On the BKS pairing for Kähler quantizations of the cotangent bundle of a Lie group

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• 作者：Carlos Florentino ; Pedro Matias ; Jos&#xe9 ; Mour&#xe3 ; o and Jo&#xe3 ; o P. Nunes
• 关键词：Blattner– ; Kostant– ; Sternberg pairing ; Geometric quantization ; Coherent state transform for Lie groups
• 刊名：Journal of Functional Analysis
• 出版年：2006
• 出版时间：1 May 2006
• 年：2006
• 卷：234
• 期：1
• 页码：180-198
• 全文大小：188 K

文摘

A natural one-parameter family of Kähler quantizations of the cotangent bundle ve&_udi=B6WJJ-4J4HH8G-1&_mathId=mml1&_user=10&_cdi=6880&_rdoc=7&_handle=V-WA-A-W-E-MsSAYZW-UUW-U-AACVZYZAWD-AACWWZZEWD-EYUUEVZAB-E-U&_acct=C000050221&_version=1&_userid=10&md5=5c7613a8d593fde5cd533f5400c9cddb"" title=""Click to view the MathML source"">TK of a compact Lie group K, taking into account the half-form correction, was studied in [C. Florentino, P. Matias, J. Mourão, J.P. Nunes, Geometric quantization, complex structures and the coherent state transform, J. Funct. Anal. 221 (2005) 303–322]. In the present paper, it is shown that the associated Blattner–Kostant–Sternberg (BKS) pairing map is unitary and coincides with the parallel transport of the quantum connection introduced in our previous work, from the point of view of [S. Axelrod, S. Della Pietra, E. Witten, Geometric quantization of Chern–Simons gauge theory, J. Differential Geom. 33 (1991) 787–902]. The BKS pairing map is a composition of (unitary) coherent state transforms of K, introduced in [B.C. Hall, The Segal–Bargmann coherent state transform for compact Lie groups, J. Funct. Anal. 122 (1994) 103–151]. Continuity of the Hermitian structure on the quantum bundle, in the limit when one of the Kähler polarizations degenerates to the vertical real polarization, leads to the unitarity of the corresponding BKS pairing map. This is in agreement with the unitarity up to scaling (with respect to a rescaled inner product) of this pairing map, established by Hall.