Free-fermion realisation of the level-$h^{\vee}$ string functions $c^{\hat\rho}_{\hat\rho}$ of $\widehat{\mathfrak{sl}}(n+1)_{n+1}$: Han's Conjectures 2.2 and 2.6 at general rank

Han's 2023 thesis observed that the $\mathfrak{sl}(n+1)$ adjoint-representation free fermions, which realise the conformal embedding $\widehat{\mathfrak{sl}}(n+1)_{n+1}\subset\widehat{\mathfrak{so}}(n^2+2n)_1$, encode in the $e^{\rho}$-graded component of their Ramond-sector character the string function $c^{\hat\rho}_{\hat\rho}$ of $\widehat{\mathfrak{sl}}(n+1)_{n+1}$, and that the modular image of the charge-zero Neveu-Schwarz component is a linear combination of string functions with highest-weight index $\hat\rho$ (Conjectures 2.2 and 2.6 of the thesis). The identities are proven only for ranks $n\le2$, where they reproduce the classical Kac-Peterson eta-quotients (also proved by Andrews via Hecke modular forms), and, on the fermionic side, for $n=3$, yielding the conjectural evaluation $c^{1111}_{1111}=\eta(2\tau)^{15}/(\eta(\tau)^{14}\eta(4\tau)^4)$ for $\widehat{\mathfrak{sl}}(4)_4$. The problem is to prove or refute both conjectures at arbitrary rank and to settle the resulting explicit eta-product structure of these string functions.

Background

Two-dimensional conformal field theory (CFT) has a chiral (holomorphic) sector whose symmetry algebra is infinite-dimensional; the best-understood chiral CFTs are the Wess-Zumino-Witten (WZW) models, whose current algebras are affine Kac-Moody algebras $\widehat{\mathfrak{g}}_k$, central extensions of the Lie algebra of polynomial loops on a finite-dimensional simple Lie algebra $\mathfrak{g}$, labelled by a non-negative integer level $k$. The integrable highest-weight modules $L(\Lambda)$ of $\widehat{\mathfrak{g}}_k$ are indexed by dominant weights $\Lambda$ of level $k$, conventionally written in Dynkin labels, i.e. the coefficients of $\Lambda$ on the fundamental weights of the extended (affine) Dynkin diagram; the sum of the labels equals the level.

Kac and Peterson introduced the string functions $c^{\Lambda}_{\lambda}(\tau)$, with $q=e^{2\pi i\tau}$, which record the graded weight multiplicities of an affine module: for a weight $\lambda$ of $L(\Lambda)$,

$c^{\Lambda}_{\lambda}(\tau)=q^{s_{\Lambda}(\lambda)}\sum_{m\ge0}\mathrm{mult}^{\Lambda}(\lambda-m\delta)\,q^{m},$

where $\delta$ is the imaginary root of the affine algebra, $\mathrm{mult}^{\Lambda}(\mu)$ is the multiplicity of the weight $\mu$, and the prefactor fixes the standard Kac-Peterson normalisation, namely $s_{\Lambda}(\lambda)=s_{\Lambda}-|\lambda|^{2}/(2k)$ and $s_{\Lambda}=|\Lambda+\hat\rho|^{2}/(2(k+h^{\vee}))-|\hat\rho|^{2}/(2h^{\vee})$. String functions are the branching coefficients of $L(\Lambda)$ with respect to the Cartan (Heisenberg) subalgebra and are the building blocks of the characters of the parafermion coset theories $\widehat{\mathfrak{g}}_k/\widehat{\mathfrak{u}(1)}^{\mathrm{rank}\,\mathfrak{g}}$ (quotient conformal field theories whose charged fields, the parafermions, have fractional spin). Kac and Peterson proved that they combine into vector-valued modular forms (families of functions mixing under $\mathrm{SL}_2(\mathbb{Z})$), gave a recursion for computing them, and evaluated a handful of examples in closed form as quotients of the Dedekind eta function $\eta(\tau)=q^{1/24}\prod_{m\ge1}(1-q^{m})$; notably $c^{\hat\rho}_{\hat\rho}$ for $\widehat{\mathfrak{sl}}(2)_2$ and $\widehat{\mathfrak{sl}}(3)_3$, where $\hat\rho=\Lambda_0+\Lambda_1+\cdots+\Lambda_n$ is the affine Weyl vector, a level-$h^{\vee}$ weight for $\mathfrak{sl}(n+1)$ with $h^{\vee}=n+1$ the dual Coxeter number (the shift entering the Sugawara central charge formula). Andrews re-proved these eta-product identities using Hecke modular forms (modular forms whose Fourier coefficients are indexed by ideals in a real quadratic number field). Beyond such low ranks, explicit evaluations are notoriously scarce.

