Instanton partition functions of N = 1 5d Super Yang-Mills reduced on S 1 can be engineered in type IIB string theory from the (p, q)-branes web diagram. To this diagram is superimposed a web of representations of the Ding-Iohara-Miki (DIM) algebra that acts on the partition function. In this correspondence, each segment is associated to a representation, and the (topological string) vertex is identified with the intertwiner operator constructed by Awata, Feigin and Shiraishi. We define a new intertwiner acting on the representation spaces of levels (1, n) ⊗ (0, m) → (1, n + m), thereby generalizing to higher rank m the original construction. It allows us to use a folded version of the usual (p, q)-web diagram, bringing great simplifications to actual computations. As a result, the characterization of Gaiotto states and vertical intertwiners, previously obtained by some of the authors, is uplifted to operator relations acting in the Fock space of horizontal representations. We further develop a method to build qq-characters of linear quivers based on the horizontal action of DIM elements. While fundamental qq-characters can be built using the coproduct, higher ones require the introduction of a (quantum) Weyl reflection acting on tensor products of DIM generators.

No abstract

Reflection states are introduced in the vertical and horizontal modules of the Ding-Iohara-Miki (DIM) algebra (quantum toroidal gl 1 ). Webs of DIM representations are in correspondence with (p, q)-web diagrams of type IIB string theory, under the identification of the algebraic intertwiner of Awata, Feigin and Shiraishi with the refined topological vertex. Extending the correspondence to the vertical reflection states, it is possible to engineer the N = 1 quiver gauge theory of D-type (with unitary gauge groups). In this way, the Nekrasov instanton partition function is reproduced from the evaluation of expectation values of intertwiners. This computation leads to the identification of the vertical reflection state with the orientifold plane of string theory. We also provide a translation of this construction in the Iqbal-Kozcaz-Vafa refined topological vertex formalism.

We compute the Schur index of Argyres-Douglas theories of type (A N −1 , A M −1 ) with surface operators inserted, via the Higgsing prescription proposed by D. Gaiotto, L. Rastelli and S. S. Razamat. These surface operators are obtained by turning on position-dependent vacuum expectation values of operators in a UV theory which can flow to the Argyres-Douglas theories. We focus on two series of (A N −1 , A M −1 ) theories; one with gcd(N, M ) = 1 and the other with M = N (k − 1) for an integer k ≥ 2. Our results are identified with the characters of non-vacuum modules of the associated 2d chiral algebras, which explicitly confirms a remarkable correspondence recently discovered

Recently, Maulik and Okounkov proposed an integrable lattice model where the degree of freedom at each site is identical to the Hilbert space of free boson in two dimensions. We give a brief review of their construction and explain the relation with W n algebra and Calogero-Sutherland model. As a generalization, we examine the Yangian associated with N = 1 superconformal algebra which describes a supersymmetric extension of Calogero-Sutherland model and compare it with the literature.