Generalized Rogers–Ramanujan identities for level-two string functions of simply laced Lie algebras: exceptional and Dynkin-index-two sectors, generalized Slater identities, and the RSOS lattice-model derivation

Level-two string functions of simply laced affine Lie algebras — the characters of the generalized parafermion coset conformal field theories $\mathfrak g_2/U(1)^r$ (a simple Lie algebra $\mathfrak g$ at level two, quotiented by its Cartan currents) — were conjectured by Genish and Gepner (2014) to equal explicit Rogers–Ramanujan-type fermionic multisums built from the Cartan matrix, with the summation vector restricted to a parity class fixed by the coset sector. The same paper observed that substituting any other parity class appeared to produce another character of the same coset (“generalized Slater identities”) and posed the origin of these identities in RSOS solvable lattice models as the central open physics question. Subsequent work proved the conjecture for the $A$ and $D$ series via q-diagrams (including corrected identities for the $D$-series weights of Dynkin index two), and the parafermionic-bases theorem of Butorac–Kožić–Primc established fermionic character formulas for all untwisted affine algebras at all levels for rectangular highest weights, resolving the source’s questions about higher levels and non-simply-laced algebras within that family. Three precisely stated parts remain open: (i) the exceptional algebras $E_6$, $E_7$, $E_8$ — in particular the sectors of Dynkin index two, which for $E_8$ are the only nontrivial level-two weights and lie outside every existing theorem, with no candidate fermionic sums known; (ii) a classification and proof of the generalized Slater identities; and (iii) the construction of RSOS models whose one-dimensional configuration sums realize the $D$- and $E$-series string functions, which would both prove the identities and yield exact local state probabilities. This problem asks for the completion of that program.

Background

In a two-dimensional conformal field theory (CFT) built from an affine Lie algebra $\hat{\mathfrak g}$ — the central extension of the loop algebra of a finite-dimensional simple Lie algebra $\mathfrak g$ — at positive integer level $k$ (the coefficient of the imaginary root in the highest weight, which bounds the integrable highest weights), the characters of the irreducible modules encode the spectrum of scaling dimensions. Kac and Peterson introduced the string functions $c^\Lambda_\lambda(\tau)$: the graded multiplicities with which the weights congruent to the highest weight $\Lambda$ modulo $kM$ (with $M$ the root lattice) occur in the affine module, equivalently the branching functions separating the contributions of the Cartan-subalgebra currents. Multiplying by $\eta(\tau)^r$, where $\eta$ is the Dedekind eta function and $r$ is the rank of $\mathfrak g$, gives the characters $\chi^\Lambda_\lambda$ of the coset CFT $\mathfrak g_k/U(1)^r$: the generalized parafermion theories introduced by Gepner, in which quotienting out the Cartan currents leaves fields of fractional statistics (“parafermions”).

A long line of work expresses such characters as fermionic (Rogers–Ramanujan-type) multisums: sums over non-negative integer vectors $\mathbf b$ of explicit powers of $q=e^{2\pi i\tau}$ divided by products of $q$-Pochhammer symbols $(q)_n=\prod_{m=1}^n(1-q^m)$. Lepowsky and Primc’s $Z$-algebra analysis (a presentation of the standard modules by explicit linear relations among their charged modes) of the standard modules of $A_1^{(1)}$ gave the first Lie-algebraic proof of the classical Rogers–Ramanujan identities; Kuniba, Nakanishi and Suzuki conjectured, from the thermodynamic Bethe ansatz (a method that derives the finite-size spectra of integrable models from their exact scattering data), fermionic formulas for the vacuum-sector ($\Lambda=0$) string functions of every affine algebra at every level; Georgiev constructed the parafermionic (quasi-particle) bases proving them for the type-$A$ algebras; and Butorac, Kožić and Primc extended the construction to the non-simply-laced types — completing the verification of the conjecture — and to rectangular highest weights $\Lambda=k_0\Lambda_0+k_j\Lambda_j$ with $j$ a mark-one node. Here the mark of a node is the coefficient $m_j=\Lambda_j\cdot\theta$ of the simple root $\alpha_j$ in the highest root $\theta$, also called the Dynkin index of the fundamental weight $\Lambda_j$; level $k$ integrability requires $\Lambda\cdot\theta\le k$, so at level two the fundamental weights of mark two are admissible while those of higher mark are not.

