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

Abstract

We study the ODE/IM correspondence between the linear problem associated with the supersymmetric affine Toda field equation for the twisted affine Lie superalgebra $C(2)^{(2)} = \mathfrak{osp}(2|2)^{(2)}$ and two-dimensional $\mathcal{N}=1$ superconformal field theories (SCFTs). On the ODE side, we introduce a boundary condition more suitable for the conformal limit and the subsequent WKB analysis and diagonalize the resulting Lax operator. This leads to a WKB expansion from which we extract the WKB periods and semi-local conserved quantities up to tenth order. On the IM side, we compute the eigenvalues of the local integrals of motion on the cylinder in both the Neveu-Schwarz and Ramond sectors of 2d $\mathcal{N}=1$ SCFTs. We then compare the two sides and verify, up to sixth order, that the WKB periods coincide with the eigenvalues of the local integrals of motion for highest-weight states in the Neveu-Schwarz sector.