Quantum characteristic classes, moment correspondences and the Hamiltonian groups of coadjoint orbits

TL;DR

This paper explores quantum characteristic classes and moment correspondences in coadjoint orbits, revealing new relationships between Hamiltonian groups, quantum cohomology, and spectral sequences with implications for symplectic topology.

Contribution

It determines key terms of the Savelyev-Seidel morphism for coadjoint orbits and connects it to known morphisms, providing new insights into Hamiltonian groups and quantum cohomology.

Findings

Kernel dimension bound related to semi-simple rank

Bott-Samelson cycles solve min-max problem in Hofer's functional

Alternative proof of Peterson-Woodward's quantum cohomology formula

Abstract

For any coadjoint orbit $G/L$, we determine all useful terms of the associated Savelyev-Seidel morphism defined on $H_{-*}(\Omega G)$. Immediate consequences are: (1) the dimension of the kernel of the natural map $\pi_*(G)\otimes \mathbb{Q}\rightarrow \pi_*(Ham(G/L))\otimes \mathbb{Q}$ is at most the semi-simple rank of $L$, and (2) the Bott-Samelson cycles in $\Omega G$ which correspond to Peterson elements are solutions to the min-max problem for Hofer's max-length functional on $\Omega Ham(G/L)$. The proof is based on Bae-Chow-Leung's recent computation of Ma'u-Wehrheim-Woodward morphism for the moment correspondence associated to $G/T$ where $T$ is a maximal torus, the computation of Abbondandolo-Schwarz isomorphism for $G$, and two theoretical results including the coincidence of the above Savelyev-Seidel and Ma'u-Wehrheim-Woodward morphisms, and a Leray-type spectral sequence relating Savelyev-Seidel morphisms for $G/L$ and $G/T$. These ingredients also allow us to obtain an alternative proof of Peterson-Woodward's comparison formula which relates the quantum cohomology of $G/T$ to that of $G/L$.