(p,q)-webs of DIM representations, 5d N=1 instanton partition functions and qq-characters

Abstract

Instanton partition functions of $\mathcal{N}=1$ 5d Super Yang-Mills reduced on $S^1$ can be engineered in type IIB string theory from the $(p,q)$-branes web diagram. To this diagram is superimposed a web of representations of the Ding-Iohara-Miki (DIM) algebra that acts on the partition function. In this correspondence, each segment is associated to a representation, and the (topological string) vertex is identified with the intertwiner operator constructed by Awata, Feigin and Shiraishi. We define a new intertwiner acting on the representation spaces of levels $(1,n)\otimes (0,m)\to (1,n+m)$, thereby generalizing to higher rank $m$ the original construction. It allows us to use a folded version of the usual $(p,q)$-web diagram, bringing great simplifications to actual computations. As a result, the characterization of Gaiotto states and vertical intertwiners, previously obtained by some of the authors, is uplifted to operator relations acting in the Fock space of horizontal representations. We further develop a method to build qq-characters of linear quivers based on the horizontal action of DIM elements. While fundamental qq-characters can be built using the coproduct, higher ones require the introduction of a (quantum) Weyl reflection acting on tensor products of DIM generators.

42 pages

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Cited by in corpus (46)

• Webs of W-algebras
• $(q,t)$-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces
• On (q,t)-deformation of Gaussian matrix model
• Reflection states in Ding-Iohara-Miki algebra and brane-web for D-type quiver
• A slow review of the AGT correspondence
• Bootstrapping BPS spectra of 5d/6d field theories
• $\mathcal{W}$-algebra Modules, Free Fields, and Gukov-Witten Defects
• Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations
• Blowup Equations for Refined Topological Strings
• Intersecting defects in gauge theory, quantum spin chains, and Knizhnik-Zamolodchikov equations
• Riemann-Hilbert correspondence and blown up surface defects
• Splitting of surface defect partition functions and integrable systems
• A Note on Quiver Quantum Toroidal Algebra
• Higgsed network calculus
• Wilson loops in 5d $\mathcal{N}=1$ theories and S-duality
• Refined geometric transition and $qq$-characters
• New quantum toroidal algebras from 5D $\mathcal{N}=1$ instantons on orbifolds
• On $W$-representations of $\beta$- and $q,t$-deformed matrix models
• A note on the algebraic engineering of 4D $\mathcal{N}=2$ super Yang-Mills theories
• The MacMahon R-matrix
• Quantum toroidal algebras and solvable structures in gauge/string theory
• More on topological vertex formalism for 5-brane webs with O5-plane
• Fiber-base duality from the algebraic perspective
• Mixed network calculus
• An Elliptic Vertex of Awata-Feigin-Shiraishi type for M-strings
• Web Construction of ABCDEFG and Affine Quiver Gauge Theories
• Plane Partition Realization of (Web of) W-algebra Minimal Models
• 5d AGT correspondence of supergroup gauge theories from quantum toroidal ${\mathfrak{gl}}_1$
• Refined topological vertex with ON-planes
• ABCD of qq-characters
• Bootstrapping the $S^5$ partition function
• Gauge origami and quiver W-algebras
• Argyres-Douglas Theories, S-duality and AGT Correspondence
• Shifted quantum groups and matter multiplets in supersymmetric gauge theories
• di-Langlands correspondence and extended observables
• Quantum Geometry and $\theta$-Angle in Five-Dimensional Super Yang-Mills
• Quantum integrability of $\mathcal{N}=2$ 4d gauge theories
• Gauge origami and quiver W-algebras III: Donaldson--Thomas $qq$-characters
• The Grassmannian VOA
• Non-Perturbative Schwinger-Dyson Equations for 3d $\mathcal{N}=4$ Gauge Theories
• Quantum $W_{1+\infty }$ subalgebras of BCD type and symmetric polynomials
• Elliptic algebra, Frenkel-Kac construction and root of unity limit
• A note on chiral trace relations from qq-characters
• More on 5d Wilson Loops in Higher-Rank Theories and Blowup Equations
• Free field realization of the Ding-Iohara algebra at general levels
• Intertwining operator and integrable hierarchies from topological strings