Symplectic geometry of electrical networks

Boris Bychkov, Vassily Gorbounov, Lazar Guterman, Anton Kazakov

Published 2024-06-05Version 1

In this paper we relate a well-known in symplectic geometry compactification of the space of symmetric bilinear forms on the Lagrangian Grassmannian to the specific compactifications of the space of electrical networks in the disc obtained in \cite{L}, \cite{CGS} and \cite{BGKT}. In particular, we state an explicit connection between these works and describe some of the combinatorics developed there in the language of symplectic geometry. We also show that the combinatorics of the concordance vectors forces the uniqueness of the symplectic form, such that corresponding points of the Grassmannian are isotropic. We define a notion of Lagrangian concordance which provides a construction of the compactification of the space of electrical networks in the positive part of the Lagrangian Grassmannian bypassing the construction from \cite{L}.