$$C_2$$ generalization of the van Diejen model from the minimal $$(D_5,D_5)$$ conformal matter

Abstract

We study superconformal indices of 4d compactifications of the 6d minimal $$(D_{N+3},D_{N+3})$$ ( D N + 3 , D N + 3 ) conformal matter theories on a punctured Riemann surface. Introduction of supersymmetric surface defect in these theories is done at the level of the index by the action of the finite difference operators on the corresponding indices. There exist at least three different types of such operators according to three types of punctures with $$A_N, C_N$$ A N , C N and $$\left( A_1\right) ^N$$ A 1 N global symmetries. We mainly concentrate on $$C_2$$ C 2 case and derive explicit expression for an infinite tower of difference operators generalizing the van Diejen model. We check various properties of these operators originating from the geometry of compactifications. We also provide an expression for the kernel function of both our $$C_2$$ C 2 operator and previously derived $$A_2$$ A 2 generalization of van Diejen model. Finally, we also consider compactifications with $$A_N$$ A N -type punctures and derive the full tower of commuting difference operators corresponding to this root system generalizing the result of our previous paper.

D-type minimal conformal matter: quantum curves, elliptic Garnier systems, and the 5d descendants

“…Indeed, in the paper, we will establish a connection between the Sp(N ) quantum curve and a class of elliptic Gainier systems that has been investigated recently in the integrability community. On the other hand, it is also worth mentioning that the van Diejen operator and its generalizations have been also studied in the analysis of surface defects in various 4d N = 1 theories from the compactifications of 6d Sp(N ) onto Riemann surfaces [41,46,47]. It would be very interesting to understand if these difference operators can be introduced in the context of a pure 6d setup, and correspond to what kind of codimension two and four defects.…”

“…A particularly interesting example is the compactifications of 6d N = (2, 0) SCFTs on Riemann surfaces punctured by codimension two defects, resulting in a wide range of 4d N = 2 theories known as class S theories and their dualities [22][23][24]. More recently, the construction is generalized to 6d N = (1, 0) down to 4d N = 1 [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44], and novel connections have been established between elliptic quantum difference equations and surface defects introduced in the 4d theories in terms of their superconformal index [25,41,[45][46][47]. Meanwhile, in the context of 6d N = (1, 0) or their corresponding KK theories, it has been realized that, in the case of rank one 6d theories on the tensor branch with trivial gauge groups, the elliptic quantum difference equations studied in the 4d setup are precisely the elliptic quantum SW-curves of the corresponding 6d theories, and the surface defects in 4d also become a class of important codimension four observables, the Wilson surface defects, that serve the eigenvalues of the quantum curves [48][49][50].…”

Sporadic dualities from tensor deconfinement

“…The final result has a field theory interpretation representing the low energy description of the baryons and mesons of the SU(N c + 1) gauge theory with the expected constraints from the truncation of the chiral ring and the moduli space. This duality has been referred to as Spiridonov-Warnaar-Vartanov (SWV) duality in [19], where it has been used in the study of 4d compactification of the 6d minimal (D,D) conformal matter theories on a punctured Riemann surface (see also [20]).…”

“…20) with 2ω ≡ ω 1 + ω 2 . Observe that the identity (3.20) remains valid for N f = N c + 1, that corresponds to the confining case of Aharony duality[23], where only the meson M andJHEP05(2024)188 the minimal USp(2N c ) monopole Y survive in the WZ model and they interact through the superpotential W = Y PfM .…”