Abstract
I provide an explicit construction of spectral curves for the affine E8 relativistic Toda chain. Their closed form expression is obtained by determining the full set of character relations in the representation ring of E8 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 slN Lê–Murakami–Ohtsuki invariant of the Poincaré 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 E8 introduced by Dubrovin–Zhang (equivalently, the orbifold quantum cohomology of the type-E8 polynomial CP 1 orbifold). This leads to closed-form expressions for the flat coordinates 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 P 1-orbifolds, and a mirror of the Dubrovin–Zhang construction for all Weyl groups and choices of marked roots.