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Coset conformal field theory and instanton counting on C^2/Z_p
High Energy Physics - Theory · Submitted 17 Jun 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study conformal field theory with the symmetry algebra $\mathcal{A}(2,p)=\hat{\mathfrak{gl}}(n)_{2}/\hat{\mathfrak{gl}}(n-p)_2$. | contribution 0.82 | 0.86 | 0.06 | asserted 0.93 |
| s1 | 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. | contribution 0.62 | 0.90 | 0.13 | speculative 0.49 |
| s2 | We show that the algebra $\mathcal{A}(2,p)$ can be realized in two ways. | result 0.80 | 0.85 | 0.09 | asserted 1.00 |
| s3 | The first realization is connected with the cross-product of $p$ Virasoro and $p$ Heisenberg algebras: $\mathcal{H}^{p}\times \textrm{Vir}^{p}$. | result 0.85 | 0.72 | 0.07 | asserted 1.00 |
| s4 | The second realization is connected with: $\mathcal{H}^{p}\times \hat{\mathfrak{sl}}(p)_2\times (\hat{\mathfrak{sl}}(2)_p \times \hat{\mathfrak{sl}}(2)_{n-p}/\hat{\mathfrak{sl}}(2)_n)$. | result 0.86 | 0.76 | 0.08 | asserted 1.00 |
| s5 | The equivalence of these two realizations provides the non-trivial identity for the characters of $\mathcal{A}(2,p)$. | result 0.74 | 0.83 | 0.23 | asserted 1.00 |
| s6 | The moduli space of instantons on $\mathbb{C}^{2}/\mathbb{Z}_{p}$ admits two different compactifications. | result 0.70 | 0.39 | 0.05 | asserted 1.00 |
| s7 | This leads to two different bases for the representations of $\mathcal{A}(2,p)$. | result 0.68 | 0.42 | 0.07 | asserted 1.00 |
| s8 | We use this fact to explain the existence of two forms of the instanton pure partition functions. | interpretation 0.88 | 0.87 | 0.18 | asserted 0.97 |
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The paper
Original abstract
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.
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References
How the literature uses it
17 arXiv papers cite this in 21 sentences
Papers citing it · 14 sentences without a confident label
Compares with it 1 paper
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Our BRST definition (and the gluing construction in the case of [ref] ) seems to produce the same (or closely related) algebras to those discussed in [ref] .
Webs of W-algebras · Outlook · 1.00
Cites it as background 5 papers
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Moreover similar correspondence has been found and examined [ref] .
$q$-Virasoro/W Algebra at Root of Unity and Parafermions · Introduction · 0.93
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For works in this direction, see for example [ref] .
Elliptic algebra, Frenkel-Kac construction and root of unity limit · Introduction · 0.97
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The list of papers which discuss subjects closely related to that of this subsection include [ref] .
Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models · more recent developments · 1.00
What it cites · 17 sentences without a confident label
Builds on 5 papers
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The present work can be considered as the continuation of the line of studies started in [ref] .
Ramond sector of super Liouville theory from instantons on an ALE space · Introduction · 0.85
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The present work can be considered as the continuation of the line of studies started in [ref] .
Parafermionic Liouville field theory and instantons on ALE spaces · Introduction · 0.85
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The present work can be considered as the continuation of the line of studies started in [ref] .
Instanton moduli spaces and bases in coset conformal field theory · Introduction · 0.85
Uses data or values from 2 papers
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The partition function in this approach was calculated in [ref] : [equation] where $a⃗=(a,-a)$, $a⃗^(σ)_s= (a^(σ)_s,-a^(σ)_s)$, and $a^(σ)_s = a + (n_σ+1+d_σ+1^s)ϵ_1^(σ) + (n_σ+d_σ^s)ϵ_2^(σ)$, and regularization parameters are $ϵ_1^(σ)=(p-σ)ϵ_1-σϵ_2$, $ϵ_2^(σ)=(σ+1-p)ϵ_1+(σ+1)ϵ_2$.
Gauge Theories on ALE Space and Super Liouville Correlation Functions · First compactification · 0.85
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The partition function in this approach was calculated in [ref] : [equation] where $a⃗=(a,-a)$, $a⃗^(σ)_s= (a^(σ)_s,-a^(σ)_s)$, and $a^(σ)_s = a + (n_σ+1+d_σ+1^s)ϵ_1^(σ) + (n_σ+d_σ^s)ϵ_2^(σ)$, and regularization parameters are $ϵ_1^(σ)=(p-σ)ϵ_1-σϵ_2$, $ϵ_2^(σ)=(σ+1-p)ϵ_1+(σ+1)ϵ_2$.
N=2 gauge theories on toric singularities, blow-up formulae and W-algebrae · First compactification · 0.85
Cites as background 12 papers
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The original so-called AGT relation proposed in [ref] states the equality between the correlation functions in the Liouville field theory and the partition function of the $𝒩=2$ supersymmetric Yang-Mills theory with the $SU(2)$ gauge group (for the generalizations of the AGT correspondence for the other gauge groups and conformal field theories see [ref] ).
Liouville Correlation Functions from Four-dimensional Gauge Theories · Introduction · 0.97
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The original so-called AGT relation proposed in [ref] states the equality between the correlation functions in the Liouville field theory and the partition function of the $𝒩=2$ supersymmetric Yang-Mills theory with the $SU(2)$ gauge group (for the generalizations of the AGT correspondence for the other gauge groups and conformal field theories see [ref] ).
A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories · Introduction · 0.97
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The original so-called AGT relation proposed in [ref] states the equality between the correlation functions in the Liouville field theory and the partition function of the $𝒩=2$ supersymmetric Yang-Mills theory with the $SU(2)$ gauge group (for the generalizations of the AGT correspondence for the other gauge groups and conformal field theories see [ref] ).
On AGT relation in the case of U(3) · Introduction · 0.97
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