Adapted complex structures and the geodesic flow

Hall, Brian C.; Kirwin, William D.

doi:10.1007/s00208-010-0564-9

Cited by 28 publications

(46 citation statements)

References 27 publications

(45 reference statements)

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“…This family contains the one-parameter deformations of the standard Kähler structure that are considered in [FMMN05,FMMN05,HK11,LS10]. The Kähler structures will be constructed using a certain class of functions h : T * K → R which generalize the standard Hamiltonian of a free particle moving on K (i.e.…”

Section: Families Of Kähler Structures On T * Kmentioning

confidence: 99%

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Complex time evolution in geometric quantization and generalized coherent state transforms

Kirwin

Mourão²,

Nunes³

2013

Journal of Functional Analysis

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For the cotangent bundle T * K of a compact Lie group K, we study the complextime evolution of the vertical tangent bundle and the associated geometric quantization Hilbert space L 2 (K) under an infinite-dimensional family of Hamiltonian flows. For each such flow, we construct a generalized coherent state transform (CST), which is a unitary isomorphism between L 2 (K) and a certain weighted L 2 -space of holomorphic functions. For a particular set of choices, we show that this isomorphism is naturally decomposed as a product of a Heisenberg-type evolution (for complex time −τ ) within L 2 (K), followed by a polarization-changing geometric quantization evolution (for complex time +τ ). In this case, our construction yields the usual generalized Segal-Bargmann transform of Hall. We show that the infinite-dimensional family of Hamiltonian flows can also be understood in terms of Thiemann's "complexifier" method (which generalizes the construction of adapted complex structures). We will also investigate some properties of the generalized CSTs, and discuss how their existence can be understood in terms of Mackey's generalization of the Stone-von Neumann theorem.

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Section: Families Of Kähler Structures On T * Kmentioning

confidence: 99%

“…As discussed in the introduction, we will refer to h as a (Thiemann) complexifier function. The standard complex structure on T * K and the one-parameter family of deformations of it which are studied in [FMMN05,FMMN05,HK11,LS10] are all associated to the "kinetic energy" complexifier…”

Section: Families Of Kähler Structures On T * Kmentioning

confidence: 99%

Complex time evolution in geometric quantization and generalized coherent state transforms

Kirwin

Mourão²,

Nunes³

2013

Journal of Functional Analysis

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“…See also Theorem 15 in [HK08] and the description of the adapted complex structure in Section 1.1 of [Zel]. When β = 0, the expressions π(Φ i (x, p)) and π(Φ 1 (x, ip)) are no longer equal.…”

Section: Resultsmentioning

confidence: 99%

“…14]. Although the proof given in [HK08] does not apply in the untwisted case, we give here a simple proof, which works also in the untwisted case. We continue to assume that (M, g) is a compact, real-analytic Riemannian manifold and that β is a closed, real-analytic 2-form on M.…”

Section: Holomorphic Functionsmentioning

confidence: 89%

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Complex structures adapted to magnetic flows

Hall

Kirwin

2015

Journal of Geometry and Physics

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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.

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

2015

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Let P be a Delzant polytope. We show that the quantization of the corresponding toric manifold X P in toric Kähler polarizations and in the toric real polarization are related by analytic continuation of Hamiltonian flows evaluated at time t = √ −1s. We relate the quantization of X P in two different toric Kähler polarizations by taking the time-√ −1s Hamiltonian "flow" of strongly convex functions on the moment polytope P . By taking s to infinity, we obtain the quantization of X P in the (singular) real toric polarization.Recall that X P has an open dense subset which is biholomorphic to (C * ) n . The quantization of X P in a toric Kähler polarization can also be described by applying the complexified Hamiltonian flow of the Abreu-Guillemin symplectic potential g, at time t = √ −1, to an appropriate finite-dimensional subspace of quantum states in the quantization of T * T n in the vertical polarization. By taking other imaginary times, t = k √ −1, k ∈ R, we describe toric Kähler metrics with cone singularities along the toric divisors in X P .For convex Hamiltonian functions and sufficiently negative imaginary part of the complex time, we obtain degenerate Kähler structures which are negative definite in some regions of X P . We show that the pointwise and L 2 -norms of quantum states are asymptotically vanishing on negative-definite regions.

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Adapted complex structures and the geodesic flow

Cited by 28 publications

References 27 publications

Complex time evolution in geometric quantization and generalized coherent state transforms

Complex time evolution in geometric quantization and generalized coherent state transforms

Complex structures adapted to magnetic flows

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

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