Elliptic algebra, Frenkel-Kac construction and root of unity limit

Abstract

We argue that the level-$1$ elliptic algebra $U_{q,p}(\hat{\mathfrak{g}})$ is a dynamical symmetry realized as a part of 2d/5d correspondence where the Drinfeld currents are the screening currents to the $q$-Virasoro/W block in the 2d side. For the case of $U_{q,p}(\widehat{\mathfrak{sl}}(2))$, the level-$1$ module has a realization by an elliptic version of the Frenkel-Kac construction. The module admits the action of the deformed Virasoro algebra. In a $r$-th root of unity limit of $p$ with $q^2\to 1$, the ${\mathbb{Z}}_r$-parafermions and a free boson appear and the value of the central charge that we obtain agrees with that of the 2d coset CFT with para-Virasoro symmetry, which corresponds to the 4d $\mathcal{N}=2$ $SU(2)$ gauge theory on ${\mathbb{R}}^4/{\mathbb{Z}}_r$.

22 pages; v2: references added

References in corpus (9)

• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• Toric Calabi-Yau threefolds as quantum integrable systems. R-matrix and RTT relations
• Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity
• $q$-Virasoro/W Algebra at Root of Unity and Parafermions
• Construction of Gaiotto states with fundamental multiplets through Degenerate DAHA
• N=2 quiver gauge theories on A-type ALE spaces
• Parafermionic polynomials, Selberg integrals and three-point correlation function in parafermionic Liouville field theory
• Elliptic Quantum Group U_{q,p}(\hat{sl}_2) and Vertex Operators
• Elliptic Algebra U_{q,p}(g^) and Quantum Z-algebras

Cited by in corpus (1)

• Twisted reduction of quiver W-algebras