S-duality and the $ \mathcal{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. June 2013 J m (β) = J (x −m β) , (6.5) and then J m+1 (β) = J m (x −1 β) , J m−1 (β) = J m (xβ) . (6.6)

Supersymmetric Casimir energy and the anomaly polynomial

“…We expect that there is a generalization of our results to supersymmetric Casimir energies on manifolds S 1 × M with M other than M = S D−1 . Two prominent examples for which this can be explored further are the 4d superconformal index on the Lens spaces M = L(p, q), studied in [61][62][63][64], and the partition functions with M some 5d Sasaki-Einstein manifold, analyzed in [65,66].…”

N = 2 $$ \mathcal{N}=2 $$ supersymmetric gauge theories on S2 × S2 and Liouville Gravity

“…(5.1) with β = b and α = a − b), which is the matter sector of the LG. Explicitly 16) up to renormalization of the vertices analogously to (5.15), once the v.e.v. a and µ f are assumed to be purely imaginary.…”

“…A rigorous mathematical framework for A n singularities has been presented in [13,14]. Supersymmetric gauge theories with N = 2 on curved four manifolds have been considered for example in [15][16][17][18][19][20][21]. The specific case of S 2 × S 2 will also be considered in [22].…”

Down the rabbit hole with theories of class S $$ \mathcal{S} $$

“…Theories of this class are obtained by compactifying 6d (2, 0) theory on a punctured Riemann surface. The supersymmetric partition functions on S 3 × S 1 [3][4][5], and more generally on S 3 /Z r × S 1 [6,7], of class S theories of A N −1 type corresponding to Riemann surface C g,s of genus g and having s punctures has a very robust and mathematically interesting structure. 1 It can be written in the following form,…”

“…supersymmetric indices on lens spaces. It can be argued [5,7,14] that the partition functions on S 3 /Z r × S 1 of the theory T C deformed by certain surface defects can be obtained by acting with a difference operator on the partition function without the defect. This surface defect spans the temporal S 1 and sits on one of the equators of S 3 (see section 3 for more details).…”