$\widehat{\mathfrak{sl}}(n{)}_N$ WZW conformal blocks from $SU(N)$ instanton partition functions on ${\mathbb{C}}^2/{\mathbb{Z}}_n$

Abstract

Generalizations of the AGT correspondence between 4D $\mathcal{N}=2$ $SU(2)$ supersymmetric gauge theory on ${\mathbb{C}}^2$ with $\Omega$-deformation and 2D Liouville conformal field theory include a correspondence between 4D $\mathcal{N}=2$ $SU(N)$ supersymmetric gauge theories, $N=2,3,\dots$, on ${\mathbb{C}}^2/{\mathbb{Z}}_n$, $n=2,3,\dots$, with $\Omega$-deformation and 2D conformal field theories with ${\mathcal{W}}_{N,n}^{para}$ ($n$-th parafermion ${\mathcal{W}}_N$) symmetry and $\widehat{\mathfrak{sl}}(n{)}_N$ symmetry. In this work, we trivialize the factor with ${\mathcal{W}}_{N,n}^{para}$ symmetry in the 4D $SU(N)$ instanton partition functions on ${\mathbb{C}}^2/{\mathbb{Z}}_n$ (by using specific choices of parameters and imposing specific conditions on the $N$-tuples of Young diagrams that label the states), and extract the 2D $\widehat{\mathfrak{sl}}(n{)}_N$ WZW conformal blocks, $n=2,3,\dots$, $N=1,2,\dots .$

47 pages, 1 figure; v2: minor changes, section 5 improved, appendix A added, references added; v3: minor corrections, published version

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Cited by in corpus (4)

• A slow review of the AGT correspondence
• Anyonic Defect Branes and Conformal Blocks in Twisted Equivariant Differential (TED) K-theory
• $n$-th parafermion ${\mathcal{W}}_N$ characters from $U(N)$ instanton counting on ${\mathbb{C}}^2/{\mathbb{Z}}_n$
• The Grassmannian VOA