Schur Connections: Chord Counting, Line Operators, and Indices

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.

69 pages + appendices

References in corpus (87)

• Gapless Spin-Fluid Ground State in a Random Quantum Heisenberg Magnet
• Comments on the Sachdev-Ye-Kitaev model
• Generalized Global Symmetries
• Liouville Correlation Functions from Four-dimensional Gauge Theories
• N=2 dualities
• An Index for 4 dimensional Super Conformal Theories
• Chaos in AdS${}_2$ holography
• Black Holes and Random Matrices
• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• Counting chiral primaries in N=1 d=4 superconformal field theories
• Reading between the lines of four-dimensional gauge theories
• Integrability and Seiberg-Witten Exact Solution
• Loop and surface operators in N=2 gauge theory and Liouville modular geometry
• Fermionic Localization of the Schwarzian Theory
• Partition functions of N=(2,2) gauge theories on S^2 and vortices
• Gauge Theories and Macdonald Polynomials
• Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N=1 Dual Theories
• The complete superconformal index for N=6 Chern-Simons theory
• Exact Results in D=2 Supersymmetric Gauge Theories
• Sachdev-Ye-Kitaev Model as Liouville Quantum Mechanics
• Quantization of Integrable Systems and Four Dimensional Gauge Theories
• Solving the Schwarzian via the Conformal Bootstrap
• The 4d Superconformal Index from q-deformed 2d Yang-Mills
• Index for three dimensional superconformal field theories with general R-charge assignments
• Indices for Superconformal Field Theories in 3,5 and 6 Dimensions
• R-Twisting and 4d/2d Correspondences
• Holomorphic Blocks in Three Dimensions
• The Virtue of Defects in 4D Gauge Theories and 2D CFTs
• Schur Indices, BPS Particles, and Argyres-Douglas Theories
• The Quantum Gravity Dynamics of Near Extremal Black Holes
• Gauge Theory Loop Operators and Liouville Theory
• Superconformal Indices for $\mathcal{N}=6$ Chern Simons Theories
• Towards a full solution of the large N double-scaled SYK model
• Factorisation of N = 2 theories on the squashed 3-sphere
• The Schwarzian Theory - Origins
• The bulk Hilbert space of double scaled SYK
• The Schwarzian Theory - A Wilson Line Perspective
• Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction
• An exact quantization of Jackiw-Teitelboim gravity
• Dual Boundary Conditions in 3d SCFT's
• Holomorphic reduction of N=2 gauge theories, Wilson-'t Hooft operators, and S-duality
• Infrared Computations of Defect Schur Indices
• Menagerie of AdS${}_2$ boundary conditions
• Fine Structure of Jackiw-Teitelboim Quantum Gravity
• Factorisation and holomorphic blocks in 4d
• Line operators on S^1xR^3 and quantization of the Hitchin moduli space
• 3-Manifolds and 3d Indices
• Line Operator Index on S1 $\times$ S3
• A symmetry algebra in double-scaled SYK
• The supersymmetric index in four dimensions
• Localization of 3d $\mathcal{N}=2$ Supersymmetric Theories on $S^1\times D^2$
• Towards a bulk description of higher spin SYK
• The algebra of Wilson-'t Hooft operators
• Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space
• Duality walls and defects in 5d N=1 theories
• On q-deformed gl(l+1)-Whittaker function II
• A slow review of the AGT correspondence
• Holomorphic blocks and the 5d AGT correspondence
• Superconformal Index with Duality Domain Wall
• Complex Sachdev-Ye-Kitaev model in the double scaling limit
• On 't Hooft Defects, Monopole Bubbling and Supersymmetric Quantum Mechanics
• The $\mathcal{N}=4$ Schur index with Polyakov loops
• Exact $\mathcal{N}=2^{\ast }$ Schur line defect correlators
• Large $N$ and large representations of Schur line defect correlators
• Semi-Chiral Operators in 4d $\mathcal{N}=1$ Gauge Theories
• On the 3d compactifications of 5d SCFTs associated with SU(N+1) gauge theories
• ABCD of 't Hooft operators
• Excitations of bubbling geometries for line defects
• 5d gauge theories on orbifolds and 4d 't Hooft line indices
• Double-scaled SYK, Chords and de Sitter Gravity
• A Panorama Of Physical Mathematics c. 2022
• Compactifying 5d superconformal field theories to 3d
• Whittaker functions on quantum groups and q-deformed Toda operators
• On q-deformed gl(l+1)-Whittaker function
• $G_2$-Manifolds from 4d N=1 Theories, Part I: Domain Walls
• Large p SYK from chord diagrams
• Giant graviton expansion for general Wilson line operator indices
• $\mathcal{N}=4$ SYM line defect Schur index and semiclassical string
• A Cordial Introduction to Double Scaled SYK
• Framed BPS States
• $\mathcal{N}=2$ Schur index and line operators
• SYK-Schur duality: Double scaled SYK correlators from $N=2$ supersymmetric gauge theory
• Giant graviton expansions for line operator index
• Schur line defect correlators and giant graviton expansion
• Commuting Line Defects At $q^N=1$
• Brane expansions for anti-symmetric line operator index
• Leading Giant graviton expansion of Schur correlators in large representations