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Parafermionic Liouville field theory and instantons on ALE spaces · Around

2011

Parafermionic Liouville field theory and instantons on ALE spaces

Alfimov, M. N., Tarnopolsky, G. M.

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In this paper we study the correspondence between the $\hat{\textrm{su}}(n)_{k}\oplus \hat{\textrm{su}}(n)_{p}/\hat{\textrm{su}}(n)_{k+p}$ 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.

Built on

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F. Fucito, J. F. Morales, and R. Poghossian, Multi instanton calculus on ALE spaces
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