Critical values of moment maps on quantizable manifolds

TL;DR

This paper investigates the critical values of moment maps on quantizable symplectic manifolds with torus actions, establishing relationships between fixed points, weights, and partition functions, revealing symmetry properties in their structure.

Contribution

It introduces a novel partitioning of fixed points based on isotropy weights and proves the existence of a map linking these partitions, along with symmetry relations for associated partition functions.

Findings

Existence of a map linking fixed point partitions with specific moment map value differences.

Symmetry relation between sums of partition functions over different fixed point sets.

Partition functions exhibit equality for large lattice elements, indicating deep structural symmetry.

Abstract

Let $M$ be a quantizable symplectic manifold acted on by $T=(S^1)^r$ in a Hamiltonian fashion and $J$ a moment map for this action. Suppose that the set $M^{T}$ of fixed points is discrete and denote by ${\alpha}_{pj}\in{\mathbb Z}^r$ the weights of the isotropy representation at $p$. By means of the $\alpha_{pj}$'s we define a partition ${\mathcal Q}_+$, ${\mathcal Q}_-$ of $M^T$. (When $r=1$, ${\mathcal Q}_{\pm}$ will be the set of fixed points such that the half of the Morse index of $J$ at them is even (odd)). We prove the existence of a map $\pi_{\pm}:{\mathcal Q}_{\pm}\to{\mathcal Q}_{\mp}$ such that $J(q)-J(\pi_{\pm}(q))\in I_{\mp}$, for all $q\in {\mathcal Q}_{\pm}$, where $I_{\pm}$ is the lattice generated by the $\alpha_{pj}$'s with $p\in{\mathcal Q}_{\pm}.$ We define partition functions $N_p$ similar to the ones of Kostant \cite{Gui} and we prove that $\sum_{p\in{\mathcal Q}_+}N_p(l)=\sum_{p\in{\mathcal Q}_-}N_p(l)$, for any $l\in{\mathbb Z}^r$ with $|l|$ sufficiently large.