At level two, Genish and Gepner computed all string functions of every simply laced algebra in closed form, via “ladder” coset constructions that identify $\mathfrak g_2/U(1)^r$ with products and $\mathbb Z_2$-orbifolds of free-boson torus theories, and conjectured a strikingly uniform identity: for $\Lambda=0$ or a mark-one fundamental weight,

$$\chi^\Lambda_\lambda(\tau)=q^{\Delta^\Lambda_\lambda}\sum_{\mathbf b}\frac{q^{\mathbf b C\mathbf b^{t}/4-\mathbf b\cdot\Lambda/2}}{(q)_{b_1}\cdots(q)_{b_r}}$$

where $C$ is the Cartan matrix of $\mathfrak g$, the sum runs over $\mathbf b=(b_1,\dots,b_r)\in\mathbb Z_{\ge0}^r$ with each component in a fixed parity class $b_i\equiv Q_i\pmod 2$, the class $Q$ is determined by $\Lambda-\lambda=\sum_k Q_k\alpha_k\bmod 2M$, and $\Delta$ is a normalization exponent. They verified this by computer algebra to orders $q^{10}$–$q^{20}$ (systematically for the $D$ series, and through examples covering the mark-one weights for $E_6$ and $E_7$ and two vacuum-sector identities for $E_8$), and proved it for the $A$ and $D$ series in two sequels using q-diagrams — a diagrammatic calculus organizing the required q-series identities and incorporating a Bailey-lemma mechanism (a standard q-series device converting between product and sum forms). The $D$-series sequel moreover found that for the “inner-node” weights of Dynkin index two the naive form of the conjecture fails, and derived new diagrammatic identities covering them. For the exceptional algebras the identities remained unproven, and $E_8$ has no mark-one node at all, so its nontrivial level-two weights necessarily have Dynkin index two.

Two further phenomena define the problem. First, replacing the parity class $Q$ in the multisum by any other class modulo two again produced a character of the same coset — the “generalized Slater identities”, generalizing classical Ising-model identities of the Rogers–Ramanujan type collected by Slater — a remarkable empirical structure that was never classified or proved. Second, in the tradition connecting CFT characters to solvable lattice models — first noted in the exact solution of the hard-hexagon model, whose local state probabilities are one-dimensional configuration sums (weighted sums over admissible height sequences) matching Rogers–Ramanujan sums, and developed for the $A_{n-1}^{(1)}$ family by Jimbo, Miwa and Okado — the authors conjectured that RSOS (restricted solid-on-solid) lattice models exist whose configuration sums realize these string functions, and posed the construction of these models, with the resulting proofs of the identities and exact local state probabilities, as the most interesting open question from the physics point of view.

Problem Statement

Let $\mathfrak g$ be a simply laced simple Lie algebra of rank $r$ with Cartan matrix $C$, root lattice $M$, highest root $\theta$, and affine extension $\hat{\mathfrak g}$ at level $k=2$; let $\chi^\Lambda_\lambda=\eta^r c^\Lambda_\lambda$ be the characters of the generalized parafermion coset $\mathfrak g_2/U(1)^r$, labeled by integrable level-two highest weights $\Lambda$ and sectors $\lambda\bmod 2M$ with $\Lambda-\lambda\in M$, subject to the coset field identifications (the equivalences $\chi^\Lambda_\lambda=\chi^{\sigma(\Lambda)}_{\sigma(\lambda)}$ under automorphisms $\sigma$ of the affine Dynkin diagram at level two, together with the analogous string-function symmetries, which reduce the sector list to the vacuum and the fundamental weights). The problem is to complete the generalized Rogers–Ramanujan program for these characters. It has three linked parts with one common objective — that every level-two sector of every simply laced algebra admit a proven fermionic multisum identity, with the parity structure classified and the RSOS lattice-model origin established:

