Abstract We study various duality webs involving the 3d FT[SU(N)] theory, a close relative of the T[SU(N)] quiver tail.We first map the partition functions of FT[SU(N)] and its 3d spectral dual to a pair of spectral dual q-Toda conformal blocks.Then we show how to obtain the FT[SU(N)] partition function by Higgsing a 5d linear click here quiver gauge theory, or equivalently from the refined topological string partition function on a certain toric Calabi-Yau three-fold.
3d spectral duality in this context descends from 5d spectral duality.Finally we discuss the 2d reduction of the 3d spectral dual pair and study the corresponding limits on the q-Toda side.In particular we obtain a new direct map between the partition function of here the 2d FT[SU(N)] GLSM and an (N + 2)-point Toda conformal block.