Complex structures adapted to magnetic flows

Let MM be a compact real-analytic manifold, equipped with a real-analytic Riemannian metric gg, and let ββ be a closed real-analytic 2-form on MM, interpreted as a magnetic field. Consider the Hamiltonian flow on T∗MT∗M that describes a charged particle moving in the magnetic field ββ. Following an idea of T. Thiemann, we construct a complex structure on a tube inside T∗MT∗M by pushing forward the vertical polarization by the Hamiltonian flow “evaluated at time ii”. This complex structure fits together with ω−π∗βω−π∗β to give a Kähler structure on a tube inside T∗MT∗M. When β=0β=0, our magnetic complex structure is the adapted complex structure of Lempert–Szőke and Guillemin–Stenzel.We describe the magnetic complex structure in terms of its (1,0)(1,0)-tangent bundle, at the level of holomorphic functions, and via a construction using the embeddings of Whitney–Bruhat and Grauert. We describe an antiholomorphic intertwiner between this complex structure and the complex structure induced by −β−β, and we give two formulas for local Kähler potentials, which depend on a local choice of vector potential 11-form for ββ. Finally, we compute the magnetic complex structure explicitly for constant magnetic fields on R2R2 and S2S2.