Diamond of triads

TL;DR

The paper extends the triad framework of Noumi–Shiraishi to elliptic and bi-elliptic settings, introducing the Shiraishi function and the ELS triad as elliptic lifts with two independent polynomial reductions that generate infinite families of symmetric polynomials and Baker–Akhiezer functions. It demonstrates bispectral and Poincaré dualities, and formulates non-stationary elliptic Ruijsenaars (and KS) Hamiltonians governing these objects, clarifying the link between gauge theory (6d Seiberg-Witten) and integrable systems. The constructions systematically relate symmetric reductions to BA-type reductions through Weyl-group averaging, and extend to the elliptic regime with theta-functions, preserving key reduction relations. Together, these developments place the bi-elliptic triad at the apex of polynomial systems associated with Seiberg-Witten theory with adjoint matter, while highlighting open questions toward a non-degenerate DELL realization.

Abstract

The triad refers to embedding of two systems of polynomials, symmetric ones and those of the Baker-Akhiezer type into a power series of the Noumi-Shiraishi type. It provides an alternative definition of Macdonald theory and its extensions. The basic triad is associated with the vector representation of the Ding-Iohara-Miki (DIM) algebra. We discuss lifting this triad to two elliptic generalizations and further to the bi-elliptic triad. At the algebraic level, it corresponds to elliptic and bi-elliptic DIM algebras. This completes the list of polynomials associated with Seiberg-Witten theory with adjoint matter in various dimensions.