Spectral curves of polygons and triangulated tori

我们基于“离散曲面”上的全纯线束的概念，研究了一种离散共态的方法，即在带有黑色和白色面的三角形曲面的顶点集上。作为一个特例，我们重新解释了Dynnikov和Novikov对S2 = CP1的共形映射的方法，揭示了它是离散全纯理论的第一个例子，它同时是Möbius-invariant和由线性方程控制的。我们在S4 = HP1中引入离散曲面任意浸入的Darboux变换，该变换可以解释为浸入空间上的时间离散Davey-Stewartson流。对于具有正则组合的离散环面的一般浸没，我们证明了达布变换的空间可以解广义化为紧致的黎曼曲面（谱曲线），从而从代数完全可积系统的理论中获得了强有力的方法。在第二阶段，除了三角曲面的孤子理论之外，我们的研究将集中于开发“离散黎曼曲面”浸入的共形性的定义。此外，我们计划研究一类新的“离散最小曲面”，这种曲面在我们的研究中自然出现。

We investigate an approach to discrete conformality based on the notion of holomorphic line bundles over "discrete surfaces", that is, over vertex sets of triangulated surfaces with black and white colored faces. As a special case, we give a reinterpretation of Dynnikov's and Novikov's approach to conformal maps to S2 = CP1 which reveals it as the first example of a theory of discrete holomorphicity that is at the same time Möbius-invariant and governed by linear equations.We introduce Darboux transformations for arbitrary immersions of discrete surfaces into S4 = HP1 which can be interpreted as a time discrete Davey-Stewartson flow on the space of immersions. For generic immersions of discrete tori with regular combinatorics, we show that the space of Darboux transformations can be desingularized to a compact Riemann surface (the spectral curve) thus making available powerful methods from the theory of algebraically completely integrable systems.In the second period, beyond the soliton theory of triangulated surfaces, our investigations will concentrate on developing a definition of conformality for immersions of "discrete Riemann surfaces". Moreover, we plan to study a new class of "discrete minimal surfaces" that appears naturally in the context of our investigations.