Extending the inner-product construction of Q-functions in the ODE/IM correspondence to complex Bethe roots, coset potentials, and the massive case

The ODE/IM (ordinary differential equation / integrable model) correspondence relates the spectral theory of the linear problems of modified affine Toda field equations to two-dimensional quantum integrable models. Ito, Kondo, Kuroda and Shu (J. Phys. A 54 (2021) 044001) expressed the Q-functions of untwisted affine Lie algebras as inner products of solutions of a dual linear problem and the subdominant solution at infinity, and computed their zeros - the Bethe roots - numerically via Cheng's algorithm, but only for monomial potentials, ground states, and real Bethe roots. The paper left open four directions: twisted affine Lie algebras and excited-state spectra (both involving complex Bethe roots), general polynomial potentials including $p(x,E)=(x^{hM/K}-E)^K$ expected to correspond to GKO coset conformal field theories, and an inner-product formulation of the Q-function for the massive ODE/IM correspondence, where holomorphic and anti-holomorphic equations must be treated simultaneously. All four directions remain open.

Background

The ODE/IM correspondence is an exact relation between the spectral analysis of ordinary differential equations (ODEs) and the functional-relation approach to two-dimensional quantum integrable models (IMs); it was discovered in the late 1990s and is reviewed in Dorey, Dunning and Tateo, arXiv:hep-th/0703066. In its modern form one starts from the modified affine Toda field equation based on an affine Lie algebra $\hat{\mathfrak{g}}$ - the affine Kac-Moody algebra obtained by extending the Dynkin diagram of a simple Lie algebra $\mathfrak{g}$ by one extra node - and studies its auxiliary linear problem, a first-order system of differential equations. In the massless limit this reduces to an ODE with an irregular singularity at infinity and a regular singularity at the origin, whose potential has the monomial form $p(x,E)=x^{hM}-E$, where $h$ is the Coxeter number of $\mathfrak{g}$ (the number of roots divided by the rank), $M$ is a positive integer, and $E$ is the spectral parameter.

On the integrable-model side, the states of a two-dimensional quantum integrable model are characterized by Baxter's Q-functions, whose zeros (the Bethe roots) solve the Bethe ansatz equations (BAEs) and determine the spectrum. The ODE/IM correspondence states that these Q-functions are encoded in the connection coefficients between the bases of solutions of the ODE at its two singularities; equivalently, Wronskians of solutions satisfy the $\psi$-system, a set of quadratic relations among subdeterminants of the solution matrix from which the BAEs follow. A standard non-perturbative way to solve the BAEs is to convert them into non-linear integral equations (NLIEs) of Destri-de Vega type and iterate numerically.

In the source paper, Ito, Kondo, Kuroda and Shu (J. Phys. A 54 (2021) 044001) obtained, for untwisted affine Lie algebras (the affine algebras $X_r^{(1)}$ obtained by the trivial one-node extension of a simple Lie algebra), including the exceptional types, a new representation of the Q-functions as the inner product of a solution of the dual linear problem (the linear system built on the dual representation, expanded around the origin) and the subdominant solution at infinity (in each Stokes sector - an angular region of the complex $x$-plane where the WKB exponential solutions have a definite decay ordering - there is a unique fastest-decaying solution). The power-series solution at the origin was generated by Cheng's algorithm, a classical recursive series method for differential equations near a regular singularity introduced by Hung Cheng in 1962 in S-matrix theory. This made the zeros of the Q-functions directly and efficiently computable from the ODE side; the results agree with NLIE solutions for the simply-laced algebras, and the construction was extended to the non-simply-laced untwisted algebras $B_r^{(1)}$, $C_r^{(1)}$, $F_4^{(1)}$ and $G_2^{(1)}$ by folding Dynkin diagrams (identifying nodes related by a symmetry of the diagram).

