Coherent states in quantum ${\mathcal{W}}_{1+\infty }$ algebra and qq-character for 5d Super Yang-Mills

Abstract

The instanton partition functions of $\mathcal{N}=1$ 5d super Yang-Mills are built using elements of the representation theory of quantum ${\mathcal{W}}_{1+\infty }$ algebra: Gaiotto state, intertwiner, vertex operator. This algebra is also known under the names of Ding-Iohara-Miki and quantum toroidal $\widehat{\mathfrak{gl}}(1)$ algebra. Exploiting the explicit action of the algebra on the partition function, we prove the regularity of the 5d qq-characters. These characters provide a solution to the Schwinger-Dyson equations, and they can also be interpreted as a quantum version of the Seiberg-Witten curve.

45 pages, 2 figures (v2: references added)

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• Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations
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• A Note on Quiver Quantum Toroidal Algebra
• Higgsed network calculus
• Nambu bracket and M-theory
• Wilson loops in 5d $\mathcal{N}=1$ theories and S-duality
• The MacMahon R-matrix
• New quantum toroidal algebras from 5D $\mathcal{N}=1$ instantons on orbifolds
• 5d/6d Wilson loops from blowups
• More on topological vertex formalism for 5-brane webs with O5-plane
• Maulik-Okounkov's R-matrix from Ding-Iohara-Miki algebra
• Fiber-base duality from the algebraic perspective
• Mixed network calculus
• Web Construction of ABCDEFG and Affine Quiver Gauge Theories
• An Elliptic Vertex of Awata-Feigin-Shiraishi type for M-strings
• 5d AGT correspondence of supergroup gauge theories from quantum toroidal ${\mathfrak{gl}}_1$
• AGT relations for sheaves on surfaces
• ABCD of qq-characters
• Quantum integrability from non-simply laced quiver gauge theory
• Gauge origami and quiver W-algebras
• Argyres-Douglas Theories, S-duality and AGT Correspondence
• Modular properties of 6d (DELL) systems
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• Quantum integrability of $\mathcal{N}=2$ 4d gauge theories
• Defects and type D relativistic Toda lattice for some 5d gauge theories
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• Elliptic Quantum Toroidal Algebras, Z-algebra Structure and Representations
• Free field realization of the Ding-Iohara algebra at general levels
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• Intertwining operator and integrable hierarchies from topological strings