Abstract
Quantum toroidal algebras are obtained from quantum affine algebras by a further affinization, and, like the latter, can be used to construct integrable systems. These algebras also describe the symmetries of instanton partition functions for 5D mathcal{N} = 1 supersymmetric quiver gauge theories. We consider here the gauge theories defined on an orbifold S1× ℂ2/ℤp where the action of ℤp is determined by two integer parameters (ν1, ν2). The corresponding quantum toroidal algebra is introduced as a deformation of the quantum toroidal algebra of mathfrak{gl} (p). We show that it has the structure of a Hopf algebra, and present two representations, called vertical and horizontal, obtained by deforming respectively the Fock representation and Saito’s vertex representations of the quantum toroidal algebra of mathfrak{gl} (p). We construct the vertex operator intertwining between these two types of representations. This object is identified with a (ν1, ν2)-deformation of the refined topological vertex, allowing us to reconstruct the Nekrasov partition function and the qq-characters of the quiver gauge theories.
Highlights
Non-perturbative dynamics of supersymmetric gauge theories is a prolific research subject in theoretical physics
We show that it has the structure of a Hopf algebra, and present two representations, called vertical and horizontal, obtained by deforming respectively the Fock representation and Saito’s vertex representations of the quantum toroidal algebra of gl(p)
The Nekrasov partition function has been a powerful tool to investigate the correspondences of four-dimensional N = 2 supersymmetric quiver gauge theories with other objects in mathematical physics, i.e., quantum integrable systems [3–5], two-dimensional CFTs [6–12], flat connections on Riemann surfaces [13, 14], and isomonodromic deformations of Fuchsian systems [15–17]
Summary
Non-perturbative dynamics of supersymmetric gauge theories is a prolific research subject in theoretical physics. The refined topological vertex is identified with an intertwiner between vertical and horizontal representations, that is the toroidal version of the vertex operator introduced in [34] for the quantum group Uq(sl(2)) In this way, the Nekrasov partition function is written as a purely algebraic object using the quantum toroidal algebra, just like conformal blocks with W-algebras [35, 36]. The constraints take an even more elegant form in the algebraic construction described above as they express the invariance of an operator T under the adjoint action of the quantum toroidal algebra [43] This operator is obtained by gluing intertwiners along the edges of the (p, q)-branes web, and its vacuum expectation value reproduces the 5D Nekrasov instanton partition. The section four is dedicated to the algebraic construction of gauge theories observables, giving the expression of the (ν1, ν2)-colored refined topological vertex and a few examples of application
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