A second classical thread is the realisation of WZW models by free fermions. A system of $d$ real free fermions carries the current algebra $\widehat{\mathfrak{so}}(d)_1$; if the fermions transform in a representation $R$ of a subalgebra $\mathfrak{g}\subset\mathfrak{so}(d)$, their bilinears generate an affine subalgebra $\widehat{\mathfrak{g}}$ at a level fixed by the Dynkin index of $R$ (a representation-theoretic invariant measuring the rescaling of the invariant form on $\mathfrak{g}$ induced by $R$). When the Sugawara stress tensor of $\widehat{\mathfrak{g}}$ (the stress tensor built quadratically from the currents) coincides with the fermionic stress tensor, the central charges agree and one has a conformal embedding. For the adjoint representation of $\mathfrak{sl}(n+1)$, $d=\dim\mathfrak{sl}(n+1)=n^{2}+2n$ and the induced level is $h^{\vee}=n+1$: the bilinears generate $\widehat{\mathfrak{sl}}(n+1)_{n+1}$ and the central charges match ($n(n+2)/2=d/2$), giving the conformal embedding $\widehat{\mathfrak{sl}}(n+1)_{n+1}\subset\widehat{\mathfrak{so}}(n^{2}+2n)_1$, a case of the free-fermion/WZW equivalences analysed in the 1980s (Fuchs 1987). Free fermions have two sectors: the Neveu-Schwarz (NS) sector with half-integer modes, and the Ramond (R) sector with integer modes including zero modes; the R-sector ground state of $d$ real fermions has conformal dimension $d/16$, which coincides with the conformal dimension $h(\hat\rho)=\dim\mathfrak{g}/16$ of the affine Weyl vector at level $h^{\vee}$ (a consequence of the Freudenthal-de Vries strange formula $\langle\rho,\rho\rangle=h^{\vee}\dim\mathfrak{g}/12$, applied to the standard expression $h(\Lambda)=\langle\Lambda,\Lambda+2\rho\rangle/(2(k+h^{\vee}))$). This numerical coincidence is what motivates the problem.

Han's thesis exploits it. Introducing variables $z_i=e^{\alpha_i}$ ($\alpha_i$ the simple roots) to track the Cartan charges of the adjoint fermions, it writes the R-sector graded character (equation (2.8) of the thesis)

$\chi^{(n)}_{R}(z_1,\dots,z_n;q)=q^{-(n^{2}+2n)/48}\,e^{\rho}\prod_{\alpha\in\Delta_+}(1+e^{-\alpha})\prod_{m=1}^{\infty}(1+q^{m})^{n}\prod_{\alpha\in\Delta_+}\bigl(1+e^{\alpha}q^{m}\bigr)\bigl(1+e^{-\alpha}q^{m}\bigr),$

where $\Delta_+$ is the set of positive roots and $\rho=\tfrac12\sum_{\alpha\in\Delta_+}\alpha$ the Weyl vector; the factor $(1+q^{m})^{n}$ counts the $n$ Cartan fermions, the remaining factors the root fermions, and the prefactor $e^{\rho}\prod_{\alpha>0}(1+e^{-\alpha})$ (inspired by the Macdonald-Weyl denominator identity, i.e. the affine Weyl denominator formula $\sum_{w\in\hat W}\det(w)e^{w\hat\rho}=e^{\hat\rho}\prod_{\hat\alpha\in\hat\Delta_+}(1-e^{-\hat\alpha})^{\mathrm{mult}(\hat\alpha)}$) is the R-sector zero-mode structure. The thesis proves, using the classical Jacobi triple product identity (the product expansion $\prod_{m\ge1}(1-q^{m})(1+zq^{m-1/2})(1+z^{-1}q^{m-1/2})=\sum_{k\in\mathbb{Z}}z^{k}q^{k^{2}/2}$), that the coefficient of $e^{\rho}$ in $\chi^{(n)}_{R}$ equals $q^{-(n^{2}+2n)/48}c^{\hat\rho}_{\hat\rho}$ for $n=1$ and $n=2$, its values being $q^{-1/16}\eta(2\tau)/\eta(\tau)^{2}$ for $\widehat{\mathfrak{sl}}(2)_2$ and $q^{-1/6}\eta(2\tau)^{3}\eta(3\tau)^{2}/(\eta(\tau)^{6}\eta(6\tau))$ for $\widehat{\mathfrak{sl}}(3)_3$ (the Dedekind-eta quotient in each value being the Kac-Peterson string function $c^{\hat\rho}_{\hat\rho}$); and, with considerably more work (2-dissections of Ramanujan's theta function, i.e. separations of a q-series into its even- and odd-indexed parts), that for $n=3$ the same coefficient equals $q^{-5/16}\eta(2\tau)^{15}/(\eta(\tau)^{14}\eta(4\tau)^{4})$, conjectured on the basis of its q-expansion to be $q^{-5/16}c^{1111}_{1111}$ of $\widehat{\mathfrak{sl}}(4)_4$. In the NS sector the thesis introduces, alongside (2.8), the two branch generating functions (equation (2.12) of the thesis; the sign choice is the branch, and the thesis works with the minus branch)

