Dimer models, integrable systems and quantum Teichmüller space
Dimer models, integrable systems and quantum Teichmüller space
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二聚体模型、可积系统和量子 Teichmüller 空间
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发表时间：
2011
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通讯作者：
S. Franco
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作者：
S. Franco
我们介绍了二聚体模型(以及由此产生的超共形箭图)与通过镜像对称与之相联系的黎曼曲面的量子Teichmüler空间之间的对应关系。通过展开贴图，每个薄膜平铺都会产生黎曼曲面的平铺，面部围绕着穿孔。我们解释了如何通过对偶这种平铺来获得理想的三角剖分。为了做到这一点，价大于3的瓦片节点(在相应的箭图规范理论中等价于大于3的级数的超势项)必须通过引入2价节点来分解。从箭袋规范理论的角度来看，这一操作对应于在大质量场中积分。Teichmüler空间中的Fock坐标与箭图中的手征场一一对应。我们给出了多个显式的例子，包括无穷族的理论，说明了如何通过这个过程产生正确数量的Fock坐标。最后，我们解释了在量子可积系统的背景下，Chekhov和Fock坐标之间的对易关系如何产生与Goncharov和Kenyon的二聚体模型相关联的交换子。对于一般的二聚体模型(即那些包含非三价节点的模型)，这种匹配需要引入Chekhov和Fock规则的自然推广。我们还解释了如何将原始地砖中的城市更新(箭袋的Seiberg二元性)映射到理想三角剖分的翻转。
We introduce a correspondence between dimer models (and hence superconformal quivers) and the quantum Teichmüller space of the Riemann surfaces associated to them by mirror symmetry. Via the untwisting map, every brane tiling gives rise to a tiling of the Riemann surface with faces surrounding punctures. We explain how to obtain an ideal triangulation by dualizing this tiling. In order to do so, tiling nodes of valence greater than 3 (equivalently superpotential terms of order greater than 3 in the corresponding quiver gauge theories) must be decomposed by the introduction of 2-valent nodes. From a quiver gauge theory perspective, this operation corresponds to integrating-in massive fields. Fock coordinates in Teichmüller space are in one-to-one correspondence with chiral fields in the quiver. We present multiple explicit examples, including infinite families of theories, illustrating how the right number of Fock coordinates is generated by this procedure. Finally, we explain how Chekhov and Fock commutation relations between coordinates give rise to the commutators associated to dimer models by Goncharov and Kenyon in the context of quantum integrable systems. For generic dimer models (i.e. those containing nodes that are not 3-valent), this matching requires the introduction of a natural generalization of Chekhov and Fock rules. We also explain how urban renewal in the original brane tiling (Seiberg duality for the quivers) is mapped to flips of the ideal triangulation.