A note on the algebraic engineering of 4D <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi mathvariant="script">N</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math> super Yang–Mills theories

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.

Section: Jhep05(2020)127mentioning

confidence: 99%

mentioning

confidence: 99%

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

Bourgine

Jeong

2020

“…Parallel to the rational case of W ∞ there is an ongoing research in the q-deformed setting [81][82][83][84][85][86][87][88][89][90]. Since the q-deformed theory is currently undoubtedly more developed, it could perhaps be easier to search for the conjectural four-parametric algebra in this setting first.…”

Section: Jhep09(2020)150mentioning

confidence: 99%

The Grassmannian VOA

Eberhardt

Procházka²

2020

We study the 3-parametric family of vertex operator algebras based on the Grassmannian coset CFT $$ \mathfrak{u} $$ u (M + N )k /($$ \mathfrak{u} $$ u (M )k×$$ \mathfrak{u} $$ u (N )k ). This VOA serves as a basic building block for a large class of cosets and generalizes the $$ {\mathcal{W}}_{\infty } $$ W ∞ algebra. We analyze representations and their characters in detail and find surprisingly simple character formulas for the representations in the generic parameter regime that admit an elegant combinatorial formulation. We also discuss truncations of the algebra and give a conjectural formula for the complete set of truncation curves. We develop a theory of gluing for these algebras in order to build more complicated coset and non-coset algebras. We demonstrate the power of this technology with some examples and show in particular that the $$ \mathcal{N} $$ N = 2 supersymmetric Grassmannian can be obtained by gluing three bosonic Grassmannian algebras in a loop. We finally speculate about the tantalizing possibility that this algebra is a specialization of an even larger 4-parametric family of algebras exhibiting pentality symmetry. Specializations of this conjectural family should include both the unitary Grassmannian family as well as the Lagrangian Grassmannian family of VOAs which interpolates between the unitary and the orthosymplectic cosets.

“…The form of this algebra directly follows from the gauge theory background R 2 1 × R 2 2 × S 1 R , and the parameters are identified as (q 1 , q 2 ) = (e R 1 , e R 2 ). Deformations of this algebra have been introduced to treat different backgrounds: an elliptic deformation for 6D gauge theories [41], a higher rank version (quantum toroidal gl n ) for the 5D background with orbifold [42], and a degenerate version for 4D N = 2 gauge theories [43].…”

Section: 1 Presentation Of the Ding-iohara-miki Algebramentioning

confidence: 99%

“…Very recently, the algebraic engineering has been extended to 4D N = 2 quiver gauge theories in [43]. At first sight, the automorphism S seems to be lost in the degenerate version of DIM algebra used in the 4D case.…”

Section: Perspectivesmentioning

confidence: 99%

Fiber-base duality from the algebraic perspective

Bourgine

2019

Self Cite

Quiver 5D N = 1 gauge theories describe the low-energy dynamics on webs of (p, q)-branes in type IIB string theory. S-duality exchanges NS5 and D5 branes, mapping (p, q)-branes to branes of charge (−q, p), and, in this way, induces several dualities between 5D gauge theories. On the other hand, these theories can also be obtained from the compactification of topological strings on a Calabi-Yau manifold, for which the S-duality is realized as a fiber-base duality. Recently, a third point of view has emerged in which 5D gauge theories are engineered using algebraic objects from the Ding-Iohara-Miki (DIM) algebra. Specifically, the instanton partition function is obtained as the vacuum expectation value of an operator T constructed by gluing the algebra's intertwiners (the equivalent of topological vertices) following the rules of the toric diagram/brane web. Intertwiners and Toperators are deeply connected to the co-algebraic structure of the DIM algebra. We show here that S-duality can be realized as the twist of this structure by Miki's automorphism.