2d-4d Connection between q-Virasoro/W Block at Root of Unity Limit and Instanton Partition Function on ALE Space

Abstract

We propose and demonstrate a limiting procedure in which, starting from the q-lifted version (or K-theoretic five dimensional version) of the (W)AGT conjecture to be assumed in this paper, the Virasoro/W block is generated in the r-th root of unity limit in q in the 2d side, while the same limit automatically generates the projection of the five dimensional instanton partition function onto that on the ALE space R^4/Z_r. This circumvents case-by-case conjectures to be made in a wealth of examples found so far. In the 2d side, we successfully generate the super-Virasoro algebra and the proper screening charge in the q -> -1, t -> -1 limit, from the defining relation of the q-Virasoro algebra and the q-deformed Heisenberg algebra. The central charge obtained coincides with that of the minimal series carrying odd integers of the N=1 superconformal algebra. In the r-th root of unity limit in q in the 2d side, we give some evidence of the appearance of the parafermion-like currents. Exploiting the q-analysis literatures, q-deformed su(n) block is readily generated both at generic q, t and the r-th root of unity limit. In the 4d side, we derive the proper normalization function for general (n, r) that accomplishes the automatic projection through the limit.

38 pages, 4 figures; v2: minor changes, references added; v3: minor changes, a comment (Note added) and three references added, published version

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• Non-Lagrangian Theories from Brane Junctions
• Fiber-Base Duality and Global Symmetry Enhancement
• 5D partition functions, q-Virasoro systems and integrable spin-chains
• $(q,t)$-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces
• A slow review of the AGT correspondence
• An M-Theoretic Derivation of a 5d and 6d AGT Correspondence, and Relativistic and Elliptized Integrable Systems
• $q$-Virasoro/W Algebra at Root of Unity and Parafermions
• Crystallization of deformed Virasoro algebra, Ding-Iohara-Miki algebra and 5D AGT correspondence
• Construction of Gaiotto states with fundamental multiplets through Degenerate DAHA
• $q$-Virasoro modular double and 3d partition functions
• Holomorphic field realization of SH${}^c$ and quantum geometry of quiver gauge theories
• A brief review of the 2d/4d correspondences
• Argyres-Douglas theories and Liouville Irregular States
• Matching branches of non-perturbative conformal block at its singularity divisor
• q-Vertex Operator from 5D Nekrasov Function
• Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models
• N=2 gauge theories, instanton moduli spaces and geometric representation theory
• The integral representation of solutions of KZ equation and a modification by $\mathcal{K}$ operator insertion
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• Twisted reduction of quiver W-algebras
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