The level-2 parafermion conformal field theories PF2(slN)\mathrm{PF}_2(\mathfrak{sl}_N) admit a realisation by coupled free fermions, and their characters take 'fermionic' form: universal chiral partition functions UCPF(G;a)\mathrm{UCPF}(\mathbf{G};\mathbf{a}), multisum qq-series whose statistical interaction matrix G\mathbf{G} encodes the fractional exclusion statistics of the coupled fermions. Bouwknegt–Chern–Han (2024) proved UCPF\mathrm{UCPF}-to-character identities, η\eta-function string-function formulae, and Rogers–Ramanujan-type product identities for N=2,3,4N=2,3,4, but the matrices G3\mathbf{G}_3 and G4\mathbf{G}_4 entering them are conjectural working matrices found case by case, and the authors identify the determination of G\mathbf{G} for general higher rank as the principal obstacle to extending the method. This problem asks for a systematic, provably correct construction of GN\mathbf{G}_N for higher-rank level-2 parafermion theories, together with the resulting higher-rank character identities, the extension of the qq-hypergeometric transform machinery to the general quadratic forms, and the expected partition-theoretic interpretations of the multisum sides. The construction of G\mathbf{G} for higher rank remains an open problem.

Background

Setting. A conformal field theory (CFT) carries a chiral algebra acting on its Hilbert space; its modules are organised by characters — graded traces that are essentially generating functions of the energy spectrum. A rich class of rational CFTs comes from affine Lie algebras g^k\hat{\mathfrak{g}}_k (a finite-dimensional simple Lie algebra g\mathfrak{g} extended by a central 'level' kk) through the Wess–Zumino–Witten (WZW) model. The coset (GKO) construction divides such a model by a subtheory; its characters are branching functions, the graded multiplicities of the subtheory's modules within the parent theory's modules. Dividing g^k\hat{\mathfrak{g}}_k by its abelian u^(1)\hat{u}(1) currents yields Gepner's parafermion theory PFk(g)\mathrm{PF}_k(\mathfrak{g}), whose fields include parafermions — fractional-spin fields whose operator products contain fractional powers of z−wz-w. At level k=2k=2 with g\mathfrak{g} simply laced, the fractional powers are half-integers and the parafermion theory is realised by finitely many real free fermions with non-trivial mutual operator products, called a coupled free fermion system.

Fermionic character forms. Haldane's fractional exclusion statistics generalises the Pauli principle: quasi-particle species labelled i=1,…,mi=1,\dots,m occupy states subject to mutual statistical interactions encoded by a matrix G\mathbf{G}, whose entry GijG_{ij} measures how each particle of species jj reduces the number of available one-particle states of species ii. The grand partition function of such a gas takes the universal chiral partition function (UCPF) form UCPF(G;a)  =  qδ∑N1,…,Nm≥0q12N TGN−a⋅N∏i(q;q)Ni,\mathrm{UCPF}(\mathbf{G};\mathbf{a}) \;=\; q^\delta \sum_{N_1,\dots,N_m\ge 0} \frac{q^{\frac12\mathbf{N}^{\,T}\mathbf{G}\mathbf{N}-\mathbf{a}\cdot\mathbf{N}}}{\prod_i (q;q)_{N_i}}, where (q;q)n=∏j=1n(1−qj)(q;q)_n=\prod_{j=1}^n(1-q^j), N=(N1,…,Nm)T\mathbf{N}=(N_1,\dots,N_m)^T is the vector of occupation numbers, a\mathbf{a} is a shift vector and δ\delta a prefactor exponent. For WZW models the statistical interaction matrix can be derived from the root system, and an algorithm exists in principle (via 'affinized projective varieties'/monomial ideals); however, that algorithm is documented to become extremely cumbersome from rank four upward, and in the parafermion coset case the coupled fermions make the problem subtler. A necessary consistency condition on any candidate G\mathbf{G} is a dilogarithm identity relating the central charge cc of the CFT to the solutions of ξa=∏b(1−ξb)Gab\xi_a=\prod_b(1-\xi_b)^{G_{ab}}.

