Tropical geometry and integrable cellular automata II
Bilinear form and Tropical Fay's identity
The ultra-discrete Toda lattice (UD-Toda) is essentially equivalent to the integrable Box and Ball system, and considered to be a fundamental object in ultra-discrete integrable systems. In this talk, we construct the general solution of the UD-Toda with periodic boundary condition, by using the tropical theta function and the bilinear form. For the proof, we introduce a tropical analogue of Fay's trisecant identity for the tropical spectral curves of the periodic UDToda. As a result, we can also prove that the general isolevel set of the periodic UD-Toda is isomorphic to the tropical Jacobian.