Universität zu Köln

Kirwin, William D., Mourao, Jose M. and Nunes, Joao P. (2016). Complex symplectomorphisms and pseudo-Kahler islands in the quantization of toric manifolds. Math. Ann., 364 (1-2). S. 1 - 29. HEIDELBERG: SPRINGER HEIDELBERG. ISSN 1432-1807

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Abstract

Let P be a Delzant polytope. We show that the quantization of the corresponding toric manifold X-P in toric Kahler polarizations and in the toric real polarization are related by analytic continuation of Hamiltonian flows evaluated at time t = root-1s. We relate the quantization of X-P in two different toric Kahler polarizations by taking the time t = root-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 Kahler polarization can also be described by applying the complexified Hamiltonian flow of the Abreu-Guillemin symplectic potential g, at time t = root-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 root-1, k is an element of R, we describe toric Kahler 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 Kahler 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.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Kirwin, William D. UNSPECIFIED UNSPECIFIED UNSPECIFIED Mourao, Jose M. UNSPECIFIED UNSPECIFIED UNSPECIFIED Nunes, Joao P. UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-284917
DOI:	10.1007/s00208-015-1205-0
Journal or Publication Title:	Math. Ann.
Volume:	364
Number:	1-2
Page Range:	S. 1 - 29
Date:	2016
Publisher:	SPRINGER HEIDELBERG
Place of Publication:	HEIDELBERG
ISSN:	1432-1807
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language MONGE-AMPERE EQUATION; COHERENT-STATE TRANSFORM; GEOMETRIC-QUANTIZATION; RIEMANNIAN-MANIFOLDS; CAUCHY-PROBLEM; GRAUERT TUBES; LIE-GROUPS; METRICS; VARIETIES; ORBIFOLDS Multiple languages Mathematics Multiple languages
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/28491