}^{m,m';(N)}(q)=\sum_{k=-\infty}^{\infty}\Big[q^{k(kmm'+m'r-ms)}T^{(N)}_{km'+\frac{b-a}{2}}(q)-q^{(km+r)(km'+s)}T^{(N)}_{km'+\frac{b+a}{2}}(q)\Big], where Tk(N)(q)=∑j=0Nqj(j+k)(Nj,j+k)qT_k^{(N)}(q)=\sum_{j=0}^{N}q^{j(j+k)}\binom{N}{j,j+k}_q is the qq-trinomial coefficient, built from the qq-multinomial coefficient (nℓ,m)q=(q)n/[(q)ℓ(q)m(q)n−ℓ−m]\binom{n}{\ell,m}_q=(q)_n/[(q)_\ell(q)_m(q)_{n-\ell-m}] with (q)n=∏i=1n(1−qi)(q)_n=\prod_{i=1}^n(1-q^i), taken to be zero unless ℓ,m≥0\ell,m\ge0 and ℓ+m≤n\ell+m\le n; this is the paper's equation (4.4), verified only numerically, and only for ground-state boundary data with N≤12N\le12. (The qq-trinomial coefficient is symmetric, Tk(N)(q)=T−k(N)(q)T_k^{(N)}(q)=T_{-k}^{(N)}(q).)

(ii) The source itself states, as a further open direction beyond the central conjecture (i): "It would be of interest to obtain the bosonic expressions for Xabc(N)(q)X_{abc}^{(N)}(q) more generally for c=b,b±2c=b,b\pm2 and to prove that these expressions satisfy the CTM recursions (2.36)." For these remaining allowed boundary data — c=bc=b for general allowed bb (beyond the ground-state heights b=c=ρμ(r)b=c=\rho_\mu(r) of part (i)) and c=b±2c=b\pm2 — no candidate formula currently exists. Should explicit bosonic qq-trinomial expressions be supplied for them and be shown, together with those of (i), to satisfy the corner-transfer-matrix recursion Xabc(N)(q)=∑d∼bqNH(d,b,c) Xadb(N−1)(q)X_{abc}^{(N)}(q)=\sum_{d\sim b} q^{NH(d,b,c)}\,X_{adb}^{(N-1)}(q) with boundary conditions Xabc(0)(q)=δa,bX_{abc}^{(0)}(q)=\delta_{a,b}, Xa0c(N)(q)=Xam′c(N)(q)=0X_{a0c}^{(N)}(q)=X_{am'c}^{(N)}(q)=0, an induction on NN would establish the full identification. The primary target of this problem is the conjectured identity (i); the extension (ii) records the source's own stated direction of further interest and is not a required component of a solution.

The identity is an identity of polynomials in qq for each fixed NN, in the exact conventions of the source paper: heights on Am′−1A_{m'-1} with 2×22\times2 adjacency σj+1−σj=0,±2\sigma_{j+1}-\sigma_j=0,\pm2, the local energies H(d,a,b)H(d,a,b) in the conjugate-modulus gauge of the source (taking the values 0,12,10,\tfrac12,1 in the source's nonnegative gauge, tabulated there for the cases needed), and the normalized finitized characters ch^r,sm,m′;(N)=Xabcm,m′;(N)\widehat{\mathrm{ch}}_{r,s}^{m,m';(N)}=X_{abc}^{m,m';(N)}. The N→∞N\to\infty limit then reproduces the Virasoro characters of M(m,m′)\mathcal{M}(m,m') and the logarithmic limit m,m′→∞m,m'\to\infty with m′/m→p′/pm'/m\to p'/p approached from above those of the logarithmic minimal models LM(p,p′){\cal LM}(p,p'), but the problem asked here is the exact finite-NN statement. The wider questions the same source leaves open — the level-reduction mechanism at general fusion level n>2n>2 (where shaded nn-bands are absent when nm<(n−1)m′nm<(n-1)m'), a fully rigorous lattice derivation of the M(m,m′)\mathcal{M}(m,m') scaling limit, the Jordan-cell mechanism in the logarithmic limit, and a classification of energy-preserving bijections of Jacob–Mathieu type — are context and motivation; resolving them is not required to close this problem.

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