The construction is limited in four ways, each explicitly flagged by the authors as an open direction. First, twisted affine Lie algebras - the affine algebras $A_{2n}^{(2)}$, $A_{2n-1}^{(2)}$, $D_{n+1}^{(2)}$, $E_6^{(2)}$ and $D_4^{(3)}$ obtained by folding with a nontrivial Dynkin-diagram automorphism - are excluded; their linear problems can be obtained by folding procedures and the corresponding BAEs were constructed earlier, but in these cases the Bethe roots are complex rather than real, and the numerical method, which relies on real zeros and the asymptotic behavior of the Q-functions in the complex $E$-plane, has not been extended to them. Second, only ground-state spectra are treated; excited-state spectra are deferred because the Bethe roots become complicated and the NLIE approach is rather difficult to apply (for the simplest, second-order ODE case, the excited-state theory was developed by Bazhanov, Lukyanov and Zamolodchikov through higher-level Q-operators, Adv. Theor. Math. Phys. 7 (2003) 711). Third, the potential is a monomial; general polynomial potentials, in particular $p(x,E)=(x^{hM/K}-E)^K$, are expected to correspond to the GKO (Goddard-Kent-Olive) coset construction - which builds unitary minimal conformal field theories as cosets of affine current algebras - at level $K$, and the BAEs for these cases are known for classical Lie algebras, but the ODE-side construction is missing. Fourth, only the massless linear problem is treated; the massive ODE/IM correspondence - relating massive integrable quantum field theories such as the sinh-Gordon model to radial linear problems, established by Lukyanov and Zamolodchikov and generalized to other affine Toda theories - has been formulated through $\psi$-systems, Wronskians and NLIEs, but an inner-product representation of the Q-function, which would require the holomorphic and anti-holomorphic linear equations to be treated simultaneously, remains to be found.

Problem Statement

Extend the inner-product construction of Q-functions and the associated numerical determination of their zeros from the linear problems of modified affine Toda equations - established by Ito, Kondo, Kuroda and Shu for untwisted affine Lie algebras with monomial potentials $p(x,E)=x^{hM}-E$ and real, ground-state Bethe roots - to the settings that paper explicitly excluded. The common objective is to make the ODE-side linear problem a direct, constructive and numerically verifiable source of the Bethe-root spectra of the corresponding integrable models, in the classes where this is currently unavailable:

1. Twisted affine Lie algebras: construct the Q-functions of the twisted linear problems (obtainable by the folding procedures of Masoero, Raimondo and Valeri) as inner products of dual-linear-problem solutions and subdominant solutions, and determine their zeros - the complex Bethe roots - numerically, validating against the known twisted Bethe ansatz equations.

2. Excited states: extend the construction to excited-state spectra (higher-level Q-functions, i.e. the Q-functions at the higher levels of the nested hierarchy of functional relations, whose zeros describe the excited states), where the Bethe roots are again complex and the non-linear-integral-equation approach is difficult to apply.

3. Polynomial potentials: generalize the potential from the monomial $p(x,E)=x^{hM}-E$ to general polynomials, in particular $p(x,E)=(x^{hM/K}-E)^K$, construct the corresponding Q-functions and their zeros, and test the conjectured correspondence with the GKO coset model at level $K$.

4. Massive correspondence: find an inner-product representation of the Q-function for the massive ODE/IM correspondence, in which the holomorphic and anti-holomorphic linear equations are treated simultaneously, reducing in the massless limit to the known construction.

A complete answer to the problem as stated requires the inner-product Q-function construction and the numerical determination of its zeros to be obtained for all four classes below; results covering only some of the classes are partial progress, not a complete solution.

