Theta-quotient complexity, odd levels, and singular specializations in the double-sum evaluation of $A_1^{(1)}$ string functions at arbitrary integral levels

The level-$N$ string functions $C_{m,\ell}^{N}(q)$ of the affine Kac–Moody algebra $A_1^{(1)}$ — the theta-coefficients of characters of integrable highest-weight modules and the configuration sums behind RSOS local state probabilities — are Hecke-type double-sums $f_{1,N+1,1}$, and the double-sum formulas of Hickerson–Mortenson evaluate them through Appell–Lerch sums plus a correction sum of exactly $N^{2}$ theta quotients. Mortenson–Postnova–Solovyev reduced this burden for even levels $N=2K$: their Theorem 1.1 expresses the symmetric combinations $C_{m,\ell}^{2K}\pm C_{2K-m,\ell}^{2K}$ as single $f_{K+1,K+1,1}$ sums whose decompositions involve only $K+1$ theta quotients, yielding direct, non-modular proofs of the classical Kac–Peterson eta-product identities at levels 2, 4, 6, 8, 10; they explicitly flag two recurring obstacles, the theta-quotient count and potential singularities at specializations, which are handled only by ad hoc limiting arguments. The open problem is to extend this framework to a uniform treatment of all integral levels: an analogue of the even-level reduction for odd levels (and for individual string functions at general level), a uniform evaluation scheme for every level $N$ and weight pair $(m,\ell)$ with theta-quotient complexity comparable to the even-level case, and a systematic analytic theory of the singular specializations. Follow-up work has settled levels $N\le4$ completely and moved to admissible fractional levels, but no odd-level or general-$N$ reduction and no general singular-specialization framework is known.

Background

Let $q=e^{2\pi i\tau}$ with $|q|<1$. The affine Kac–Moody algebra $A_1^{(1)}$ is the central extension of the polynomial loop algebra of $\mathfrak{sl}_2$; its integrable highest-weight modules $L(\Lambda)$ are graded by a non-negative integer $N$ called the level. The character of $L(\Lambda)$ decomposes as a finite sum $\sum C_{m,\ell}^{N}(q)\,\Theta_{m,N}(z,q)$ over Jacobi theta functions $\Theta_{m,N}(z,q)$ of degree $N$, and the coefficients $C_{m,\ell}^{N}(q)$ — indexed by the maximal weight $m$ and highest weight $\ell$ with $m\equiv\ell\pmod 2$, $0\le m<2N$, $0\le\ell\le N$ — are the level-$N$ string functions. They are the one-dimensional configuration sums that give local state probabilities of the restricted solid-on-solid (RSOS) lattice models of Andrews–Baxter–Forrester, and they are the basic building blocks from which the characters are assembled. Kac and Peterson computed many of them in terms of the Dedekind eta function $\eta(\tau)=q^{1/24}(q;q)_{\infty}$, where $(a;q)_{\infty}=\prod_{k\ge0}(1-aq^{k})$ denotes the infinite $q$-shifted factorial, proving the identities by global modular arguments: the string functions transform under the full modular group, so it suffices to compare the first few Fourier coefficients and use vanishing at cusps. Write $J_{m}=(q^{m};q^{m})_{\infty}$, $J_{a,m}=j(q^{a};q^{m})$, and $j(x;q)=\sum_{n\in\mathbb{Z}}(-1)^{n}q^{\binom{n}{2}}x^{n}$ (a Jacobi theta function; by the Jacobi triple product $j(x;q)=0$ exactly when $x$ is an integral power of $q$). The Appell–Lerch function $m(x,q,z)$ is a bilateral Lambert series whose translates form the building blocks of Ramanujan's classical mock theta functions; the Hecke-type double-sums $f_{a,b,c}(x,y,q)=(\sum_{r,s\ge0}-\sum_{r,s<0})(-1)^{r+s}x^{r}y^{s}q^{a\binom{r}{2}+brs+c\binom{s}{2}}$ are the hybrid objects that connect the two. Schilling–Warnaar's conjugate Bailey-pair machinery, completed into a self-contained proof by Mortenson–Postnova–Solovyev, identifies the normalization $\mathcal{C}_{m,\ell}^{N}(q)$ of $C_{m,\ell}^{N}$ with $J_{1}^{-3}f_{1,N+1,1}(q^{1+(m+\ell)/2},q^{1-(m-\ell)/2},q)$. Hickerson–Mortenson's double-sum formulas express $f_{a,b,c}$ in terms of Appell–Lerch sums and a finite correction sum of theta quotients — ratios of products of $j$-functions; for $f_{1,n+1,1}$ the correction sum contains $n^{2}$ theta quotients, so a level-$N$ string function naively costs $N^{2}$ terms (the level-10 identity for $f_{1,11,1}(q^{4},q^{3},q)$ costs one hundred). Two further facts define the state of the art. First, for even levels $N=2K$ the new symmetries of Mortenson–Postnova–Solovyev (their Theorem 1.1) give $C_{m,\ell}^{2K}(q)\pm C_{2K-m,\ell}^{2K}(q)=J_{1}^{-3}q^{s(m,\ell,2K)}(f_{K+1,K+1,1}(\pm q^{1+(K+\ell)/2},q^{1+(m+\ell)/2},q)\pm q^{(K-\ell)/2}f_{K+1,K+1,1}(\pm q^{1+(3K-\ell)/2},q^{1+K+(m-\ell)/2},q))$ with $s(m,\ell,N)=-\frac18+\frac{(\ell+1)^{2}}{4(N+2)}-\frac{m^{2}}{4N}$, and the $f_{K+1,K+1,1}$ decompositions involve only $K+1$ theta quotients; this is what made direct, elementary (non-modular) proofs of the classical Kac–Peterson identities possible at levels 2, 4, 6, 8, 10. Second, the authors state that 'there are two recurring issues that must be kept in mind': the number of theta quotients to be computed, and potential singularities — at the parameter values where string functions live, several factors $j(\cdot\,;q)$ vanish identically, so the generic decomposition formulas do not apply directly and case-by-case limits are required, as in their evaluation of the level-2 string function $c_{11}^{11}=\eta(\tau)^{-2}\eta(2\tau)$, which proceeds through the limit $\lim_{x\to q}f_{2,2,1}(x^{2},x,q)$ obtained from an Appell–Lerch functional equation. What the framework does not provide is any analogue of the even-level reduction for odd levels, a uniform treatment of all integral levels, or a systematic substitute for the ad hoc limiting arguments.

