A two-dimensional conformal field theory (2D CFT) with a holomorphic U(1) current possesses an infinite family of commuting conserved charges, the quantum Benjamin-Ono2_2 (qBO2_2) hierarchy, and its generalized thermalization is described by a Generalized Gibbs Ensemble (GGE) with one chemical potential per charge. Novaes (JHEP 05 (2021) 276) computed this GGE at finite central charge when, among the higher qBO2_2 charges, only the chemical potential η2\eta_2 of the first non-trivial one is non-zero (purely imaginary, on the branch where all charges are real) and found that the descendant contribution to the partition function is exactly the square of an infinite product of partial theta functions. The paper explicitly left open the systematic study of this object: its modular transformation properties, its asymptotics, and its thermodynamic implications, including the suggested connections to false theta functions, quantum modular forms, anisotropic and non-relativistic (Lifshitz) CFTs, and the question whether thermal correlators of qBO2_2 charges obey modular differential equations analogous to the KdV case. The problem remains open: the neighboring Virasoro/KdV, W3\mathcal{W}_3, symplectic-fermion, and universal vertex-operator-algebra cases progressed in 2025-2026, but the U(1)-charged partial-theta GGE itself is untouched, and Downing and Karimi (SciPost Phys. 18 (2025) 085) state that studying the modular properties of BO2_2 GGEs remains an open question.

Background

A two-dimensional conformal field theory (2D CFT) is a quantum field theory invariant under local angle-preserving transformations; its symmetry is the Virasoro algebra with a central charge cc counting effective degrees of freedom, parametrized in the Liouville convention by c=1+6(b+b−1)2c=1+6(b+b^{-1})^{2} with bb the Liouville parameter. Many 2D CFTs of interest also carry a holomorphic U(1) current J(z)J(z), a conserved spin-one symmetry current whose zero mode measures an internal charge. Such a theory possesses, besides the Hamiltonian, an infinite family of mutually commuting conserved charges QkQ_k (k=0,1,2,…k=0,1,2,\dots, with Q0Q_0 the U(1) charge and Q1Q_1 the energy) known as the quantum Benjamin-Ono2_2 (qBO2_2) integrable hierarchy. This hierarchy arises naturally in the Alday-Gaiotto-Tachikawa (AGT) correspondence between 2D Liouville-type CFTs and 4D supersymmetric gauge theory: descendants of a primary are labeled by two Young diagrams (λ,μ)(\lambda,\mu) (the standard graphical encoding of integer partitions) and the Alba-Fateev-Litvinov-Tarnopolskiy (AFLT) basis of such descendants diagonalizes all qBO2_2 charges simultaneously. The qBO2_2 hierarchy is related through the quantum intermediate-long-wave (qILW2_2) hierarchy (Litvinov 2013) to the quantum KdV hierarchy, which is built from the stress tensor alone.

When a quantum system has many conserved quantities, its long-time equilibration is described not by the canonical ensemble but by a Generalized Gibbs Ensemble (GGE), in which every independent charge QkQ_k enters with its own chemical potential ηk\eta_k: Z(η)=Tr exp⁡(∑kηkQk)Z(\boldsymbol{\eta})=\mathrm{Tr}\,\exp(\sum_k \eta_k Q_k). For 2D CFTs quantized on a torus with spatial circumference ℓ\ell and inverse temperature β\beta, the modular parameter is τ0=iπℓ/β\tau_0=i\pi\ell/\beta, and the modular SS-transform τ0→−1/τ0\tau_0\to-1/\tau_0 exchanges the thermal and spatial circles. For ordinary partition functions this modularity underlies Cardy-type state-counting asymptotics and exact thermodynamics; inserting higher-spin charges destroys holomorphicity, and the modular behavior of the resulting ensembles is a long-standing problem. In the qKdV case, thermal correlation functions of the charges were shown by Maloney, Ng, Ross and Tsiares to obey modular differential equations (differential equations in the modular parameter that transform covariantly under the modular group), and the modular SS-transform of KdV-charged GGEs was recently obtained asymptotically by Downing and Karimi (SciPost Phys. 18 (2025) 085) by expanding the GGE into thermal correlators of the zero modes of quasi-primary fields (fields that transform covariantly, though not as primaries, under conformal transformations), transforming each, and resumming; the resummed transformed GGE is a formal, generally divergent object that they regularized through thermodynamic Bethe ansatz (TBA) equations (a system of integral equations encoding the finite-size spectrum) in the Lee-Yang model (the simplest non-unitary minimal-model CFT). The W3\mathcal{W}_3 analogue was proposed by Downing et al. (JHEP 06 (2026) 114); exact SS-transforms were derived for symplectic-fermion GGEs by Karimi and Watts; a universal asymptotic formula for partition functions with zero modes of holomorphic currents inserted was proven by Ashok et al. (JHEP 08 (2026) 168); and the general modularity of vertex-operator-algebra correlators with zero modes was established by Addabbo and Keller in the rational C2C_2-cofinite setting. None of these covers a theory with a U(1) charge and qBO2_2 charges.