References

1. Katsushi Ito, Takayasu Kondo, Kohei Kuroda, Hongfei Shu, 'ODE/IM correspondence for affine Lie algebras: A numerical approach', Journal of Physics A: Mathematical and Theoretical 54 (2021) 044001, DOI:10.1088/1751-8121/abd21e, arXiv:2004.09856, https://arxiv.org/abs/2004.09856
2. Patrick Dorey, Clare Dunning, Roberto Tateo, 'The ODE/IM Correspondence', arXiv:hep-th/0703066, https://arxiv.org/abs/hep-th/0703066
3. P. Dorey, C. Dunning, D. Masoero, J. Suzuki, R. Tateo, 'Pseudo-differential equations, and the Bethe ansatz for the classical Lie algebras', Nuclear Physics B 772 (2007) 249, DOI:10.1016/j.nuclphysb.2007.02.029, arXiv:hep-th/0612298, https://arxiv.org/abs/hep-th/0612298
4. Juanjuan Sun, 'Polynomial relations for q-characters via the ODE/IM correspondence', SIGMA 8 (2012) 028, DOI:10.3842/SIGMA.2012.028, arXiv:1201.1614, https://arxiv.org/abs/1201.1614
5. Davide Masoero, Andrea Raimondo, Daniele Valeri, 'Bethe Ansatz and the Spectral Theory of Affine Lie Algebra-Valued Connections I. The simply-laced Case', Communications in Mathematical Physics 344 (2016) 719, DOI:10.1007/s00220-016-2643-6, arXiv:1501.07421, https://arxiv.org/abs/1501.07421
6. Davide Masoero, Andrea Raimondo, Daniele Valeri, 'Bethe Ansatz and the Spectral Theory of Affine Lie algebra-Valued Connections II: The Non Simply-Laced Case', Communications in Mathematical Physics 349 (2017) 1063, DOI:10.1007/s00220-016-2744-2, arXiv:1511.00895, https://arxiv.org/abs/1511.00895
7. Katsushi Ito, Christopher Locke, 'ODE/IM correspondence and Bethe ansatz for affine Toda field equations', Nuclear Physics B 896 (2015) 763, DOI:10.1016/j.nuclphysb.2015.05.016, arXiv:1502.00906, https://arxiv.org/abs/1502.00906
8. S. L. Lukyanov, A. B. Zamolodchikov, 'Quantum Sine(h)-Gordon Model and Classical Integrable Equations', JHEP 07 (2010) 008, DOI:10.1007/JHEP07(2010)008, arXiv:1003.5333, https://arxiv.org/abs/1003.5333
9. P. Dorey, S. Faldella, S. Negro, R. Tateo, 'The Bethe Ansatz and the Tzitzeica-Bullough-Dodd equation', Philosophical Transactions of the Royal Society A 371 (2013) 20120052, DOI:10.1098/rsta.2012.0052, arXiv:1209.5517, https://arxiv.org/abs/1209.5517
10. Panagiota Adamopoulou, Clare Dunning, 'Bethe Ansatz equations for the classical $A_n^{(1)}$ affine Toda field theories', Journal of Physics A 47 (2014) 205205, DOI:10.1088/1751-8113/47/20/205205, arXiv:1401.1187, https://arxiv.org/abs/1401.1187
11. Katsushi Ito, Hongfei Shu, 'Massive ODE/IM Correspondence and Non-linear Integral Equations for $A_r^{(1)}$-type modified Affine Toda Field Equations', Journal of Physics A 51 (2018) 385401, DOI:10.1088/1751-8121/aad63f, arXiv:1805.08062, https://arxiv.org/abs/1805.08062
12. V. V. Bazhanov, S. L. Lukyanov, A. B. Zamolodchikov, 'Higher-level eigenvalues of Q-operators and Schrodinger equation', Advances in Theoretical and Mathematical Physics 7 (2003) 711, DOI:10.4310/ATMP.2003.v7.n4.a4, arXiv:hep-th/0307108, https://arxiv.org/abs/hep-th/0307108
13. Hung Cheng, 'Meromorphic Property of the S Matrix in the Complex Plane of Angular Momentum', Physical Review 127 (1962) 647, DOI:10.1103/PhysRev.127.647, https://doi.org/10.1103/PhysRev.127.647
14. Davide Masoero, Andrea Raimondo, 'Opers for higher states of quantum KdV models', Communications in Mathematical Physics 378 (2020) 1, DOI:10.1007/s00220-020-03792-3, arXiv:1812.00228, https://arxiv.org/abs/1812.00228
