Parafermionic Liouville field theory and instantons on ALE spaces

Abstract

In this paper we study the correspondence between the $\widehat{\text{su}}(n{)}_k\oplus \widehat{\text{su}}(n{)}_p/\widehat{\text{su}}(n{)}_{k+p}$ coset conformal field theories and $\mathcal{N}=2$ SU(n) gauge theories on ${\mathbb{R}}^4/{\mathbb{Z}}_p$. Namely we check the correspondence between the SU(2) Nekrasov partition function on ${\mathbb{R}}^4/{\mathbb{Z}}_4$ and the conformal blocks of the $S_3$ parafermion algebra (in $S$ and $D$ modules). We find that they are equal up to the U(1)-factor as it was in all cases of AGT-like relations. Studying the structure of the instanton partition function on ${\mathbb{R}}^4/{\mathbb{Z}}_p$ we also find some evidence that this correspondence with arbitrary $p$ takes place up to the U(1)-factor.

21 pages, 6 figures, misprints corrected, references added, version to appear in JHEP

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