S-duality and the N=2 Lens Space Index

Abstract

We discuss some of the analytic properties of lens space indices for 4d N=2 theories of class S. The S-duality properties of these theories highly constrain the lens space indices, and imply in particular that they are naturally acted upon by a set of commuting difference operators corresponding to surface defects. We explicitly identify the difference operators to be a matrix-valued generalization of the elliptic Ruijsenaars-Schneider model. In a special limit these difference operators can be expressed naturally in terms of Cherednik operators appearing in the double affine Hecke algebras, with the eigenfunctions given by non-symmetric Macdonald polynomials.

33 pages, 3 figures, harvmac. v2: minor corrections

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• Orbifolds and Exact Solutions of Strongly-Coupled Matrix Models
• N=1 theories of class S_k
• $SL(3,\mathbb{Z})$ Modularity and New Cardy Limits of the $\mathcal{N}=4$ Superconformal Index
• 4d $\mathcal{N}=3$ indices via discrete gauging
• $q$-Virasoro modular double and 3d partition functions
• Four-dimensional Lens Space Index from Two-dimensional Chiral Algebra
• Equivariant Verlinde formula from fivebranes and vortices
• Equivariant Verlinde algebra from superconformal index and Argyres-Seiberg duality
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• Modularity of supersymmetric partition functions
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• $\mathcal{N}=2$ supersymmetric gauge theories on $S^2\times S^2$ and Liouville Gravity
• Conformal Quantum Mechanics, Holomorphic Factorisation, and Ultra-Spinning Black Holes
• Equivariant U(N) Verlinde algebra from Bethe/Gauge correspondence
• An infrared bootstrap of the Schur index with surface defects
• Ground state wavefunctions of elliptic relativistic integrable Hamiltonians
• Wilson punctured network defects in 2D q-deformed Yang-Mills theory
• Lens space index and global properties for 4d N = 2 models
• Orbifold Schur Index and IR formula
• A note on discrete dynamical systems in theories of class $S$
• Aspects of 4d supersymmetric dynamics and geometry
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