New quantum toroidal algebras from 5D $$ \mathcal{N} $$ = 1 instantons on orbifolds

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 N = 1 supersymmetric quiver gauge theories. We consider here the gauge theories defined on an orbifold S 1 ×C 2 /Z p where the action of Z p is determined by two integer parameters (ν 1 , ν 2). The corresponding quantum toroidal algebra is introduced as a deformation of the quantum toroidal algebra of 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 gl(p). We construct the vertex operator intertwining between these two types of representations. This object is identified with a (ν 1 , ν 2)-deformation of the refined topological vertex, allowing us to reconstruct the Nekrasov partition function and the qq-characters of the quiver gauge theories.

More on topological vertex formalism for 5-brane webs with O5-plane

“…Furthermore, web diagrams each of which are made by gluing three or four (dual) toric diagrams have been constructed in [19] and the method developed there computes the partition functions of SO(2N ) gauge theories and also the pure E 6 , E 7 , E 8 gauge theories. Recently the topolgical vertex formalims has been also extended in other directions [20][21][22][23][24][25][26][27][28][29].…”

Riemann-Hilbert correspondence and blown up surface defects

“…In the geometric engineering of N = 2 gauge theories [47], the insertion of half-BPS surface defects corresponds to incorporating open strings [51]. In the point of view of the algebraic engineering [3], where the gauge theory partition function is represented as a correlation function of operators which intertwine representations of quantum toroidal algebra, the insertion of the surface defect can be realized by properly extending the quantum toroidal algebra [4]. It would be nice to formulate the blowup formula in these contexts and study its implications.…”

“…A mystery is that even though the both results connect the N = 2 gauge theories to the Riemann-Hilbert problem and isomonodromic deformations of Fuchsian systems, the field theory settings in which the correspondence arises are rather different. The computation of the partition function of N = 2 gauge theories on the non-compact C 2 involves a regularization implemented by the Ω-background, weakly gauging the maximal torus of the spacetime isometry U(1) ε 1 × U(1) ε 2 ⊂ SO (4). In [7], the isomonodromic tau function for a Fuchsian system is expressed as an infinite sum of the gauge theory partition functions with shifted Coulomb moduli, subject to the self-dual limit ε 2 = −ε 1 of the Ω-background.…”

Intertwining operator and integrable hierarchies from topological strings

“…This observation led to a number of important results in this field. For example, we can mention the extension of the topological vertex technique to various theories [17][18][19][20][21][22][23][24] and observables [25][26][27], the derivation of proofs for the q-deformed AGT correspondence [28][29][30], or the description of the fiber-base duality [30][31][32][33].…”

“…The algebraic description of topological string theory has been extended to different algebras and geometric backgrounds. Some of these algebras should possess an SL(2, Z) subgroup of automorphisms, like the quantum toroidal gl(p) algebras [18], their elliptic deformations [19,62] or even the fully deformed algebra of [23]. In all these cases, we expect our construction to apply, producing tau functions of different integrable hierarchies in specific limits.…”