arXiv subjects

Luiz Frederic Wagner

Publications and source records attributed to Luiz Frederic Wagner.

3 recordsLinked to original sources

arXiv2025

Pseudo-Holomorphic Hamiltonian Systems and Kähler Duality of Complex Coadjoint Orbits

This thesis is split up into two parts: The first one concerns (pseudo)-holomorphic Hamiltonian systems, while the second part is about Kähler structures of complex coadjoint orbits. We begin the first part by investigating basic properties of holomorphic Hamiltonian systems (HHSs) like maximal holomorphic trajectories and holomorphic Hamiltonian foliations. Afterwards, we use these notions to combine HHSs with two structures frequently studied in geometry, namely Lefschetz and almost toric fibrations, leading us to the notion of a holomorphic symplectic Lefschetz fibration. Following this examination, we formulate action functionals for HHSs. These action functionals are well-suited to find periodic orbits. However, it turns out that HHSs rarely exhibit periodic orbits. One possible obstruction for a HHS to possess periodic orbits is the integrability of the underlying complex structure. To circumvent this obstruction, we introduce the notion of pseudo-holomorphic Hamiltonian systems (PHHSs) which allow us to describe classical mechanics on almost complex manifolds. We show that PHHSs satisfy almost the same properties as HHSs. In the second part, we study coadjoint orbits of complex Lie groups and show that they also exhibit, next to their Hyperkähler structure which was introduced by Kronheimer and Kovalev in the 90s, a holomorphic Kähler structure. A holomorphic Kähler structure can be thought of as a complexification of a usual Kähler structure. We call the fact that a space admits both Hyperkähler and holomorphic Kähler structures ''Kähler duality''. In this thesis, we suspect that the Kähler duality of complex coadjoint orbits can be traced back to double cotangent bundles. Precisely speaking, we conjecture that double cotangent bundles naturally exhibit Kähler duality and that this Kähler duality transfers via reduction to complex coadjoint orbits.

math.SG↗

arXiv2023

Pseudo-Holomorphic Hamiltonian Systems

In this paper, we first explore holomorphic Hamiltonian systems. In particular, we define action functionals for those systems and show that holomorphic trajectories obey an action principle, i.e., that they can be understood - in some sense - as critical points of these action functionals. As an application, we use holomorphic Hamiltonian systems to establish a relation between Lefschetz fibrations and almost toric fibrations. During the investigation of action functionals for holomorphic Hamiltonian systems, we observe that the complex structure $J$ corresponding to a holomorphic Hamiltonian system poses strong restrictions on the existence of certain trajectories. For instance, no non-trivial holomorphic trajectories with complex tori as domains can exist in $\mathbb{C}^{2n}$ due to the maximum principle. To lift this restriction, we generalize the notion of holomorphic Hamiltonian systems to systems with non-integrable almost complex structures $J$ leading us to the definition of pseudo-holomorphic Hamiltonian systems. We show that these systems exhibit properties very similar to their holomorphic counterparts, notably, that they are also subject to an action principle. Furthermore, we prove that the integrability of $J$ is equivalent to the closedness of the "pseudo-holomorphic symplectic" form. Lastly, we show that, aside from dimension four, the set of proper pseudo-holomorphic Hamiltonian systems is open and dense in the set of pseudo-holomorphic Hamiltonian systems by considering deformations of holomorphic Hamiltonian systems. This implies that proper pseudo-holomorphic Hamiltonian systems are generic.

math.SG↗

arXiv2021

Revisiting Hadronic Mass Relations from First Order Flavor Symmetry Breaking

The Gell-Mann$-$Okubo (GMO) mass relations following from $\text{SU}(3)$-flavor symmetry breaking in the strong sector have been proven to be of great success for describing hadron masses and classifying them into multiplets. Nowadays, it is widely believed that baryons being fermions have to obey GMO relations which are linear in the baryon masses, whilst mesons as bosons follow quadratic GMO relations. In this work, I challenge the distinction of GMO relations for baryons and mesons and conclude that both linear and quadratic GMO relations apply to both baryons and mesons in most instances (exceptions are the pseudoscalar meson octet and the mass relations following from heavy quark symmetry). I support this claim by presenting an in-depth analysis of the GMO relations on a theoretical and experimental level. On the theoretical side, two approaches to the derivation of the GMO relations are given and isospin symmetry breaking, electromagnetic contributions, and heavy quark symmetry are incorporated into the mass relations. Both linear and quadratic mass relations are checked against experimentally determined hadron masses. Furthermore, the mass relations are used to classify unassigned resonances like $Λ^+_c(2940)$ and $Ξ_c(3123)$ into multiplets and to predict the masses of missing particles in yet incomplete multiplets.

hep-ph↗