Lefschetz fibrations on adjoint orbits

TL;DR

This paper demonstrates that adjoint orbits of semisimple Lie algebras can be structured as symplectic Lefschetz fibrations, analyzing their topology and computing related categories for specific cases.

Contribution

It establishes the symplectic Lefschetz fibration structure on adjoint orbits and computes their topological invariants and Fukaya--Seidel categories for particular examples.

Findings

Adjoint orbits admit symplectic Lefschetz fibrations.

Topological invariants like Betti numbers are computed.

Fukaya--Seidel categories are explicitly calculated for sl(2,C).

Abstract

We prove that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We then describe the topology of the regular and singular fibres, in particular calculating their middle Betti numbers. For the example of sl(2,C) we compute the Fukaya--Seidel category of Lagrangian vanishing cycles.