Bosonic finitized character formulas and the CTM recursion for the one-dimensional sums of the $2\times2$ fused RSOS$(m,m')$ models with $m'>2m$

For coprime integers $2\le m<m'$ with $m'>2m$, the $2\times2$ fused Forrester–Baxter restricted solid-on-solid models $\mathrm{RSOS}(m,m')_{2\times2}$, with crossing parameter $\lambda=(m'-m)\pi/m'\in(\pi/2,\pi)$, are believed to lie in the universality classes of the nonunitary Virasoro minimal models $\mathcal{M}(m,m')$. The evidence is Baxter's corner-transfer-matrix one-dimensional sums $X_{abc}^{(N)}(q)$, which Fehér, Pearce and Vittorini-Orgeas (2018) argued are finitized characters of $\mathcal{M}(m,m')$: they conjectured explicit bosonic closed forms as alternating sums of finitized $q$-trinomial coefficients, but verified them only numerically (up to system size $N=12$, and only for ground-state boundary data $b=c$), and left open whether such expressions exist for general boundary data and whether they satisfy the corner-transfer-matrix recursion that defines the sums. The problem posed here is to settle these polynomial $q$-series identities: derive the closed bosonic forms for general boundary data ($c=b$ and $c=b\pm2$), prove that they satisfy the CTM recursion with its boundary conditions, and thereby establish rigorously, at the level of the one-dimensional sums, the finitized-character content of the identification of the scaling limit of $\mathrm{RSOS}(m,m')_{2\times2}$ with $\mathcal{M}(m,m')$ for all $m'>2m$.

Background

The Andrews–Baxter–Forrester (ABF) restricted solid-on-solid (RSOS) models are exactly solved two-dimensional lattice models in which an integer height $a\in\{1,\dots,m'-1\}$ sits at every site of a square lattice subject to the restriction that nearest-neighbour heights differ by $\pm1$, so that the heights walk on the $A_{m'-1}$ Dynkin diagram. Forrester and Baxter generalized the elliptic Boltzmann face weights to the family $\mathrm{RSOS}(m,m')$ with coprime $2\le m<m'$ and crossing parameter $\lambda=(m'-m)\pi/m'$; these weights satisfy the Yang–Baxter equation, so the models are exactly solvable both at and away from criticality. In Regime III (the massless critical regime of the phase diagram) the continuum scaling limit of $\mathrm{RSOS}(m,m')$ is the nonunitary minimal model $\mathcal{M}(m,m')$ of conformal field theory, with central charge $c=1-6(m'-m)^2/(mm')$ and Virasoro characters labelled by Kac labels $(r,s)$, $1\le r\le m-1$, $1\le s\le m'-1$, having conformal weights $\Delta_{r,s}^{m,m'}=((rm'-sm)^2-(m'-m)^2)/(4mm')$ and bosonic (Rocha-Caridi) character formula

$$\mathrm{ch}^{m,m'}_{r,s}(q)=\frac{q^{-c/24+\Delta_{r,s}^{m,m'}}}{(q)_\infty}\sum_{k=-\infty}^{\infty}\Big[q^{k(kmm'+m'r-ms)}-q^{(km+r)(km'+s)}\Big],\qquad (q)_\infty=\prod_{j=1}^{\infty}(1-q^j).$$

Fusion constructs larger integrable lattice models $\mathrm{RSOS}(m,m')_{n\times n}$ from $n\times n$ blocks of elementary faces; at fusion level $n=2$ the allowed height steps are $0,\pm2$ rather than $\pm1$. Tartaglia and Pearce conjectured, with one-dimensional-sum evidence, that for $\lambda<\pi/n$ (equivalently $nm>(n-1)m'$) the continuum limit of the $n\times n$ fused models is the higher-level coset minimal model $\mathcal{M}(M,M',n)$ with $(M,M')=(nm-(n-1)m',m')$. For $nm<(n-1)m'$ the lattice lacks the contiguous shaded bands (a band is the region between two adjacent height values; bands are shaded according to where the ground-state paths live) needed to support the level-$n$ ground states, and the scaling limit is expected to default to a lower effective fusion level $n'<n$ — the level-reduction expectation. The interval $\pi/2<\lambda<\pi$, i.e. $m'>2m$, is the first nontrivial such instance: shaded 2-bands are absent and only shaded 1-bands exist.

Baxter's corner transfer matrix (CTM) method reduces local partition functions of an exactly solved model to one-dimensional sums. For the fused models these are the generating polynomials

$$X_{abc}^{(N)}(q)=\sum_{\sigma}q^{E(\sigma)},\qquad E(\sigma)=\sum_{j=1}^{N}j\,H(\sigma_{j-1},\sigma_j,\sigma_{j+1}),$$

over fused RSOS paths $\sigma$ of length $N$ on $A_{m'-1}$ with boundary heights $\sigma_0=a$, $\sigma_N=b$, $\sigma_{N+1}=c$, where the local energy functions $H$ are extracted from the elliptic face weights in the low-temperature limit (tabulated explicitly for $n\le3$ in a suitable gauge). The sums are uniquely characterized by the CTM recursion

$$X_{abc}^{(N)}(q)=\sum_{d\sim b}q^{N H(d,b,c)}X_{adb}^{(N-1)}(q),\qquad X_{a0c}^{(N)}(q)=X_{a\,m'c}^{(N)}(q)=0,\qquad X_{abc}^{(0)}(q)=\delta_{a,b},$$

where $d\sim b$ runs over the heights adjacent to $b$ on the fused height graph ($d=b,b\pm2$ at level 2, with the self-edge $d=b$ excluded at the end heights $1$ and $m'-1$). A finitized character is a polynomial in $q$ that truncates a Virasoro character at conformal energies of order $N$ and converges to it as $N\to\infty$. For the unfused ABF models such sums are finitized characters in a proven sense: Schilling proved polynomial bosonic forms for the branching functions of the integer-level cosets $\widehat{su(2)}_M\times\widehat{su(2)}_N/\widehat{su(2)}_{M+N}$ (which realize the unitary minimal models), and Warnaar proved Melzer's polynomial identities for the ABF one-dimensional sums by unifying thermodynamic Bethe ansatz and CTM fermionic methods; Schilling and Warnaar also proved bosonic-equals-fermionic identities for the associated $q$-supernomials, whose finitizations coincide with the $q$-multinomial expressions used below in certain sectors. None of these proofs covers the nonunitary $m'>2m$ regime, in which the shaded 2-bands that would support superconformal ground states are absent.

