CONNECTING RANDOM MATRIX AND TOPOLOGICAL LG MODELS VIA INTEGRABILITY

Abstract

We prove that a bi-Hamiltonian system (e.g., a KdV system) is integrable if and only if its Krichever–Moser–Jacobi matrix L has a double eigenvalue, i.e., if (the characteristic polynomial of) L has rational double point singularities. Therefore, L (precisely, its characteristic polynomial) can be identified with the potential of a Landau–Ginzburg (LG) topological model with the same (rational double point) singularity. A rational curve is naturally defined in the corresponding Kummer variety and this explains the appearance (or non-appearance) of the doubling of the string equation as well as a phenomenon observed by Eguchi et al. Finally, we point out a parallelism between rational two-dimensional theories and rational singularities.