What is known. Bouwknegt–Chern–Han analysed PF2(slN)\mathrm{PF}_2(\mathfrak{sl}_N) (the coset sl^(N)2/u^(1)N−1\widehat{\mathfrak{sl}}(N)_2/\hat{u}(1)^{N-1}) for N=2,3,4N=2,3,4. From the intertwiner structure of the coupled fermions — accessible through generalised commutation relations — they conjectured the statistical interaction matrices: G2=(1)\mathbf{G}_2=(1) is trivial, G3=12(211121112)\mathbf{G}_3=\frac12\begin{pmatrix}2&1&1\\1&2&1\\1&1&2\end{pmatrix}, and a 6×66\times 6 matrix G4\mathbf{G}_4. They then proved that the corresponding UCPFs equal (linear combinations of) the parafermionic characters, reproduce characters of a Z2\mathbb{Z}_2-orbifold of the scaled root lattice (1/2)A2(1/\sqrt2)A_2, yield closed Dedekind-η\eta expressions for affine-algebra string functions (the qq-series cλΛ(τ)c^\Lambda_\lambda(\tau) encoding how affine-Lie-algebra modules branch over their Heisenberg subalgebra — generally hard to compute), and imply Rogers–Ramanujan-type identities, i.e. statements that a multisum qq-series equals an infinite product, in the tradition of the classical Rogers–Ramanujan identities whose sum sides count partitions with gap conditions.

What is missing. For N≥5N\ge5 nothing is known. There is no algorithmic or representation-theoretic method that produces GN\mathbf{G}_N; the low-rank matrices were found case by case, and the authors state that determining G\mathbf{G} for higher rank 'analogously' is the obstacle to generalising the whole method, anticipating that additional algebraic tools are needed. Already at N=4N=4 the qq-hypergeometric transforms converting the sextuple multisums into products could only be obtained with reduced generality, and no partition-theoretic (gap-condition) interpretation of any of the multisum sides is known.

Problem Statement

Extend the coupled-free-fermion / universal-chiral-partition-function construction of parafermionic qq-identities from the treated ranks sl2,sl3,sl4\mathfrak{sl}_2,\mathfrak{sl}_3,\mathfrak{sl}_4 to general higher-rank level-2 parafermion theories PF2(slN)\mathrm{PF}_2(\mathfrak{sl}_N) — at least all N≥5N\ge5, and ideally general higher-rank affine Lie algebras — as one program with the following components:

  1. Statistical interaction matrix (the crux). Give a systematic — algorithmic or representation-theoretic — construction of the statistical interaction matrix GN\mathbf{G}_N governing the fractional exclusion statistics of the coupled free fermions realising PF2(slN)\mathrm{PF}_2(\mathfrak{sl}_N), together with a correctness argument. The construction must reproduce the known G2,G3,G4\mathbf{G}_2,\mathbf{G}_3,\mathbf{G}_4 (the latter two of which are currently only conjectured working matrices in the source) as instances, rather than producing them by case-by-case ansatz.

  2. Character identities and string functions. Prove, for the higher-rank theories, the corresponding identities between UCPF(GN;a)\mathrm{UCPF}(\mathbf{G}_N;\mathbf{a}) and the parafermionic characters (branching functions), including closed-form Dedekind-η\eta expressions for the associated affine string functions where they exist.

  3. qq-hypergeometric transforms. Extend the transform machinery to the general quadratic forms N TGNN\mathbf{N}^{\,T}\mathbf{G}_N\mathbf{N} that arise, and derive the corresponding Rogers–Ramanujan-type product identities. As in the source's own N=4N=4 treatment, transforms obtained with a reduced number of free summation parameters — the reduction already necessary there to bring out valid transforms — are acceptable; removing that reduction is not a requirement.

  4. Partition-theoretic interpretation. Characterise the multisum sides partition-theoretically: exhibit explicit difference-at-a-distance conditions (successive parts required to differ by at least a fixed amount) and congruence ('gap') conditions on integer partitions whose generating functions equal the multisum sides, completing the Rogers–Ramanujan-type character of the identities as the source expects.

The common scientific objective is that the method work at general rank as it does for sl2\mathfrak{sl}_2–sl4\mathfrak{sl}_4: component 1 is the enabling core (the authors' stated obstacle) and components 2 and 3 are its direct consequences; together, components 1–3 constitute this problem's completion boundary. Component 4 records the source's expectation that the summation sides of Rogers–Ramanujan-type identities 'would have a similar meaning as partitions, perhaps under analogous but more complicated gap conditions for the parts'; it is a source-discussed expectation rather than a mandatory completion requirement. Two further open problems flagged by the same source — proving the Kanade–Russell modulo-9 partition conjectures, and giving a structural representation-theoretic proof of the equivalence between the PF2(sl4)\mathrm{PF}_2(\mathfrak{sl}_4) coset and the Z2\mathbb{Z}_2-orbifold lattice constructions — are related but are not part of this problem's completion boundary.

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