Symplectic geometry of electrical networks

TL;DR

This paper explores the symplectic geometric structure underlying electrical networks, connecting compactifications of network spaces with Lagrangian Grassmannian geometry and introducing a new notion of Lagrangian concordance.

Contribution

It establishes explicit links between symplectic geometry and electrical network compactifications, and introduces the concept of Lagrangian concordance for network analysis.

Findings

Connection between symplectic compactification and electrical network spaces.

Uniqueness of symplectic form enforced by combinatorics of concordance vectors.

Definition of Lagrangian concordance for network space compactification.

Abstract

In this paper we relate a well-known in symplectic geometry compactification of the space of symmetric bilinear forms considered as a chart of 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}.