从矩阵模型和量子场到Hurwitz空间和绝对伽罗瓦群
From Matrix Models and quantum fields to Hurwitz space and the absolute Galois group
摘要：
我们证明了具有一般势的厄米单矩阵模型的变换子可以映射到某些三重排列的计数，从而映射到从世界单到球目标的具有三个分支点的全纯映射的计数.这使得使用旧的矩阵模型的结果，推导出新的显式公式一类的Hurwitz数。根据Belyi定理，具有三个分支点的全纯映射与定义在代数数$\bmQ$上的曲线和映射有关。这表明在一般耦合下的单矩阵模型的弦论对偶具有在代数数和目标空间$ \mP^1 (\bmQ)$上定义的世界表。绝对伽罗瓦群$ Gal (\bmQ / \mQ) $作用在1-矩阵模型的Feynman图上，它与Grothendieck的Dessins d 'Enfants有关.多矩阵模型的相关器被映射到由置换群的子群定义的等价置换的三元组的计数。这与Grothendieck Dessins边缘的着色有关。色边Dessins是描述Grothendieck Dessins上的$ Gal (\bmQ / \mQ) $作用的一些已知不变量和定义新不变量的有用工具。
英文摘要：
We show that correlators of the hermitian one-Matrix model with a general potential can be mapped to the counting of certain triples of permutations and hence to counting of holomorphic maps from world-sheet to sphere target with three branch points on the target. This allows the use of old matrix model results to derive new explicit formulae for a class of Hurwitz numbers. Holomorphic maps with three branch points are related, by Belyi's theorem, to curves and maps defined over algebraic numbers $\bmQ$. This shows that the string theory dual of the one-matrix model at generic couplings has worldsheets defined over the algebraic numbers and a target space $ \mP^1 (\bmQ)$. The absolute Galois group $ Gal (\bmQ / \mQ) $ acts on the Feynman diagrams of the 1-matrix model, which are related to Grothendieck's Dessins d'Enfants. Correlators of multi-matrix models are mapped to the counting of triples of permutations subject to equivalences defined by subgroups of the permutation groups. This is related to colorings of the edges of the Grothendieck Dessins. The colored-edge Dessins are useful as a tool for describing some known invariants of the $ Gal (\bmQ / \mQ) $ action on Grothendieck Dessins and for defining new invariants.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
