Maurer-Cartan方程渐近分析的精细散射图和θ函数
Refined scattering diagrams and theta functions from asymptotic analysis of Maurer-Cartan equations
摘要：
我们进一步发展的渐近分析方法的散射图的研究。我们这样做，通过分析的Maurer-Cartan元素的渐近行为的微分分次李代数构造从（不一定热带）么半群分次李代数。在这个框架中，我们给出了另一种微分几何证明的一致完成的散射图，最初证明了Kontsevich-Soibelman，Gross-Siebert和Bridgeland。我们还给出了θ函数及其跨壁的几何解释。在热带环境中，我们解释Maurer-Cartan元素，因此一致的散射图，在热带磁盘的精细计数。我们还描述了θ函数，在他们的热带和霍尔代数设置，在平坦的部分的Maurer-Cartan变形微分。特别是，这使我们能够给出霍尔代数θ函数的非退化斜对称欧拉形式的非循环箭图的组合描述。
英文摘要：
We further develop the asymptotic analytic approach to the study of scattering diagrams. We do so by analyzing the asymptotic behavior of Maurer-Cartan elements of a differential graded Lie algebra constructed from a (not-necessarily tropical) monoid-graded Lie algebra. In this framework, we give alternative differential geometric proofs of the consistent completion of scattering diagrams, originally proved by Kontsevich-Soibelman, Gross-Siebert and Bridgeland. We also give a geometric interpretation of theta functions and their wall-crossing. In the tropical setting, we interpret Maurer-Cartan elements, and therefore consistent scattering diagrams, in terms of the refined counting of tropical disks. We also describe theta functions, in both their tropical and Hall algebraic settings, in terms of flat sections of the Maurer-Cartan-deformed differential. In particular, this allows us to give a combinatorial description of Hall algebra theta functions for acyclic quivers with non-degenerate skew-symmetrized Euler forms.
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