Schur Connections: Chord Counting, Line Operators, and Indices

TL;DR

This paper generalizes the connection between 4d $SU(2)$ supersymmetric Yang-Mills line operator indices and combinatorial models, linking algebraic structures, quantum integrable systems, and combinatorics for $SU(N)$ theories.

Contribution

It derives the algebra of line operators for $SU(N)$ theories, describes the half-index via $ ext{q}$-oscillators, and connects these to quantum Toda chains and combinatorial chord counting.

Findings

Algebra of line operators expressed in $ ext{q}$-Weyl and $ ext{q}$-oscillator forms

Half-index interpreted as expectation value in $ ext{q}$-oscillator Fock space

Connection established between half-index, quantum Toda chain, and combinatorial chord counting

Abstract

Recently, an intriguing correspondence was conjectured in arXiv:2409.11551 between Schur half-indices of pure 4d $SU(2)$ $\mathcal{N}=2$ 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 $SU(N)$ $\mathcal{N}=2$ SYM theories. We begin by deriving the algebra of line operators, $\mathcal{A}_{\text{Schur}}$, representing it both in terms of the $\mathfrak{q}$-Weyl algebra and $\mathfrak{q}$-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 $\mathfrak{q}$-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.