New quantum toroidal algebras from 5D N $$ \mathcal{N} $$ = 1 instantons on orbifolds
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Springer
Received: September 23, Revised: April 9, Accepted: May 4, Published: May 26,
2019 2020 2020 2020
Jean-Emile Bourginea and Saebyeok Jeongb a
Korea Institute for Advanced Study (KIAS), Quantum Universe Center (QUC), 85 Hoegiro, Dongdaemun-gu, Seoul, South Korea b C.N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY 11794-3840, U.S.A.
E-mail: [email protected], [email protected] 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 ×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 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 (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. Keywords: Nonperturbative Effects, Quantum Groups, Supersymmetric Gauge Theory, Topological Strings ArXiv ePrint: 1906.01625
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2020)127
JHEP05(2020)127
New quantum toroidal algebras from 5D N = 1 instantons on orbifolds
Contents 1 Introduction
1
2 Instantons on orbifolds 2.1 Action of the abelian group Zp on the ADHM data 2.2 Instantons partition function 2.3 Y-observables
3 3 6 8 9 9 12 14
4 Algebraic engineering 4.1 Vertex operators 4.2 Partition functions and qq-characters
16 17 18
5 Concluding remarks
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A Quantum toroidal algebra of gl(p) A.1 Definition A.2 Horizontal representation A.3 Vertical representations A.4 Deformation of the algebra
23 23 25 27 28
B Shell formula
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C Representations of the extended algebra C.1 Vertical representation C.2 Horizontal representation
30 30 31
D Automorphisms, gradings and modes expansion D.1 Automorphisms and gradings D.2 Modes expansion D.3 Coproduct D.4 Vertical representation D.5 Horizontal representation
33 33 34 36 37 37
E Relation with the quantum toroidal gl(p) algebra
38
F Derivation of the vertex operators F.1 Definition of the vacuum components F.2 Solution of intertwining relations
40 40 42
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JHEP05(2020)127
3 New quantum toroidal algebras 3.1 Definition of the algebra 3.2 Vertical representation 3.3 Horizontal representation
G Example of qq-characters G.1 U(1) gauge group G.2 U(2) gauge group
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44 44 45
Introduction
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