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Pseudo-differential equations, and the Bethe Ansatz for the classical Lie algebras

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arxiv hep-th/0612298 v2 pith:7OL67ZW5 submitted 2006-12-29 hep-th math-phmath.MP

Pseudo-differential equations, and the Bethe Ansatz for the classical Lie algebras

classification hep-th math-phmath.MP
keywords equationsansatzbetheclassicalgeneralisedpseudo-differentialalgebrascorrespondence
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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The correspondence between ordinary differential equations and Bethe ansatz equations for integrable lattice models in their continuum limits is generalised to vertex models related to classical simple Lie algebras. New families of pseudo-differential equations are proposed, and a link between specific generalised eigenvalue problems for these equations and the Bethe ansatz is deduced. The pseudo-differential operators resemble in form the Miura-transformed Lax operators studied in work on generalised KdV equations, classical W-algebras and, more recently, in the context of the geometric Langlands correspondence. Negative-dimension and boundary-condition dualities are also observed.

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Cited by 5 Pith papers

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  1. The ODE/IM Correspondence between $C(2)^{(2)}$-type Linear Problems and 2d $\mathcal{N}=1$ SCFT

    hep-th 2026-04 unverdicted novelty 7.0

    WKB periods from the C(2)^{(2)} linear problem match eigenvalues of local integrals of motion in the Neveu-Schwarz sector of 2d N=1 SCFTs up to sixth order.

  2. Modular Properties of Symplectic Fermion Generalised Gibbs Ensemble

    hep-th 2026-03 unverdicted novelty 7.0

    Exact modular S-transforms are derived for GGEs in the symplectic fermion theory, agreeing with conjectures for the W3 zero mode and mirroring free-fermion results for the KdV subset.

  3. Integrals of motion in $WE_6$ CFT and the ODE/IM correspondence

    hep-th 2026-04 unverdicted novelty 6.0

    Period integrals from the E6 ODE WKB expansion match eigenvalues of WE6 CFT integrals of motion up to sixth order.

  4. Integrals of motion in $WE_6$ CFT and the ODE/IM correspondence

    hep-th 2026-04 conditional novelty 6.0

    The WKB periods of the E_6^(1) linear problem agree with the integrals of motion of the W E6 CFT on highest-weight states up to spin 6 under the standard parameter dictionary.

  5. The ODE/IM Correspondence between $C(2)^{(2)}$-type Linear Problems and 2d $\mathcal{N}=1$ SCFT

    hep-th 2026-04 accept novelty 5.5

    WKB periods of the fully diagonalized C(2)^{(2)} Lax operator coincide with NS-sector local IoM eigenvalues of N=1 SCFT up to sixth order under a fixed parameter dictionary.