Schur connections: Chord counting, line operators, and indices

Recently, an intriguing correspondence was conjectured in [1] between Schur halfindices of pure 4d SU(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) N = 2 SYM theories. We begin by deriving the algebra of line operators, ASchur, representing it both in terms of the q-Weyl algebra and 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 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) SYM, and their eigenfunctions correspond to the function basis appearing in the half-index.