Complex structures adapted to magnetic flows

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 $T^{\ast }M$ 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 $T^{\ast }M$ by pushing forward the vertical polarization by the Hamiltonian flow "evaluated at time $i$." This complex structure fits together with $\omega -{\pi }^{\ast }\beta$ to give a Kaehler structure on a tube inside $T^{\ast }M$. 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-Szőke 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 Krötz-Thangavelu-Xu on heat kernel analysis on the Heisenberg group.

29 pages

References in corpus (3)

• Pluri-potential theory on Grauert tubes of real analytic Riemannian manifolds, I
• The exponential map of the complexification of {\em Ham} in the real-analytic case
• On complexified analytic Hamiltonian flows and geodesics on the space of Kahler metrics

Cited by in corpus (1)

• Coherent states for a 2-sphere with a magnetic field