Bloch and Okounkov (2000) studied the character of the infinite wedge (fermionic Fock space) representation of the central extension of the Lie algebra of differential operators on the circle. For the half-integer shift u=−12u=-\tfrac12 they proved the character is quasimodular, and they computed its nn-point correlation functions --- qq-expectations of products of shifted power sums over partitions --- as permutation sums of determinants built from a genus-1 theta function and its derivatives. They explicitly left two conceptual questions open: they could find no precise connection between these determinantal theta formulas and any vertex-operator, conformal-field-theory, or QFT construction, and they could give no conceptual reason why the particular shift u=−12u=-\tfrac12 yields quasimodularity, beyond observing that half-integer shifts give the underlying module a nondegenerate symmetric bilinear form. Subsequent work forms a 2000-2026 lineage: Milas reformulated the nn-point functions as constant terms of genus-1 (torus) free-fermion vertex-algebra traces; Cheng-Wang and coauthors interpreted and generalized the formulas in terms of higher-level and twisted modules; Zagier gave a short proof and an sl2\mathfrak{sl}_2-equivariant reformulation; van Ittersum and others developed the theory of quasimodular algebras of qq-brackets; and Ittersum-Ruzza and Masoero-Ruzza connected the shifted-symmetric framework to quantum KdV spectra. None of this derives the determinantal theta formula itself from field-theoretic principles, and none explains the special role of u=−12u=-\tfrac12; the problem therefore remains open, with a documented partial narrowing of its first part.

Background

Let D\mathcal{D} be the Lie algebra of differential operators on the circle, with basis tmDnt^m D_n (m∈Zm\in\mathbb{Z}, n≥0n\ge0), where Dn=(t d/dt)nD_n=(t\,d/dt)^n acts on the ring C[t,t−1]\mathbb{C}[t,t^{-1}] of Laurent polynomials. It contains the Witt algebra of vector fields on the punctured plane (spanned by tmD1t^mD_1), whose central extension is the Virasoro algebra, and it admits a nontrivial graded central extension D^\widehat{\mathcal{D}} (Kac-Radul). A basic projective representation of D^\widehat{\mathcal{D}} lives on the infinite wedge space Λ∞/2V\Lambda^{\infty/2}V (also called the half-infinite wedge or fermionic Fock space): the semi-infinite exterior power of V=C[t,t−1]tuV=\mathbb{C}[t,t^{-1}]t^u, spanned by Dirac-sea wedges. The charge-zero subspace has a natural basis indexed by integer partitions λ\lambda (non-increasing sequences λ1≥λ2≥⋯≥0\lambda_1\ge\lambda_2\ge\cdots\ge0 with finite sum ∣λ∣|\lambda|), placed at the exponents λi−i+12\lambda_i-i+\tfrac12 when u=−12u=-\tfrac12; the energy grading of these exponents recovers ∣λ∣|\lambda|, so traces over the charge-zero sector become sums over partitions weighted by q∣λ∣q^{|\lambda|}.

Bloch and Okounkov studied the character of this representation: the graded trace Ω(q0,q1,…)=Trace(q0D~0q1D~1⋯ )\Omega(q_0,q_1,\ldots)=\mathrm{Trace}(q_0^{\widetilde D_0}q_1^{\widetilde D_1}\cdots) of lifted commuting operators from an abelian subalgebra of D\mathcal{D}, normalized by anomaly factors q2r+1−ξ(−2r−1)q_{2r+1}^{-\xi(-2r-1)} (r≥0r\ge0, odd indices only) with ξ(s)=∑n≥1(n−12)−s=(2s−1)ζ(s)\xi(s)=\sum_{n\ge1}(n-\tfrac12)^{-s}=(2^s-1)\zeta(s), which zeta-regularize the divergent sums and are uniquely forced by quasimodularity. A quasimodular form is a holomorphic function on the upper half plane (with q=e2πiτq=e^{2\pi i\tau}) that transforms under SL2(Z)\mathrm{SL}_2(\mathbb{Z}) up to an explicit polynomial in (cτ+d)−1(c\tau+d)^{-1}; equivalently, an element of the ring C[G2,G4,G6]\mathbb{C}[G_2,G_4,G_6] generated by Eisenstein series, the weight-2 generator G2G_2 being the prototypical quasimodular object. The backdrop: characters of rational conformal field theories (modules of the Virasoro algebra, or more generally of vertex operator algebras --- algebras of meromorphic fields whose graded traces are genus-1 conformal blocks, per Zhu's theorem) are modular forms; Dijkgraaf, motivated by mirror symmetry of elliptic curves, suggested and Kaneko-Zagier proved in special cases that partition-counting series attached to the differential-operator algebra are quasimodular. Bloch and Okounkov proved the general statement: for u=−12u=-\tfrac12 the character is quasimodular (weight 0 for Ω\Omega, weight −12-\tfrac12 for its partition specialization), which in modern language says that the qq-bracket ⟨f⟩q=∑λf(λ)q∣λ∣/∑λq∣λ∣\langle f\rangle_q=\sum_\lambda f(\lambda)q^{|\lambda|}/\sum_\lambda q^{|\lambda|} of every shifted symmetric function --- a function on partitions symmetric in the shifted variables λi−i\lambda_i-i, the algebra generated by the shifted power sums --- is a quasimodular form. This is now called the Bloch-Okounkov theorem.

