Schur connections: Chord counting, line operators, and indices

Recently, an intriguing correspondence was conjectured in [D. Gaiotto and H. Verlinde, J. High Energy Phys. 06, 163 (2025)] between Schur half-indices of pure 4d SU(2) S U ( 2 ) \mathcal{N}=2 𝒩 = 2 supersymmetric Yang-Mills (SYM) theory with line operator insertions and partition functions of the double scaling limit of the Sachdev-Ye-Kitaev model (DSSYK). Motivated by this, we explore a generalization to SU(N) S U ( N ) \mathcal{N}=2 𝒩 = 2 SYM theories. We begin by deriving the algebra of line operators, \mathcal{A}_{\text{Schur}} 𝒜 Schur , representing it both in terms of the \mathfrak{q} 𝔮 -Weyl algebra and \mathfrak{q} 𝔮 -deformed harmonic oscillators, respectively. In the latter framework, the half-index admits a natural description as an expectation value in the Fock space of the oscillators. This \mathfrak{q} 𝔮 -oscillator perspective further suggests an interpretation in terms of generalized colored chord counting, and maps the half-index to a purely combinatorial quantity. Finally, we establish a connection with the quantum Toda chain, which is an integrable model whose commuting Hamiltonians can be identified with the Wilson lines of the SU(N) S U ( N ) SYM, and their eigenfunctions correspond to the function basis appearing in the half-index.