(1) Exceptional and Dynkin-index-two sectors. Prove fermionic multisum identities for the level-two string functions of $E_6$, $E_7$ and $E_8$. This includes the mark-one weights of $E_6$ ($\omega_1,\omega_6$) and $E_7$ ($\omega_7$) in the multisum form displayed in the Background, which are discharged either by a direct proof or by an explicit derivation from the rectangular-weight parafermionic-basis theorems; and, as the essential new content, the Dynkin-index-two sectors — the fundamental weights at the mark-two nodes (numbered as in the standard Bourbaki labeling of the Dynkin diagrams), namely $\omega_2,\omega_3,\omega_5$ of $E_6$, $\omega_1,\omega_2,\omega_6$ of $E_7$, and $\omega_1,\omega_8$ of $E_8$ (up to field identifications) — for which the naive conjecture is known to fail in the analogous $D$-series inner-node cases and for which no candidate fermionic sums are known at all in the exceptional types (for $E_8$ these two weights are the only nontrivial level-two sectors). Where no candidate sum is known, the task includes finding the correct summation structure, as the $SO(2r)$ sequel did for the $D$-series inner nodes.

(2) Generalized Slater identities. Prove and classify the observed parity phenomenon: for every simply laced $\mathfrak g$ and every level-two $\Lambda$, determine precisely which coset character $\chi^{\Lambda'}_{\lambda'}$ (if any) equals the multisum when the parity class $Q$ is replaced by any other class $Q'\bmod 2$; prove the resulting identities; and give the structural explanation, expected to involve the coset field identifications and Weyl symmetries (the reflection symmetries of the root system of $\mathfrak g$).

(3) RSOS derivation. Identify or construct the solvable RSOS-type lattice models associated with the $D$- and $E$-series level-two cosets, in parallel with the $A_{n-1}^{(1)}$ analysis of Jimbo, Miwa and Okado: the admissible-height structure, the Boltzmann weights, and the one-dimensional configuration sums whose scaling limits are the string functions $\eta^r c^\Lambda_\lambda$; use the finite configuration-sum recursions to prove the finite (and thence limiting) fermionic identities; and compute the local state probabilities of these models.

No particular proof technique is required — q-diagrams, Bailey lemmas, parafermionic or quasi-particle bases, vertex-algebraic methods, or RSOS recursions are all admissible — and where the same identity is required by more than one part, a single proof of it satisfies each part that requires it — but a complete answer must resolve all three parts as stated: the exceptional and index-two identities, the parity classification, and the RSOS construction for the $D$ and $E$ cosets.

