Complex structures adapted to magnetic flows

TL;DR

This paper constructs a magnetic-adapted complex structure on the cotangent bundle of a real-analytic manifold, extending known structures to include magnetic fields and providing explicit formulas and examples.

Contribution

It introduces a novel magnetic complex structure on cotangent bundles, generalizing the adapted complex structure to incorporate magnetic fields and analyzing its properties.

Findings

Constructed a magnetic complex structure using Hamiltonian flow at imaginary time.

Described the structure in terms of holomorphic functions and embeddings.

Explicitly computed the structure for constant magnetic fields on  and Sb2.

Abstract

Let $M$ be a compact real-analytic manifold, equipped with a real-analytic Riemannian metric $g,$ and let $\beta$ be a closed real-analytic 2-form on $M$, interpreted as a magnetic field. Consider the Hamiltonian flow on $T^*M$ that describes a charged particle moving in the magnetic field $\beta$. Following an idea of T. Thiemann, we construct a complex structure on a tube inside $T^*M$ by pushing forward the vertical polarization by the Hamiltonian flow "evaluated at time $i$." This complex structure fits together with $\omega-\pi^*\beta$ to give a Kaehler structure on a tube inside $T^*M$. We describe this magnetic complex structure in terms of its $(1,0)$-tangent bundle, at the level of holomorphic functions, and via a construction using the embeddings of Whitney-Bruhat and Grauert, which is a magnetic analogue to the analytic continuation of the geometric exponential map. We describe an antiholomorphic intertwiner between this complex structure and the complex structure induced by $-\beta$, and we give two formulas for local Kaehler potentials, which depend on a local choice of vector potential 1-form for $\beta$. When $\beta=0$, our magnetic complex structure is the adapted complex structure of Lempert-Sz\H{o}ke and Guillemin-Stenzel. We compute the magnetic complex structure explicitly for constant magnetic fields on $\mathbb{R}^{2}$ and $S^{2}.$ In the $\mathbb{R}^{2}$ case, the magnetic adapted complex structure for a constant magnetic field is related to work of Kr\"otz-Thangavelu-Xu on heat kernel analysis on the Heisenberg group.