Fehér, Pearce and Vittorini-Orgeas treated precisely this regime. They identified the ground states (supported on shaded 1-bands at the heights $\rho_\mu(r)$: the union of the two sequences $\rho(r)=\lfloor rm'/m\rfloor$ and $\rho(r)+1$, $r=1,\dots,m-1$, splits into its even members $\rho_0(r)$ ($\mu=0$) and odd members $\rho_1(r)$ ($\mu=1$)), verified that the ground-state one-dimensional sums produce finitized characters of the level-1 minimal models $\mathcal{M}(m,m')$ with $m'>2m$, and conjectured the bosonic closed form

$$\widehat{\mathrm{ch}}^{m,m';(N)}_{r,s}(q)=X^{m,m';(N)}_{abc}(q)=\sum_{k=-\infty}^{\infty}\Big[q^{k(kmm'+m'r-ms)}T^{(N)}_{km'+\frac{b-a}{2}}(q)-q^{(km+r)(km'+s)}T^{(N)}_{km'+\frac{b+a}{2}}(q)\Big],$$

with $a=s$, $b=c=\rho_\mu(r)$, $\mu=s\bmod2$, in terms of the finitized $q$-trinomial coefficients $T_k^{(N)}(q)=\sum_{j\ge0}q^{j(j+k)}[N;j,j+k]_q$, where $[N;j,j+k]_q$ is a $q$-multinomial coefficient (a $q$-analogue of the multinomial counting of groups of sizes $j$, $j+k$ and $N-2j-k$). This is the natural finitization of the classical bosonic minimal-model character formula quoted above: the alternating $k$-sum mirrors the Rocha-Caridi structure, and the $q$-trinomials truncate the factor $1/(q)_\infty$. They checked it by computer up to $N=12$ for $(m,m')=(2,5),(2,7),(3,7),(3,8)$ ($N=11$ for $(2,9),(4,9)$, $N=10$ for $(3,10)$, $N=9$ for $(2,11),(3,11),(4,11),(5,11)$). For $m'=2m+1$ the sums agree with the half-integer RSOS path sums of Jacob and Mathieu. The paper states that it would be of interest to obtain the bosonic expressions for $X_{abc}^{(N)}(q)$ more generally for $c=b,b\pm2$ and to prove that these expressions satisfy the CTM recursion; no proof, of either the conjectured formulas or of any extension, is known.

Problem Statement

Settle the finitized-character conjecture of Fehér–Pearce–Vittorini-Orgeas for the $2\times2$ fused RSOS models, as follows.

(1) Prove, or refute, the conjectured bosonic finitized character formula quoted in the Background (eq. (4.4) of the source): for every coprime pair $2\le m<m'$ with $m'>2m$, every sector $(r,s)$ of $\mathcal{M}(m,m')$, and every system size $N$, the ground-state one-dimensional sum $X^{m,m';(N)}_{abc}(q)$ with $a=s$, $b=c=\rho_\mu(r)$, $\mu=s\bmod2$, equals the stated alternating sum of finitized $q$-trinomial coefficients. A single verified instance of disagreement (a concrete $m,m',r,s,N$ together with the two differing $q$-polynomials) refutes the conjecture. A verified refutation of part (1) alone resolves the task; parts (2) and (3) presuppose that the part-(1) formula holds and are required only conditional on part (1) being proved.

(2) Obtain the corresponding closed bosonic expressions (alternating sums of finitized $q$-trinomial or equivalent $q$-multinomial coefficients) for the one-dimensional sums $X_{abc}^{(N)}(q)$ for general boundary data — in particular for $c=b$ and $c=b\pm2$ with arbitrary allowed $a,b$ — for the same models.

(3) Prove that all these bosonic expressions satisfy the CTM recursion

$$X_{abc}^{(N)}(q)=\sum_{d\sim b}q^{N H(d,b,c)}X_{adb}^{(N-1)}(q),\qquad X_{a0c}^{(N)}(q)=X_{a\,m'c}^{(N)}(q)=0,\qquad X_{abc}^{(0)}(q)=\delta_{a,b},$$

so that, by induction on $N$, they coincide with the one-dimensional sums; and show that in the limit $N\to\infty$ the ground-state expressions of part (1) (the $b=c$ finitized characters) converge to the Virasoro characters $\mathrm{ch}^{m,m'}_{r,s}(q)$ of $\mathcal{M}(m,m')$ up to the conventional prefactor $q^{-c/24+\Delta_{r,s}}$; for the general boundary data of part (2) the source imposes no specific limit, so their convergence behaviour is whatever the proven recursions yield. This would establish, rigorously and at the level of Baxter's one-dimensional sums, the finitized-character content of the claim that the continuum scaling limit of $\mathrm{RSOS}(m,m')_{2\times2}$ for $m'>2m$ is the minimal model $\mathcal{M}(m,m')$.

The identities must hold for all $N$ and all coprime $m'>2m$, including the exceptional small cases such as $(m,m')=(2,5)$.

References

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