The ODE/IM Correspondence between $C (2)^{(2)}$-type Linear Problems and 2d $\mathcal{N} = 1$ SCFT
TLDR
This paper investigates the ODE/IM correspondence linking $C(2)^{(2)}$-type linear problems with 2d $\mathcal{N}=1$ superconformal field theories.
Key contributions
- Introduced a suitable boundary condition for ODE side's conformal limit and WKB analysis.
- Extracted WKB periods and non-local conserved quantities up to tenth order from Lax operator diagonalization.
- Computed local integral of motion eigenvalues for 2d $\mathcal{N}=1$ SCFTs in NS and Ramond sectors.
- Verified the ODE/IM correspondence up to sixth order by comparing WKB periods with IM eigenvalues.
Why it matters
This work extends the ODE/IM correspondence to supersymmetric affine Lie superalgebras, specifically $C(2)^{(2)}$, providing a rigorous verification. It deepens our understanding of the profound connections between differential equations and quantum field theories, potentially opening new avenues for research in both fields.
Original 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 non-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.
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