References

1. Arel Genish and Doron Gepner, “Level two string functions and Rogers Ramanujan type identities”, Nuclear Physics B 886 (2014) 554–568, DOI: 10.1016/j.nuclphysb.2014.07.009, arXiv:1405.1387, https://arxiv.org/abs/1405.1387
2. Arel Genish and Doron Gepner, “The SO(2r)_2 string functions as q-diagrams”, Nuclear Physics B 897 (2015) 179–212, DOI: 10.1016/j.nuclphysb.2015.05.020, arXiv:1504.07612, https://arxiv.org/abs/1504.07612
3. Arel Genish and Doron Gepner, “The SU(r)_2 string functions as q-diagrams”, Nuclear Physics B 907 (2016) 154–179, DOI: 10.1016/j.nuclphysb.2016.03.021, arXiv:1511.08265, https://arxiv.org/abs/1511.08265
4. Doron Gepner, “On the Characters of Parafermionic Field Theories”, Letters in Mathematical Physics 105 (2015) 769–778, DOI: 10.1007/s11005-015-0764-z, arXiv:1410.1778, https://arxiv.org/abs/1410.1778
5. Arel Genish and Doron Gepner, “Generalized Rogers–Ramanujan identities for twisted affine algebras”, Modern Physics Letters A 32 (2017) 1750110, DOI: 10.1142/S0217732317501103, arXiv:1704.02467, https://arxiv.org/abs/1704.02467
6. Doron Gepner, “Generalized Rogers–Ramanujan expressions for some non-singlet twisted affine algebras”, Modern Physics Letters A 37 (2022) 2250105, DOI: 10.1142/S021773232250105X, arXiv:2204.10601, https://arxiv.org/abs/2204.10601
7. Marijana Butorac, Slaven Kožić and Mirko Primc, “Parafermionic bases of standard modules for affine Lie algebras”, Mathematische Zeitschrift 298 (2021) 1003–1032, DOI: 10.1007/s00209-020-02639-w, arXiv:2002.00435, https://arxiv.org/abs/2002.00435
8. Galin Georgiev, “Combinatorial constructions of modules for infinite-dimensional Lie algebras, II. Parafermionic space”, arXiv:q-alg/9504024, https://arxiv.org/abs/q-alg/9504024
9. Masato Okado and Ryo Takenaka, “Parafermionic bases of standard modules for twisted affine Lie algebras of type A_{2l-1}^{(2)}, D_{l+1}^{(2)}, E_6^{(2)} and D_4^{(3)}”, arXiv:2109.08892, https://arxiv.org/abs/2109.08892
10. Ryo Takenaka, “Vertex algebraic construction of modules for twisted affine Lie algebras of type A_{2l}^{(2)}”, arXiv:2205.05271, https://arxiv.org/abs/2205.05271
11. Marijana Butorac and Slaven Kožić, “Combinatorial bases of standard modules of twisted affine Lie algebras in types A_{2l-1}^{(2)} and D_{l+1}^{(2)}: rectangular highest weights”, Communications in Algebra 51 (2023) 4012–4032, DOI: 10.1080/00927872.2023.2196337, arXiv:2211.05171, https://arxiv.org/abs/2211.05171
12. Marijana Butorac and Slaven Kožić, “Principal subspaces for the affine Lie algebras in types D, E and F”, arXiv:1902.10794, https://arxiv.org/abs/1902.10794
13. Atsuo Kuniba, Tomoki Nakanishi and Junji Suzuki, “Characters in conformal field theories from thermodynamic Bethe ansatz”, Modern Physics Letters A 8 (1993) 1649–1659, DOI: 10.1142/S0217732393001392, https://doi.org/10.1142/S0217732393001392
14. Michio Jimbo, Tetsuji Miwa and Masato Okado, “Local state probabilities of solvable lattice models: an A_{n-1}^{(1)} family”, Nuclear Physics B 300 (1988) 74–108, DOI: 10.1016/0550-3213(88)90587-1, https://doi.org/10.1016/0550-3213(88)90587-1
15. Go Hatayama, Atsuo Kuniba, Masato Okado, Taichiro Takagi and Zengo Tsuboi, “Paths, Crystals and Fermionic Formulae”, arXiv:math/0102113, https://arxiv.org/abs/math/0102113
16. James Lepowsky and Mirko Primc, “Structure of the Standard Modules for the Affine Lie Algebra A_1^{(1)}”, Contemporary Mathematics 46, American Mathematical Society (1985), DOI: 10.1090/conm/046, ISBN 9780821850480, https://doi.org/10.1090/conm/046
17. Victor G. Kac and Dale H. Peterson, “Infinite-dimensional Lie algebras, theta functions and modular forms”, Advances in Mathematics 53 (1984) 125–264, DOI: 10.1016/0001-8708(84)90032-X, https://doi.org/10.1016/0001-8708(84)90032-X
18. L. J. Slater, “Further Identities of the Rogers–Ramanujan Type”, Proceedings of the London Mathematical Society (2) 54 (1952) 147–167, DOI: 10.1112/plms/s2-54.2.147, https://doi.org/10.1112/plms/s2-54.2.147
19. Bolin Han, “Coupled Free Fermion Conformal Field Theory and Representations”, arXiv:2312.06150, https://arxiv.org/abs/2312.06150