(q,t)-KZ equation for Ding-Iohara-Miki algebra

Abstract

We derive the generalization of the Knizhnik-Zamolodchikov equation (KZE) associated with the Ding-Iohara-Miki (DIM) algebra U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1). We demonstrate that certain refined topological string amplitudes satisfy these equations and find that the braiding transformations are performed by the R-matrix of U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1). The resulting syste is the uplifting of the \widehat{\mathfrak{u}}_1 Wess-Zumino-Witten model. The solutions to the (q,t)-KZE are identified with the (spectral dual of) building blocks of the Nekrasov partition function for 5d linear quiver gauge theories. We also construct an elliptic version of the KZE and discuss its modular and monodromy properties, the latter being related to a dual version of KZE.

22 pages

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Cited by in corpus (19)

• $(q,t)$-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces
• A slow review of the AGT correspondence
• Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations
• A Note on Quiver Quantum Toroidal Algebra
• Higgsed network calculus
• On $W$-representations of $\beta$- and $q,t$-deformed matrix models
• The MacMahon R-matrix
• Quantum toroidal algebras and solvable structures in gauge/string theory
• BPS Hall Algebra of Scattering Hall States
• 5d AGT correspondence of supergroup gauge theories from quantum toroidal ${\mathfrak{gl}}_1$
• Gauge origami and quiver W-algebras
• Argyres-Douglas Theories, S-duality and AGT Correspondence
• Polynomial representations of classical Lie algebras and flag varieties
• Irreducible representations of simple Lie algebras by differential operators
• Cut-and-join operators and Macdonald polynomials from the 3-Schur functions
• MacMahon KZ equation for Ding-Iohara-Miki algebra
• Quasi-Hopf twist and elliptic Nekrasov factor
• Elliptic algebra, Frenkel-Kac construction and root of unity limit
• From superintegrability to tridiagonal representation of $\beta$-ensembles