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Integrals of motion in conformal field theory with W-symmetry and the ODE/IM correspondence
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abstract
We study the ODE/IM correspondence between two-dimensional $WA_{r}$/$WD_{r}$-type conformal field theories and the higher-order ordinary differential equations (ODEs) obtained from the affine Toda field theories associated with $A_r^{(1)}/D_r^{(1)}$-type affine Lie algebras. We calculate the period integrals of the WKB solution to the ODE along the Pochhammer contour, where the WKB expansions correspond to the classical conserved currents of the Drinfeld-Sokolov integrable hierarchies. We also compute the integrals of motion for $WA_{r}$($WD_{r}$) algebras on a cylinder. Their eigenvalues on the vacuum state are confirmed to agree with the period integrals up to the sixth order. These results generalize the ODE/IM correspondence to higher-order ODEs and can be used to predict higher-order integrals of motion.
Forward citations
Cited by 5 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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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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On W-algebras and ODE/IM correspondence
The eigenvalues of quantum KdV-type charges in Virasoro, W3, and W4 algebras are computed from Bethe roots via WKB periods of Catalan curves, verified against direct CFT diagonalization.
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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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