I provide an explicit construction of spectral curves for the affine $\mathrm{E}_8$ relativistic Toda chain. Their closed form expression is obtained by determining the full set of character relations in the representation ring of $\mathrm{E}_8$ for the exterior algebra of the adjoint representation; this is in turn employed to provide an explicit construction of both integrals of motion and the action-angle map for the resulting integrable system. I consider two main areas of applications of these constructions. On the one hand, I consider the resulting family of spectral curves in the context of the correspondences between Toda systems, 5d Seiberg-Witten theory, Gromov-Witten theory of orbifolds of the resolved conifold, and Chern-Simons theory to establish a version of the B-model Gopakumar-Vafa correspondence for the $\mathrm{sl}_N$ L\^e-Murakami-Ohtsuki invariant of the Poincar\'e integral homology sphere to all orders in $1/N$. On the other, I consider a degenerate version of the spectral curves and prove a 1-dimensional Landau-Ginzburg mirror theorem for the Frobenius manifold structure on the space of orbits of the extended affine Weyl group of type $\mathrm{E}_8$ introduced by Dubrovin-Zhang (equivalently, the orbifold quantum cohomology of the type-$\mathrm{E}_8$ polynomial $\mathbb{C} P^1$ orbifold). This leads to closed-form expressions for the flat co-ordinates of the Saito metric, the prepotential, and a higher genus mirror theorem based on the Chekhov-Eynard-Orantin recursion. I will also show how the constructions of the paper lead to a generalisation of a conjecture of Norbury-Scott to ADE $\mathbb{P}^1$-orbifolds, and a mirror of the Dubrovin-Zhang construction for all Weyl groups and choices of marked roots.

中文翻译：

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E8光谱曲线

我为仿射$ \ mathrm {E} _8 $相对论Toda链提供了光谱曲线的显式构造。通过确定伴随表示的外部代数$ \ mathrm {E} _8 $的表示环中的字符关系的全集，可以得到它们的闭式表达式。这反过来又被用来为所得到的可积分系统提供运动积分和作用角图的显式构造。我考虑了这些构造的两个主要应用领域。一方面，我考虑了Toda系统之间的对应关系，5d Seiberg-Witten理论，Gromov-Witten解析球面的球状体理论，和Chern-Simons理论建立Poincar积分同源性球的$ \ mathrm {sl} _N $ L \ ^ e-Murakami-Ohtsuki不变量的B模型Gopakumar-Vafa对应关系的一个版本$ 1 / N $。另一方面，我考虑了光谱曲线的简并形式，并证明了在$ \ mathrm {E} _8 $型扩展仿射Weyl群的轨道空间上的Frobenius流形结构的一维Landau-Ginzburg镜像定理。由Dubrovin-Zhang引入（等效地，类型为$ \ mathrm {E} _8 $多项式$ \ mathbb {C} P ^ 1 $ orbifold的单倍量子同调）。这导致了基于契ito夫-爱纳德-奥兰汀递归的齐藤度量，势能和更高属镜像定理的平坦坐标的闭式表达式。