Ward identities in the most general "network matrix model" from [1] can be described in terms of the Ding-Iohara-Miki algebras (DIM). This confirms an expectation that such algebras and their various limits/reductions are the relevant substitutes/deformations of the Virasoro/W-algebra for (q, t) and (q1, q2, q3) deformed network matrix models. Exhaustive for these purposes should be the Pagoda triple-affine elliptic DIM, which corresponds to networks associated with 6d gauge theories with adjoint matter (double elliptic systems). We provide some details on elliptic qq-characters.

Dotsenko-Fateev and Chern-Simons matrix models, which describe Nekrasov functions for SYM theories in different dimensions, are all incorporated into network matrix models with the hidden Ding-Iohara-Miki (DIM) symmetry. This lifting is especially simple for what we call balanced networks. Then, the Ward identities (known under the names of Virasoro/W-constraints or loop equations or regularity condition for qq-characters) are also promoted to the DIM level, where they all become corollaries of a single identity.

We consider T [SU (N )] and its mirror, and we argue that there are two more dual frames, which are obtained by adding flipping fields for the moment maps on the Higgs and Coulomb branch. Turning on a monopole deformation in T [SU (N )], and following its effect on each dual frame, we obtain four new daughter theories dual to each other. We are then able to construct pairs of 3d spectral dual theories by performing simple operations on the four dual frames of T [SU (N )]. Engineering these 3d spectral pairs as codimension-two defect theories coupled to a trivial 5d theory, via Higgsing, we show that our 3d spectral dual theories descends from the 5d spectral duality, or fiber base duality in topological string. We provide further consistency checks about the web of dualities we constructed by matching partition functions on S 3 b , and in the case of spectral duality, matching exactly topological string computations with holomorphic blocks.

We describe relationships between integrable systems with N degrees of freedom arising from the AGT conjecture. Namely, we prove the equivalence (spectral duality) between the N -cite Heisenberg spin chain and a reduced gl N Gaudin model both at classical and quantum level. The former one appears on the gauge theory side of the AGT relation in the Nekrasov-Shatashvili (and further the SeibergWitten) limit while the latter one is natural on the CFT side. At the classical level, the duality transformation relates the Seiberg-Witten differentials and spectral curves via a bispectral involution. The quantum duality extends this to the equivalence of the corresponding Baxter-Schrödinger equations (quantum spectral curves). This equivalence generalizes both the spectral self-duality between the 2 × 2 and N × N representations of the Toda chain and the famous AHH duality.