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 β 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 β. 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 ω − π * β to give a Kähler 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 −β, and we give two formulas for local Kähler potentials, which depend on a local choice of vector potential 1-form for β. When β = 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 R 2 and S 2 . In the 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.

Adapted complex and involutive structures

“…The construction of adapted complex structures in [Bi,HK2,Sz1,Sz4] can be obtained from Example 4.2. Those structures correspond to the choice c = i, whereas the involutive structures above are complex only if c ∈ C \ R is sufficiently small.…”

“…This difference can be overcome by a certain scaling. The second order M valued ODEs in [Bi,HK2,Sz1,Sz4] have constant maps as solutions. By rescaling, using Example 4.2 one obtains a result of the following nature: Any x ∈ N representing a constant map [−r, r] → M has a neighborhood on which the involutive structure P (i) is a complex structure.…”

“…Simultaneously Guillemin and Stenzel in [GS] investigated related complex structures on the cotangent bundle of M ; the two structures turn out to be the same if T M and T * M are identified using the metric. Subsequently Bielawski, Duchamp-Kalka, Hall-Kirwin, Szőke, and Ursitti [Bi,DK,HK1,HK2,Sz4,U] proposed generalizations that involved manifolds with connection, Finsler metrics, sub-Riemannian manifolds and Riemannian manifolds endowed with a closed 2-form. Independently, Thiemann [Th] constructed related complex structures for general analytic Hamiltonian systems.…”

Complex symplectomorphisms and pseudo-Kähler islands in the quantization of toric manifolds

“…Hamiltonian flows in complex time were studied in the context of quantum gravity by Thiemann [Th]. In [HK1,HK2], the evolution of polarizations under complexified flows on cotangent bundles was studied.…”

Quantization in singular real polarizations: Kähler regularization, Maslov correction and pairings

“…Denoting the corresponding real polarization by P µ , the proposal of [MN2] corresponds then to obtaining the Hilbert space H µ , of quantum states in the polarization P µ , as the infinite imaginary time limit √ −1s with s → +∞, of the family defined by applying the imaginary time flow of the Hamiltonian vector field of the norm square of the moment map, X ||µ|| 2 , to the Hilbert space corresponding to a starting Kähler quantization. In order to relate the Schrödinger representation to this Kähler polarization, we consider the Thiemann complexifier method [Th1,Th2] adapted to geometric quantization in [HK1,HK2,KMN2,KMN3,KMN4]. In the toric case, these families of polarizations were first introduced in [BFMN].…”