Coset conformal field theory and instanton counting on C²/Z_p

We study conformal field theory with the symmetry algebra $\mathcal{A}(2,p)=\hat{\mathfrak{gl}}(n)_{2}/\hat{\mathfrak{gl}}(n-p)_2$. In order to support the conjecture that this algebra acts on the moduli space of instantons on $\mathbb{C}^{2}/\mathbb{Z}_{p}$, we calculate the characters of its representations and check their coincidence with the generating functions of the fixed points of the moduli space of instantons. We show that the algebra $\mathcal{A}(2,p)$ can be realized in two ways. The first realization is connected with the cross-product of $p$ Virasoro and $p$ Heisenberg algebras: $\mathcal{H}^{p}\times \textrm{Vir}^{p}$. The second realization is connected with: $\mathcal{H}^{p}\times \hat{\mathfrak{sl}}(p)_2\times (\hat{\mathfrak{sl}}(2)_p \times \hat{\mathfrak{sl}}(2)_{n-p}/\hat{\mathfrak{sl}}(2)_n)$. The equivalence of these two realizations provides the non-trivial identity for the characters of $\mathcal{A}(2,p)$. The moduli space of instantons on $\mathbb{C}^{2}/\mathbb{Z}_{p}$ admits two different compactifications. This leads to two different bases for the representations of $\mathcal{A}(2,p)$. We use this fact to explain the existence of two forms of the instanton pure partition functions.