Lagrangian fields, Calabi functions, and local symplectic groupoids

TL;DR

This paper introduces a framework connecting Lagrangian fields on symplectic manifolds with local symplectic groupoids, using Calabi functions to describe their structure and interactions.

Contribution

It constructs local symplectic groupoids from transversal Lagrangian fields and relates their properties to Calabi functions derived from symplectic forms.

Findings

Constructed local symplectic groupoids from transversal Lagrangian fields.

Expressed the groupoid's n-cycle manifold using Calabi functions.

Linked Lagrangian field geometry to symplectic groupoid structures.

Abstract

A Lagrangian field on a symplectic manifold $M$ is a family $\Lambda=\{\Lambda_x|x \in M\}$ of pointed Lagrangian submanifolds of $M$. This notion is a generalization of a real Lagrangian polarization for which each $\Lambda_x$ is the leaf containing $x$. Two Lagrangian fields $\Lambda$ and $\tilde \Lambda$ are called transversal if $\Lambda_x$ intersects $\tilde\Lambda_x$ transversally at $x$ for every $x$. Two transversal Lagrangian fields determine an almost para-K\"ahler structure on $M$. We construct a local symplectic groupoid on a neighborhood of the zero section of $T^\ast M$ from two transversal Lagrangian fields on $M$. The Lagrangian manifold of $n$-cycles of this groupoid in $(T^\ast M)^n$ has a generating function whose germ around the diagonal of $M^n$ is given by the $n$-point cyclic Calabi function of a closed (1,1)-form on a neighborhood of the diagonal of $M^2$ obtained from the symplectic form on $M$.