AGT relations for abelian quiver gauge theories on ALE spaces

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• 作者：Mattia Pedrini^a ; ^b ; ^{mattia.pedro@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Francesco Sala^c ; ^d ; ^e ; ^{salafra83@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Richard J. Szabo^d ; ^e ; ^f ; ^{R.J.Szabo@hw.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：14D20 ; 14D21 ; 14J80 ; 81T13 ; 81T60
• 刊名：Journal of Geometry and Physics
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：103
• 期：Complete
• 页码：43-89
• 全文大小：785 K

文摘

We construct level one dominant representations of the affine Kac–Moody algebra 7449a1491eb254d">${\widehat{\mathfrak{g}\mathfrak{l}}}_k$ on the equivariant cohomology groups of moduli spaces of rank one framed sheaves on the orbifold compactification of the minimal resolution X_k$X_k$ of the 74d77c4ce" title="Click to view the MathML source">A_k−1$A_{k-1}$ toric singularity C²/Z_k${\mathbb{C}}^2/{\mathbb{Z}}_k$. We show that the direct sum of the fundamental classes of these moduli spaces is a Whittaker vector for 7449a1491eb254d">${\widehat{\mathfrak{g}\mathfrak{l}}}_k$, which proves the AGT correspondence for pure $\mathcal{N}=2U\left(1\right)$ gauge theory on X_k$X_k$. We consider Carlsson–Okounkov type Ext-bundles over products of the moduli spaces and use their Euler classes to define vertex operators. Under the decomposition 25f79410006805d62dc">${\widehat{\mathfrak{g}\mathfrak{l}}}_k\simeq \mathfrak{h}\oplus {\widehat{\mathfrak{s}\mathfrak{l}}}_k$, these vertex operators decompose as products of bosonic exponentials associated to the Heisenberg algebra 2511ca1ba4" title="Click to view the MathML source">h$\mathfrak{h}$ and primary fields of ${\widehat{\mathfrak{s}\mathfrak{l}}}_k$. We use these operators to prove the AGT correspondence for N=2$\mathcal{N}=2$ superconformal abelian quiver gauge theories on X_k$X_k$.