Coset conformal field theory and instanton counting on C^2/Z_p

Abstract

We study conformal field theory with the symmetry algebra $\mathcal{A}(2,p)=\widehat{\mathfrak{gl}}(n{)}_2/\widehat{\mathfrak{gl}}(n-p{)}_2$. 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: ${\mathcal{H}}^p\times {\text{Vir}}^p$. The second realization is connected with: ${\mathcal{H}}^p\times \widehat{\mathfrak{sl}}(p{)}_2\times (\widehat{\mathfrak{sl}}(2{)}_p\times \widehat{\mathfrak{sl}}(2{)}_{n-p}/\widehat{\mathfrak{sl}}(2{)}_n)$. 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.

23 pages, 1 figure

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Cited by in corpus (15)

• Webs of W-algebras
• A slow review of the AGT correspondence
• 2d-4d Connection between q-Virasoro/W Block at Root of Unity Limit and Instanton Partition Function on ALE Space
• $q$-Virasoro/W Algebra at Root of Unity and Parafermions
• A brief review of the 2d/4d correspondences
• On spectrum of ILW hierarchy in conformal field theory II: coset CFT's
• N=2 quiver gauge theories on A-type ALE spaces
• q-Vertex Operator from 5D Nekrasov Function
• Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models
• N=2 gauge theories, instanton moduli spaces and geometric representation theory
• Super-Liouville - Double Liouville correspondence
• $\widehat{\mathfrak{sl}}(n{)}_N$ WZW conformal blocks from $SU(N)$ instanton partition functions on ${\mathbb{C}}^2/{\mathbb{Z}}_n$
• $n$-th parafermion ${\mathcal{W}}_N$ characters from $U(N)$ instanton counting on ${\mathbb{C}}^2/{\mathbb{Z}}_n$
• Elliptic algebra, Frenkel-Kac construction and root of unity limit
• Coloured refined topological vertices and parafermion conformal field theories