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Hall, Brian C. and Kirwin, William D. (2015). Complex structures adapted to magnetic flows. J. Geom. Phys., 90. S. 111 - 132. AMSTERDAM: ELSEVIER SCIENCE BV. ISSN 1879-1662

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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 Kahler structure on a tube inside T*M. When beta = 0, our magnetic complex structure is the adapted complex structure of Lempert-Szoke and Guillemin-Stenzel. We describe the 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. We describe an antiholomorphic intertwiner between this complex structure and the complex structure induced by -beta, and we give two formulas for local Kahler potentials, which depend on a local choice of vector potential 1-form for beta. Finally, we compute the magnetic complex structure explicitly for constant magnetic fields on R-2 and S-2. (C) 2015 Elsevier B.V. All rights reserved.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Hall, Brian C. UNSPECIFIED UNSPECIFIED UNSPECIFIED Kirwin, William D. UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-408488
DOI:	10.1016/j.geomphys.2015.01.015
Journal or Publication Title:	J. Geom. Phys.
Volume:	90
Page Range:	S. 111 - 132
Date:	2015
Publisher:	ELSEVIER SCIENCE BV
Place of Publication:	AMSTERDAM
ISSN:	1879-1662
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language MONGE-AMPERE EQUATION; GAUGE FIELD-THEORY; COHERENT STATES GCS; GEOMETRIC-QUANTIZATION; RIEMANNIAN-MANIFOLDS; GRAUERT TUBES; TRANSFORM; BUNDLE Multiple languages Mathematics, Applied; Mathematics; Physics, Mathematical Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/40848