On Combinatorial Expansion of the Conformal Blocks Arising from AGT Conjecture
On Combinatorial Expansion of the Conformal Blocks Arising from AGT Conjecture
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DOI：
10.1007/s11005-011-0503-z
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发表时间：
2011-10-01
影响因子：
1.2
通讯作者：
Tarnopolskiy, Grigory M.
中科院分区：
文献类型：
--
作者：
Alba, Vasyl A.;Fateev, Vladimir A.;Tarnopolskiy, Grigory M.
在他们最近的论文中，Alday等人。(Lett Math Phys 91：167-197,2010)提出了N=2四维超对称规范理论与二维共形场理论之间的关系。作为猜想的一部分，他们给出了由Nekrasov配分函数的瞬子部分的精确形式启发的共形块展开的显式组合公式。在本文中，我们从CFT的角度研究了这种膨胀的起源。我们考虑相互交换的Virasoro代数和Heisenberg代数的张量积的代数A=Vir圈乘H，发现了A的最高权表示中的状态的特殊正交基。在这个基中，初级场的矩阵元具有非常简单的因式分解形式，并且与瞬子计数文献中出现的函数Z(Bif)重合。有了这样一个简单的基础，共形块的计算问题就大大简化了，并且可以证明导致了Alday等人提出的展开式。(2010)。我们发现，这个基使与Benjamin-Ono可积族相关的无限可交换运动积分系对角化。
In their recent paper, Alday et al. (Lett Math Phys 91: 167-197, 2010) proposed a relation between N = 2 four-dimensional supersymmetric gauge theories and twodimensional conformal field theories. As part of their conjecture they gave an explicit combinatorial formula for the expansion of the conformal blocks inspired by the exact form of the instanton part of the Nekrasov partition function. In this paper we study the origin of such an expansion from a CFT point of view. We consider the algebra A= Vir circle times H which is the tensor product of mutually commuting Virasoro and Heisenberg algebras and discover the special orthogonal basis of states in the highest weight representations of A. The matrix elements of primary fields in this basis have a very simple factorized form and coincide with the function called Z(bif) appearing in the instanton counting literature. Having such a simple basis, the problem of computation of the conformal blocks simplifies drastically and can be shown to lead to the expansion proposed in Alday et al. (2010). We found that this basis diagonalizes an infinite system of commuting Integrals of Motion related to Benjamin-Ono integrable hierarchy.