Problem Statement

The problem is to extend the double-sum evaluation framework of Mortenson–Postnova–Solovyev to all integral levels, removing the two obstacles that they identify. Concretely, for the level-$N$ string functions $C_{m,\ell}^{N}(q)$ of $A_1^{(1)}$ represented as Hecke-type double-sums $f_{1,N+1,1}$:

1. Odd-level (and general-level) reduction. For odd levels $N$, and more generally for arbitrary integral levels and weights not covered by the even-level symmetric combinations, find structural identities — new string-function symmetries, quasi-periodicity relations, or other decompositions — that reduce the quadratic theta-quotient count in the Hickerson–Mortenson evaluation of $C_{m,\ell}^{N}$, in the way that Theorem 1.1 reduces the cost from $(2K)^{2}$ to $K+1$ terms at even level $2K$.

2. Uniform treatment. Give a single uniform framework that, for every integral level $N$ and every admissible pair $(m,\ell)$ with $m\equiv\ell\pmod 2$, produces the explicit evaluation of $C_{m,\ell}^{N}$ in terms of theta functions and Appell–Lerch sums with theta-quotient complexity comparable to what the even-level Theorem 1.1 achieves for symmetric combinations ($K+1$ theta quotients at level $2K$) rather than the naive quadratic count. The even-level Theorem 1.1 and the complete level $N\le4$ evaluations of Mortenson (Ramanujan J. 2024) are the known instances; the framework should subsume them and cover all remaining levels.

3. Singular specializations. Develop a general analytic treatment of the singular specializations of the double-sum formulas — the parameter values at which theta factors $j(\cdot\,;q)$ vanish — that replaces the case-by-case limiting arguments (as used for the level-2 evaluation of $c_{11}^{11}$) by a systematic, level-independent procedure applicable to all string-function specializations.

The common objective is a uniform double-sum evaluation theory for all integral-level $A_1^{(1)}$ string functions with controlled theta-quotient complexity and no ad hoc singular analysis, and all three items are mandatory components of it rather than alternatives. Item 2's uniform framework is required to realize the reduction of the quadratic theta-quotient count that item 1 demands for odd and general levels, so a framework attaining the stated complexity for every level and admissible pair resolves item 1 as well; item 3, by contrast, is an independent requirement, since a systematic, level-independent treatment of the singular specializations is stronger than evaluating those specializations case by case. A result that establishes only one of the three items leaves the objective partially resolved.

References

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