15. Davide Masoero, Andrea Raimondo, 'Opers for higher states of the quantum Boussinesq model', in 'Asymptotic, Algebraic and Geometric Aspects of Integrable Systems', Springer Proceedings in Mathematics and Statistics (2020) 55, DOI:10.1007/978-3-030-57000-2_5, arXiv:1908.11559, https://arxiv.org/abs/1908.11559
16. Davide Masoero, Andrea Raimondo, 'Feigin-Frenkel-Hernandez Opers and the QQ-System', Communications in Mathematical Physics 405 (2024) 193, DOI:10.1007/s00220-024-05064-w, arXiv:2312.01955, https://arxiv.org/abs/2312.01955
17. Davide Masoero, Giulio Ruzza, 'ODE/IM Correspondence at the Free-Fermion Point. Laguerre Wronskians, Shifted Symmetric Functions, and Quantum KdV', arXiv:2605.24563, https://arxiv.org/abs/2605.24563
18. Katsushi Ito, Marcos Mariño, Hongfei Shu, 'TBA equations and resurgent Quantum Mechanics', JHEP 01 (2019) 228, DOI:10.1007/JHEP01(2019)228, arXiv:1811.04812, https://arxiv.org/abs/1811.04812
19. Katsushi Ito, Hongfei Shu, 'TBA equations for the Schrödinger equation with a regular singularity', Journal of Physics A 53 (2020) 335201, DOI:10.1088/1751-8121/ab96ee, arXiv:1910.09406, https://arxiv.org/abs/1910.09406
20. Katsushi Ito, Takayasu Kondo, Kohei Kuroda, Hongfei Shu, 'WKB periods for higher order ODE and TBA equations', JHEP 10 (2021) 167, DOI:10.1007/JHEP10(2021)167, arXiv:2104.13680, https://arxiv.org/abs/2104.13680
21. Katsushi Ito, Takayasu Kondo, Hongfei Shu, 'Wall-crossing of TBA equations and WKB periods for the third order ODE', Nuclear Physics B 979 (2022) 115788, DOI:10.1016/j.nuclphysb.2022.115788, arXiv:2111.11047, https://arxiv.org/abs/2111.11047
22. Katsushi Ito, Mingshuo Zhu, 'ODE/IM correspondence and supersymmetric affine Toda field equations', Nuclear Physics B 985 (2022) 116004, DOI:10.1016/j.nuclphysb.2022.116004, arXiv:2206.08024, https://arxiv.org/abs/2206.08024
23. Katsushi Ito, Mingshuo Zhu, 'WKB analysis of the linear problem for modified affine Toda field equations', JHEP 08 (2023) 007, DOI:10.1007/JHEP08(2023)007, arXiv:2305.03283, https://arxiv.org/abs/2305.03283
24. Katsushi Ito, Mingshuo Zhu, 'Integrals of motion in conformal field theory with W-symmetry and the ODE/IM correspondence', Nuclear Physics B 1010 (2025) 116756, DOI:10.1016/j.nuclphysb.2024.116756, arXiv:2408.12917, https://arxiv.org/abs/2408.12917
25. Daichi Ide, Katsushi Ito, Wataru Kono, 'Integrals of motion in $W E_6$ CFT and the ODE/IM correspondence', Journal of High Energy Physics 07 (2026) 173, DOI:10.1007/JHEP07(2026)173, arXiv:2604.07829, https://arxiv.org/abs/2604.07829
26. Naozumi Tanabe, 'The ODE/IM Correspondence between $C(2)^{(2)}$-type Linear Problems and 2d $\mathcal{N}=1$ SCFT', Journal of High Energy Physics 08 (2026) 150, DOI:10.1007/JHEP08(2026)150, arXiv:2604.14899, https://arxiv.org/abs/2604.14899
27. Fabrizio Aramini, Nicolò Brizio, Stefano Negro, Roberto Tateo, 'Deforming the ODE/IM correspondence with $T\bar T$', Journal of High Energy Physics 03 (2023) 084, DOI:10.1007/JHEP03(2023)084, arXiv:2212.13957, https://arxiv.org/abs/2212.13957
28. Katsushi Ito, Hongfei Shu, 'ODE/IM Correspondence and Quantum Periods', Springer, Singapore (2025), DOI:10.1007/978-981-96-0499-9, https://doi.org/10.1007/978-981-96-0499-9