Flat connections and geometric quantization. (English) Zbl 0718.53021

Using the space of holomorphic symmetric tensors on the moduli space of stable bundles over a Riemannian surface, a projectively flat connection on a vector bundle over a Teichmüller space is constructed. The fibre of the vector bundle consists of the global sections of a power of the determinant bundle on the moduli space.
Reviewer: H.Baum (Berlin)


53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
81T70 Quantization in field theory; cohomological methods
53D50 Geometric quantization
Full Text: DOI


[1] Atiyah, M.F.: Complex analytic connections in fibre bundles. Trans. Am. Math. Soc.85, 181–207 (1957) · Zbl 0078.16002
[2] Atiyah, M.F., Bott, R.: The Yang-Mills equations over Riemann surfaces. Phil. Trans. R. Soc. Lond. A308, 523–615 (1982) · Zbl 0509.14014
[3] Beauville, A., Narasimhan, M.S., Ramanan, S.: Spectral curves and the generalized theta divisor. J. Reine Angew. Math.398, 169–179 (1989) · Zbl 0666.14015
[4] Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: Wiley 1978 · Zbl 0408.14001
[5] Guillemin, V., Sternberg, S.: Geometric asymptotics. Mathematical surveys, Vol. 14. Providence, RI: Am. Math. Soc. 1977 · Zbl 0364.53011
[6] Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc.55, 59–126 (1987) · Zbl 0634.53045
[7] Hitchin, N.J.: Stable bundles and integrable systems. Duke Math. J.54, 91–114 (1987) · Zbl 0627.14024
[8] Kohno, T.: Monodromy representations of braid groups and Yang-Baxter equations. Ann. Inst. Fourier Grenoble37, 139–160 (1987) · Zbl 0634.58040
[9] Knizhnik, V.G., Zamolodchikov, A.B.: Current algebra and Wess-Zumino model in two dimensions. Nucl. Phys. B247, 83–103 (1984) · Zbl 0661.17020
[10] Nakamura, I.: On moduli of stable quasi-abelian varieties. Nagoya Math. J.58, 149–214 (1975) · Zbl 0295.14018
[11] Narasimhan, M.S., Ramanan, S.: Deformations of the moduli space of vector bundles over an algebraic curve. Ann. Math.101, 39–47 (1975) · Zbl 0314.14004
[12] Narasimhan, M.S., Seshadri, C.S.: Stable and unitary bundles on a compact Riemann surface. Ann. Math.82 540–564 (1965) · Zbl 0171.04803
[13] Nitsure, N.: Moduli space of semistable pairs on a curve. Proc. Lond. Math. Soc. (to appear) · Zbl 0733.14005
[14] Oda, T., Seshadri, C.S.: Compactification of the generalized Jacobian variety. Trans. Am. Math. Soc.253, 1–90 (1979) · Zbl 0418.14019
[15] Quillen, D.: Determinants of Cauchy-Riemann operators over a Riemann surface. Funct. Anal. Appl.19, 31–34 (1985) · Zbl 0603.32016
[16] Scheja, G.: Riemannsche Hebbarkeitsätze für Cohomologieklassen. Math. Ann.144, 345–360 (1961) · Zbl 0112.38001
[17] Segal, G.B.: Conformal field theory. In: Proceedings of International Congress of Physics, Swansea 1988 · Zbl 0657.53060
[18] Simpson, C.: Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. J. Am. Math. Soc.1, 867–918 (1988) · Zbl 0669.58008
[19] Tsuchiya, A., Ueno, K., Yamada, Y.: Conformal field theory on universal family of stable curves with gauge symmetries. Adv. Stud. Pure Math.19 (1989): Integrable systems in quantum field theory and statistical mechanics, pp. 459–566 · Zbl 0696.17010
[20] Welters, G.: Polarized abelian varieties and the heat equations. Compos. Math.49, 173–194 (1983) · Zbl 0576.14042
[21] Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys.121, 351–399 (1989) · Zbl 0667.57005
[22] Witten, E., Axelrod, S. Della Pietra, S.: To appear
[23] Woodhouse, N.: Geometric quantization. Oxford: Oxford University Press 1980 · Zbl 0458.58003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.