On W-representations of β- and q,t-deformed matrix models
On W-representations of β- and q,t-deformed matrix models
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DOI：
10.1016/j.physletb.2019.03.047
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发表时间：
2019-01
影响因子：
4.4
通讯作者：
A.Morozov
中科院分区：
文献类型：
--
作者：
A.Morozov
W-表示通过在零模背景的真空态上加入一个割并算子来实现配分函数。我们给出了最简单矩形复矩阵模型的β-和q，t-变形的这类显式公式。在后一种情况下，而不是复杂的定义在多个杰克逊积分，我们定义的配分函数作为权重2系列，从麦克唐纳多项式，这是评估在不同的轨迹在空间中的时间变量。由此产生的表达式的W运营商似乎有关的问题，简单的赫尔维茨号码（贡献也是杨图，但所有的线长度2和1）。这个问题是已知的表现出良好的可积性。然而，对W的回答似乎出乎意料地复杂，需要改进。由于矩阵模型是所有规范理论和弦理论构造的基础，我们的练习很好地说明了β-形变和q，t-形变之间复杂性的跳跃--这在卡洛格罗-鲁伊斯塞纳斯哈密顿量的偶然简单水平上并不总是能看到（两种形变都同样简单）。然而，这种复杂性在网络模型、拓扑顶点和结点的理论中非常熟悉。
W-representation realizes partition functions by an action of a cut-and-join operator on the vacuum state with a zero-mode background. We provide explicit formulas of this kind for β-and q, t-deformations of the simplest rectangular complex matrix model. In the latter case, instead of the complicated definition in terms of multiple Jackson integrals, we define partition functions as the weight-two series, made from Macdonald polynomials, which are evaluated at different loci in the space of time variables. Resulting expression for the W ˆ operator appears related to the problem of simple Hurwitz numbers (contributing are also the Young diagrams with all but one lines of length two and one). This problem is known to exhibit nice integrability properties. Still the answer for W ˆ can seem unexpectedly sophisticated and calls for improvements. Since matrix models lie at the very basis of all gauge-and string-theory constructions, our exercise provides a good illustration of the jump in complexity between β-and q, t-deformations–which is not always seen at the accidentally simple level of Calogero-Ruijsenaars Hamiltonians (where both deformations are equally straightforward). This complexity is, however, quite familiar in the theories of network models, topological vertices and knots.