$\chi^{(n)}_{NS\pm}(z_1,\dots,z_n;q)=q^{-(n^{2}+2n)/48}\prod_{m=1}^{\infty}(1\pm q^{m-1/2})^{n}\prod_{\alpha\in\Delta_+}\bigl(1\pm e^{\alpha}q^{m-1/2}\bigr)\bigl(1\pm e^{-\alpha}q^{m-1/2}\bigr),$

and verifies for $n\le3$ that the coefficient of $(z_1\cdots z_n)^{0}$ in the minus branch becomes, after the modular transformation $\tau\mapsto-1/\tau$, a linear combination of string functions with highest-weight index $\hat\rho$, for instance $3c^{1111}_{1111}(\tau)+2c^{1111}_{2200}(\tau)+3c^{1111}_{3010}(\tau)$ for $\widehat{\mathfrak{sl}}(4)_4$. Both statements are conjectured for all $n$, with the expectation that the $c^{\hat\rho}_{\hat\rho}$ are (linear combinations of) products of Dedekind eta functions at every rank, and with the suggestion that proofs might follow from dimension bounds on spaces of modular forms.

Problem Statement

Prove or refute, for every rank $n\ge1$, the two conjectures of Han's thesis, in the following self-contained form.

Let $\mathfrak{g}=\mathfrak{sl}(n+1)$ with simple roots $\alpha_1,\dots,\alpha_n$, positive roots $\Delta_+$, and Weyl vector $\rho=\tfrac12\sum_{\alpha\in\Delta_+}\alpha=\sum_{i=1}^{n}\tfrac{i(n+1-i)}{2}\alpha_i$; let $\hat\rho=\Lambda_0+\cdots+\Lambda_n$ (Dynkin labels $(1,\dots,1)$, level $n+1$); write $z_i=e^{\alpha_i}$ and $q=e^{2\pi i\tau}$; and let $c^{\Lambda}_{\lambda}(\tau)$ denote the Kac-Peterson string functions of $\widehat{\mathfrak{sl}}(n+1)_{n+1}$, normalised as $c^{\Lambda}_{\lambda}(\tau)=q^{s_{\Lambda}(\lambda)}\sum_{m\ge0}\mathrm{mult}^{\Lambda}(\lambda-m\delta)\,q^{m}$ with $s_{\Lambda}(\lambda)=s_{\Lambda}-|\lambda|^{2}/(2k)$, $s_{\Lambda}=|\Lambda+\hat\rho|^{2}/(2(k+h^{\vee}))-|\hat\rho|^{2}/(2h^{\vee})$, $k$ the level. Let

$\chi^{(n)}_{R}(z_1,\dots,z_n;q)=q^{-(n^{2}+2n)/48}\,e^{\rho}\prod_{\alpha\in\Delta_+}(1+e^{-\alpha})\prod_{m=1}^{\infty}(1+q^{m})^{n}\prod_{\alpha\in\Delta_+}\bigl(1+e^{\alpha}q^{m}\bigr)\bigl(1+e^{-\alpha}q^{m}\bigr)$

be the Ramond-sector generating function of the $n^{2}+2n$ adjoint-representation fermions, and let

$\chi^{(n)}_{NS}(z_1,\dots,z_n;q)=q^{-(n^{2}+2n)/48}\prod_{m=1}^{\infty}(1-q^{m-1/2})^{n}\prod_{\alpha\in\Delta_+}\bigl(1-e^{\alpha}q^{m-1/2}\bigr)\bigl(1-e^{-\alpha}q^{m-1/2}\bigr)$

be the (minus branch of the) Neveu-Schwarz-sector generating function, the branch the thesis works with (equations (2.8) and (2.12) of the thesis; the Ramond function is also displayed in the Background).