They also computed the nn-point correlation functions F(t1,…,tn)=⟨∏k=1n∑i≥1tk λi−i+1/2⟩q,F(t_1,\ldots,t_n)=\Big\langle\prod_{k=1}^n\sum_{i\ge1}t_k^{\,\lambda_i-i+1/2}\Big\rangle_q, the expectation of a product of nn generating series over the partition measure, and proved the closed formula F(t1,…,tn)=∑σ∈Sndet⁡(Θ(j−i+1)(tσ(1)⋯tσ(n−j))(j−i+1)!)i,j=1nΘ(tσ(1)) Θ(tσ(1)tσ(2))⋯Θ(tσ(1)⋯tσ(n)),F(t_1,\ldots,t_n)=\sum_{\sigma\in S_n}\frac{\det\Big(\dfrac{\Theta^{(j-i+1)}\big(t_{\sigma(1)}\cdots t_{\sigma(n-j)}\big)}{(j-i+1)!}\Big)_{i,j=1}^{n}}{\Theta\big(t_{\sigma(1)}\big)\,\Theta\big(t_{\sigma(1)}t_{\sigma(2)}\big)\cdots\Theta\big(t_{\sigma(1)}\cdots t_{\sigma(n)}\big)}, where Θ(t)=η(q)−3∑n∈Z(−1)nq(n+1/2)2/2tn+1/2\Theta(t)=\eta(q)^{-3}\sum_{n\in\mathbb{Z}}(-1)^n q^{(n+1/2)^2/2}t^{n+1/2} is a genus-1 (Jacobi) theta function, Θ(p)=(t d/dt)pΘ\Theta^{(p)}=(t\,d/dt)^p\Theta, and Θ(p)=0\Theta^{(p)}=0 for p<0p<0. For example F(t)=1/Θ(t)F(t)=1/\Theta(t). Both by their definition (averages of products) and by their analytic structure (permutation sums of determinants of theta derivatives, Wick-theorem style), these functions closely resemble correlation functions of a quantum field theory on a torus; the paper's own proof, however, uses solely classical analysis (theta transformation laws). The whole quasimodular statement depends on the half-integer shift u=−12u=-\tfrac12 in V=C[t,t−1]tuV=\mathbb{C}[t,t^{-1}]t^u; the paper's only structural observation about this is that for u∈Z+12u\in\mathbb{Z}+\tfrac12 the space VV carries the nondegenerate symmetric bilinear form (tn+u,tm+u)=rest=0 tn+m+2u−1dt(t^{n+u},t^{m+u})=\mathrm{res}_{t=0}\,t^{n+m+2u-1}dt, compatible with an involution of D\mathcal{D}. These are the two conceptual gaps the source itself declares, and they form one coherent objective: a structural, ideally field-theoretic, account of where the theta functions, the determinant structure, and the quasimodularity of the infinite wedge character come from.

Problem Statement

The problem, as left open by Bloch and Okounkov in 'The Character of the Infinite Wedge Representation' (Adv. Math. 149 (2000) 1-60), is to supply the conceptual account that their analytic proof lacks, for the infinite wedge representation of the central extension D^\widehat{\mathcal{D}} of the Lie algebra of differential operators on the circle, with V=C[t,t−1]tuV=\mathbb{C}[t,t^{-1}]t^u and u=−12u=-\tfrac12:

(1) Find the precise connection to quantum field theory suggested by the nn-point correlation functions F(t1,…,tn)=⟨∏k=1n∑i≥1tk λi−i+1/2⟩qF(t_1,\ldots,t_n)=\Big\langle\prod_{k=1}^n\sum_{i\ge1}t_k^{\,\lambda_i-i+1/2}\Big\rangle_q (qq-expectations over partitions, q∣λ∣q^{|\lambda|} measure) and their closed evaluations as permutation sums of determinants built from the genus-1 theta function Θ(t)=η(q)−3∑n∈Z(−1)nq(n+1/2)2/2tn+1/2\Theta(t)=\eta(q)^{-3}\sum_{n\in\mathbb{Z}}(-1)^nq^{(n+1/2)^2/2}t^{n+1/2} and its logarithmic derivatives. In particular, the question is whether an explicit vertex-operator-algebraic (or other chiral CFT, path-integral, or geometric) construction exists whose torus correlation functions produce these functions and from which the determinantal theta formula itself is derived, beyond a reformulation of the functions as constant terms of such correlators or a verification of the formula by classical analysis. The source's own words: the functions 'closely resemble correlation functions in QFT. However, we were not able to find any precise connection.'

(2) Explain conceptually, in representation-theoretic or physical terms, why the particular shift u=−12u=-\tfrac12 yields a quasimodular character --- the paper proves quasimodularity at this shift but states that 'the reason why this particular value of uu yields a quasimodular character is not clear', offering only the observation that for half-integer uu the space VV carries a nondegenerate symmetric bilinear form (tn+u,tm+u)=rest=0 tn+m+2u−1dt(t^{n+u},t^{m+u})=\mathrm{res}_{t=0}\,t^{n+m+2u-1}dt compatible with an involution of D\mathcal{D}.

The two questions form one scientific objective: a structural explanation of the quasimodular theta-determinant structure of the infinite wedge character, in place of its original purely analytic derivation --- not a re-proof of quasimodularity by new means, or a generalization of the formulas to other settings (other levels, other measure classes, other subgroups).

References

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