Symplectic $G$-capacities and integrable systems

TL;DR

This paper introduces $G$-equivariant symplectic capacities for Lie group actions, especially for integrable systems, and investigates their continuity properties within symplectic $G$-categories.

Contribution

It constructs a new class of symplectic capacities invariant under Lie group actions and explores their behavior in the context of integrable systems.

Findings

Defined $G$-equivariant symplectic capacities for specific Lie groups.

Established invariance of these capacities for integrable systems.

Analyzed the continuity properties of the capacities in symplectic $G$-categories.

Abstract

For any Lie group $G$, we construct a $G$-equivariant analogue of symplectic capacities and give examples when $G = \mathbb{T}^k\times\mathbb{R}^{d-k}$, in which case the capacity is an invariant of integrable systems. Then we study the continuity of these capacities, using the natural topologies on the symplectic $G$-categories on which they are defined.