Theta functions, geometric quantization and unitary Schottky bundles. (English) Zbl 1119.53058

Muñoz Porras, José M. (ed.) et al., The geometry of Riemann surfaces and abelian varieties. III Iberoamerican congress on geometry in honor of Professor Sevín Recillas-Pishmish’s 60th birthday, Salamanca, Spain, June 8–12, 2004. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3855-5/pbk). Contemporary Mathematics 397, 55-71 (2006).
In the paper under review the authors relate two different bases for quantization of the moduli space of flat connections on a compact Riemann surface \(X\), using a trinion decomposition of \(X\) [L. C. Jeffrey and J. Weitsman, Commun. Math. Phys. 150, 593–630 (1992; Zbl 0787.53068)]. One of these constructions comes from geometric quantization of the moduli space of flat connections on \(X\) in a real polarization and the consideration of the corresponding Bohr-Sommerfeld points [J. Weitsman, Commun. Math. Phys. 137, 175–190 (1991; Zbl 0717.53065)]. The other one relies on the construction of spin networks [J. C. Baez, Adv. Math. 117, 253–272 (1996:Zbl 0843.58012)] on the graph associated to the given trinion decomposition. Non-abelian theta functions on the moduli spaces of semistable vector bundles, generalizing the abelian setting [H. Lange and C. Birkenhake, Complex abelian varieties, Grundlehren der Mathematischen Wissenschaften. 302, Berlin: Springer (1992; Zbl 0779.14012)], are obtained by Hall’s procedure of generalized coherent state transform [B. C. Hall, J. Funct. Anal. 122, 103–151 (1994; Zbl 0838.22004)], applied already in the paper [C. Florentino, C., J. Mourão and J. P. Nunes, Tr. Mat. Inst. Steklova 246, 297–315 (2004: Zbl 1105.14058)].
An important role is played by the Schottky map, as was already pointed out [C. Florentino, Manuscr. Math. 105, 69–83 (2001: Zbl 1024.30021)]. Several results obtained in other papers [C. A. Florentino, J. Mourão and J. P. Nunes, J. Funct. Anal. 192, 410–424 (2002; Zbl 1119.22301); C. A. Florentino, J. Mourão and J. P. Nunes, J. Funct. Anal. 204, 355–398 (2003; Zbl 1081.22005)] are also summarized here. It is shown that the different bases for the quantization of the moduli spaces in real polarizations [C. Florentino, P. Matias, J. Mourão and J. P. Nunes, J. Funct. Anal. 221, 303–322 (2005; Zbl 1070.22004)] are related by the Blattner-Kostant-Sternberg pairing in geometric quantization.
For the entire collection see [Zbl 1083.14001].


53D50 Geometric quantization
14K25 Theta functions and abelian varieties
14H81 Relationships between algebraic curves and physics
14H60 Vector bundles on curves and their moduli