(1) Conjecture 2.2 (Ramond sector). Show that for every $n\ge1$ the coefficient of $e^{\rho}$ in $\chi^{(n)}_{R}(z_1,\dots,z_n;q)$, that is, of $z_1^{1/2}$ for $n=1$, of $z_1z_2$ for $n=2$, of $z_1^{3/2}z_2^{2}z_3^{3/2}$ for $n=3$, and in general of $z_1^{c_1}\cdots z_n^{c_n}$ with $c_i=i(n+1-i)/2$, equals the string function $c^{\hat\rho}_{\hat\rho}(\tau)$ multiplied by the Ramond-sector vacuum-energy factor $q^{-(n^{2}+2n)/48}$ carried by $\chi^{(n)}_{R}$, that is, $q^{(n^{2}+2n)/48}\,\mathcal{O}_{e^{\rho}}^{\chi_{R}^{(n)}}=c^{\hat\rho}_{\hat\rho}(\tau)$; or exhibit a rank at which the identity fails. (The thesis labels this monomial $(z_1\cdots z_n)^{n/2}$, which coincides with $e^{\rho}$ only for $n\le2$; the coefficient extracted in its proofs is that of $e^{\rho}$, and that is the statement to be settled.)

(2) Conjecture 2.6 (Neveu-Schwarz sector). Formulate precisely and prove the general statement that for every $n\ge1$ the coefficient of $(z_1\cdots z_n)^{0}$ in $\chi^{(n)}_{NS}$, after the modular transformation $\tau\mapsto-1/\tau$, is a finite linear combination of string functions $c^{\hat\rho}_{\lambda}(\tau)$ of $\widehat{\mathfrak{sl}}(n+1)_{n+1}$: identify which weights $\lambda$ occur and with which coefficients, and prove the resulting identities; or exhibit a rank at which the coefficient of $(z_1\cdots z_n)^{0}$ after $\tau\mapsto-1/\tau$ is not a finite linear combination of string functions $c^{\hat\rho}_{\lambda}(\tau)$.

(3) Explicit structure. As the expected outcome of (1) and (2), determine the closed-form structure of the string functions so obtained at arbitrary rank, in particular whether $c^{\hat\rho}_{\hat\rho}$ and the Neveu-Schwarz linear combinations are (linear combinations of) finite products of Dedekind eta functions, as at ranks $n\le3$ where $c^{\hat\rho}_{\hat\rho}$ takes the values $\eta(2\tau)/\eta(\tau)^{2}$, $\eta(2\tau)^{3}\eta(3\tau)^{2}/(\eta(\tau)^{6}\eta(6\tau))$ and $\eta(2\tau)^{15}/(\eta(\tau)^{14}\eta(4\tau)^{4})$ for $\widehat{\mathfrak{sl}}(2)_2$, $\widehat{\mathfrak{sl}}(3)_3$ and $\widehat{\mathfrak{sl}}(4)_4$ respectively (the last equality being conjectural, its fermionic side proved).

These parts form one coherent objective, a general-rank theory of the free-fermion realisation of the level-$h^{\vee}$ string functions, with (3) the explicit evaluation that (1) and (2) are expected to deliver. A resolution of the problem is either a proof of both conjectures at arbitrary rank, together with the explicit structure they deliver, or a refutation of either conjecture at an explicit rank, presented as a reproducible computation of the two sides that disagree; progress at a single new rank, or on only one of the two conjectures short of refutation, is partial and does not complete the problem.

References

1. Bolin Han, 'Coupled Free Fermion Conformal Field Theory and Representations', PhD thesis, Australian National University, December 2023, arXiv:2312.06150, https://arxiv.org/abs/2312.06150; thesis summary published as Bulletin of the Australian Mathematical Society 110 (2024) 170, doi:10.1017/S0004972724000224.
2. 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
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