Spectra of tropical Laplacians of classical root polytopes

Farhad Babaee and June Huh introduced the tropical Laplacian of a tropical surface and used it to disprove a generalized Hodge conjecture. We study four families of tropical surfaces arising from the root polytopes of types A ,B ,C and D. We compute the spectra of their tropical Laplacians for type A, and describe them conjecturally for types B, C and D. Our results confirm that these tropical Laplacians have exactly one negative eigenvalue, as anticipated by Babaee and Huh. Moreover, we give an explicit eigenbasis for the tropical Laplacian of An- i , proving its diagonalizability.