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arXiv 1507.03540hep-th
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AGT，N-Burge划分与W_N极小模型

AGT, N-Burge partitions and W_N minimal models

Vladimir Belavin, Omar Foda, Raoul Santachiara

备注：
34 pages. More references, same content
发表信息：
Journal of High Energy Physics 10 (2015) 073
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摘要：

在共形场论$\mathcal{M}^{\, p, \, p^{\prime}}_N \! \times \! \mathcal{M}^{\mathcal{H}}$中，令${\mathcal B}^{\, p, \, p^{\prime}, \, {\mathcal H}}_{N, n}$是一个共形块，具有$n$个连续通道$χ_ı$，$ı= 1, \cdots, n$，其中$\mathcal{M}^{\, p, \, p^{\prime}}_N$是一个$\mathcal{W}_N$最小模型，由自旋为$1, \cdots, N$的手征场生成，并由两个互质整数$p$和$p^{\prime}$，$1 < p < p^{\prime}$标记，而$\mathcal{M}^{\mathcal{H}}$是一个自由玻色子共形场论。$\mathcal{B}^{\, p, \, p^{\prime}, \mathcal{H}}_{N, n}$是初始状态和最终状态之间的顶点运算符的期望值。每个顶点算子都被一个电荷向量标记，该电荷向量存在于李代数$A_{N-1}$的权格中，由权向量$ω_1, \cdots, ω_{N-1}$张成。我们限制我们的注意力与顶点运营商的电荷矢量点沿$ω_1$共形块。标记初始和最终状态的电荷矢量可以指向任何方向。遵循$\mathcal{W}_N$ AGT对应，并使用Nekrasov的瞬子配分函数不加修改地计算$\mathcal{B}^{\, p, \, p^{\prime}, \mathcal{H}}_{N, n}$，导致定义不清的表达式。我们表明，限制的状态，流动的渠道$χ_ı$，$ı= 1, \cdots, n$，状态标记的$N$分区，满足条件，我们称之为$N$ -伯奇分区，导致定义明确的表达式，我们确定与$\mathcal{B}^{\, p, \, p^{\prime}, \, \mathcal{H}}_{N, n}$。我们检查我们的识别表明，一个特定的非平凡的共形块，我们计算，使用$N$ -伯奇条件满足预期的微分方程。

英文摘要：

Let ${\mathcal B}^{\, p, \, p^{\prime}, \, {\mathcal H}}_{N, n}$ be a conformal block, with $n$ consecutive channels $χ_ı$, $ı= 1, \cdots, n$, in the conformal field theory $\mathcal{M}^{\, p, \, p^{\prime}}_N \! \times \! \mathcal{M}^{\mathcal{H}}$, where $\mathcal{M}^{\, p, \, p^{\prime}}_N$ is a $\mathcal{W}_N$ minimal model, generated by chiral fields of spin $1, \cdots, N$, and labeled by two co-prime integers $p$ and $p^{\prime}$, $1 < p < p^{\prime}$, while $\mathcal{M}^{\mathcal{H}}$ is a free boson conformal field theory. $\mathcal{B}^{\, p, \, p^{\prime}, \mathcal{H}}_{N, n}$ is the expectation value of vertex operators between an initial and a final state. Each vertex operator is labelled by a charge vector that lives in the weight lattice of the Lie algebra $A_{N-1}$, spanned by weight vectors $ω_1, \cdots, ω_{N-1}$. We restrict our attention to conformal blocks with vertex operators whose charge vectors point along $ω_1$. The charge vectors that label the initial and final states can point in any direction. Following the $\mathcal{W}_N$ AGT correspondence, and using Nekrasov's instanton partition functions without modification, to compute $\mathcal{B}^{\, p, \, p^{\prime}, \mathcal{H}}_{N, n}$, leads to ill-defined expressions. We show that restricting the states that flow in the channels $χ_ı$, $ı= 1, \cdots, n$, to states labeled by $N$ partitions that satisfy conditions that we call $N$-Burge partitions, leads to well-defined expressions that we identify with $\mathcal{B}^{\, p, \, p^{\prime}, \, \mathcal{H}}_{N, n}$. We check our identification by showing that a specific non-trivial conformal block that we compute, using the $N$-Burge conditions satisfies the expected differential equation.

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