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On W-algebras and ODE/IM correspondence
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On W-algebras and ODE/IM correspondence
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We study the ODE/IM correspondence for two-dimensional conformal field theories with Virasoro and $\mathcal{W}_N$ symmetry. Building on earlier work establishing the correspondence, we develop a systematic algorithm for calculating the eigenvalues of local integrals of motion in terms of the Bethe roots using formal WKB expansions of wave functions associated to the differential operators. The method is demonstrated explicitly for Virasoro, $\mathcal{W}_3$, and $\mathcal{W}_4$ algebras, yielding closed expressions for the eigenvalues of the first few local quantum KdV Hamiltonians. A key geometric structure emerging from our analysis is the mirror curve, a three-punctured sphere that is naturally covered by the WKB curve. We show how the algebraic properties of the $\mathcal{W}$-symmetry algebras are reflected in the geometry of these curves, and how period integrals on these curves reproduce the spectral data of the integrable system. Applications to Argyres-Douglas minimal models allow us to test the prescription both analytically and numerically and we find complete agreement between the calculations in different triality frames. Finally, we examine large rank limits of ground state eigenvalues and show that they match the genus expansion of the topological string partition function on $\mathbb{C}^3$.
Forward citations
Cited by 6 Pith papers
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The ODE/IM Correspondence between $C(2)^{(2)}$-type Linear Problems and 2d $\mathcal{N}=1$ SCFT
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.
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Modular Properties of Symplectic Fermion Generalised Gibbs Ensemble
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.
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On the Integrable Structure of the SU(2) Wess-Zumino-Novikov-Witten Model
A new family of commuting local conserved charges in the SU(2) WZNW model is constructed for the first four spins and shown to agree with Bethe-ansatz and ODE/quantum-field-theory predictions.
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Integrals of motion in $WE_6$ CFT and the ODE/IM correspondence
Period integrals from the E6 ODE WKB expansion match eigenvalues of WE6 CFT integrals of motion up to sixth order.
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Integrals of motion in $WE_6$ CFT and the ODE/IM correspondence
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.
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The ODE/IM Correspondence between $C(2)^{(2)}$-type Linear Problems and 2d $\mathcal{N}=1$ SCFT
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.
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