The starting point of the present problem is the key finite-cc result of Novaes (JHEP 05 (2021) 276). On the branch of the spectrum where the Liouville momentum is real and the U(1)-charge parameter is imaginary (identified in the source as the only branch on which all qBO2_2 charges can be made real, and then only for purely imaginary η2\eta_2), the sum over AFLT descendants in the GGE partition function factorizes exactly: it equals the square of an infinite product over modes m≥1m\ge 1 of partial theta functions. A partial theta function is a theta-type series summed over non-negative integers only, ∑N≥0qa(N)\sum_{N\ge 0} q^{a(N)} instead of over all integers; such series are not modular forms but are closely tied to false theta functions and to quantum modular forms, whose modular transformation behavior holds only in radial asymptotics or at roots of unity (Bringmann and Milas 2015; Bringmann and Nazaroglu 2019). In the classical limit b→0b\to 0 the descendant product reduces to ∏m≥1(1−e2πiτ0(m+η~2m2/2))−1\prod_{m\ge 1}(1-e^{2\pi i\tau_0(m+\tilde\eta_2 m^{2}/2)})^{-1}, the partition function of a Lifshitz (anisotropic, non-relativistic) free boson with dynamical exponent 2, whose state-counting asymptotics were analyzed by Melnikov, Novaes, Perez and Troncoso. Novaes' paper concludes by calling for a systematic study of the modular transformations, asymptotics, and thermodynamic implications of these partial-theta contributions, and asks whether qBO2_2 thermal correlators obey modular differential equations like the qKdV ones.

Problem Statement

The problem is the systematic modular-asymptotic analysis of the finite-cc qBO2_2 Generalized Gibbs Ensemble that Novaes (JHEP 05 (2021) 276) left open: 'partial theta functions appear, which have modular properties related to false theta functions and quantum modular forms ... [these] can be relevant in the studies of anisotropic field theories and non-relativistic CFTs', and 'it would also be worthwhile to investigate if the qBO2_2 correlation functions obey similar modular equations' to the qKdV ones.

The starting object is pinned by the source formulas. With only the first non-trivial chemical potential η2\eta_2 non-zero (purely imaginary) and, besides the inverse temperature β\beta, the U(1) chemical potential η\eta also turned on (purely imaginary as well, and entering only through the primary-sector weights), the descendant contribution to the finite-cc GGE partition function is

Zdesc  =  [∏m=1∞  ∑N=0∞exp⁡(2πi τ0 (am′N+b2η~22 m N2))]2Z_{\mathrm{desc}} \;=\; \Big[\prod_{m=1}^{\infty}\;\sum_{N=0}^{\infty}\exp\Big(2\pi i\,\tau_0\,\big(a'_m N+\tfrac{b^{2}\tilde\eta_2}{2}\,m\,N^{2}\big)\Big)\Big]^{2}

where τ0=iπℓ/β\tau_0=i\pi\ell/\beta is the torus modular parameter (ℓ\ell the spatial circumference), η~2=(π/b) η2/β2\tilde\eta_2=(\pi/b)\,\eta_2/\beta^{2} is the rescaled chemical potential used in the source, bb is the Liouville parameter, and am′a'_m is the image of am=m+η~22m2a_m=m+\tfrac{\tilde\eta_2}{2}m^{2} under the change of basis a′=a⋅Pa'=a\cdot P that diagonalizes the AFLT quadratic form (PtAP=DP^{t}AP=D with Dmn=m δmnD_{mn}=m\,\delta_{mn}); the full partition function is the primary-sector sum weighted by this product. The problem asks for the following four parts, each of which is independently answerable and none of which is a prerequisite for the others, so that a resolution may settle any one part without settling the rest:

(i) the modular SS-transform of this finite-cc partition function under τ0→−1/τ0\tau_0\to-1/\tau_0 - the exact transform if one exists, and otherwise a controlled asymptotic expansion - and in particular whether the transformed object can be resummed or regularized into a well-defined dual-channel ensemble (in the Virasoro/KdV case the naive resummation is only a divergent formal GGE, regulated there by TBA equations that have no known counterpart for qBO2_2);

(ii) the asymptotics and thermodynamic implications of the partial-theta descendant contributions: the high-temperature behavior of the free energy, convergence and boundedness as a function of η~2\tilde\eta_2, and the connection to the b→0b\to 0 classical limit ∏m≥1(1−e2πiτ0(m+η~2m2/2))−1\prod_{m\ge 1}(1-e^{2\pi i\tau_0(m+\tilde\eta_2 m^{2}/2)})^{-1} and to anisotropic and non-relativistic (Lifshitz) CFTs;

(iii) the number-theoretic structure: which false-theta or quantum-modular transformation properties this specific product of partial theta functions enjoys, and what these imply for the ensemble's thermodynamics and for operator averages;

(iv) as the linked correlator counterpart of the same modular structure, whether thermal correlation functions of the qBO2_2 charges obey modular differential equations analogous to those satisfied by the qKdV charges.

References

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