Abstract
We summarize recent work ([W],[JW91a], [JW92]) on the symplectic geometry of the moduli space of flat connections on a two-manifold. This work is based on the existence in these moduli spaces of Hamiltonian torus actions. Using these torus actions and the images of the corresponding moment maps we find a simple description of the moduli spaces, and show how it can be used to compute symplectic volumes and other quantities arising in the geometry and topology of the moduli space.
Supported in part by NSF Mathematical Sciences Postdoctoral Research Fellowship DMS 88-07291
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References
M.F. Atiyah. Convexity and commuting Hamiltonians. Bull. Lond. Math. Soc. 14, (1981) 1.
M.F. Atiyah, R. Bott. The Yang Mills equations over Riemann surfaces. Phil. Trans. Roy. Soc. London A 308 (1982) 523.
A. Bertram, A. Szenes. Hilbert polynomials of moduli spaces of rank 2 vector bundles II. Harvard preprint (1991).
T. Delzant. Hamiltoniens périodiques et images convexes de l’application moment. Bull. Soc. Math. France 116, (1988) 315.
S.K. Donaldson. Gluing techniques in the cohomology of moduli spaces. Oxford preprint (1992).
J.J. Duistermaat, G. Heckman. On the variation in the cohomology of the symplectic form of the reduced phase-space. Inv. Math. 69, (1982) 259.
W. Goldman. Invariant functions on Lie groups and Hamiltonian flows of surface group representations. Inv. Math. 85, (1986) 263.
V. Guillemin, S. Sternberg. Convexity properties of the moment mapping. Inv. Math. 67, (1982) 491.
V. Guillemin, S. Sternberg. “Symplectic techniques in physics”, Cambridge University Press, 1984.
L.C. Jeffrey, J. Weitsman. Bohr-Sommerfeld orbits in the moduli space of flat connections and the Verlinde dimension formula. Commun. Math Phys. 152, (1992) 593.
L.C. Jeffrey, J. Weitsman. Half density quantization of the moduli space of flat connections and Witten’s semiclassical manifold invariants. IAS preprint IASSNS-HEP-91/94; Topology, to appear.
L.C. Jeffrey, J. Weitsman. Toric structures on the moduli space of flat connections on a Riemann surface: volumes and the moment map. Institute for Advanced Study Preprint IASSNS-HEP-92/25; Adv. Math., to appear.
F. Kirwan. The cohomology rings of moduli spaces of bundles over Riemann Surfaces. J. Amer. Math. Soc., 5, 853 (1992).
G. Moore, N. Seiberg. Classical and quantum conformal field Theory. Commun. Math. Phys. 123, (1989) 77.
M.S. Narasimhan, T. R. Ramadas. Factorization of generalized theta functions. Tata Institute Preprint, 1991.
T. Oda, “Convex Bodies and Algebraic Geometry”, Springer Verlag, New York, 1988.
T.R. Ramadas, I.M. Singer, J. Weitsman. Some comments on Chern-Simons gauge theory. Commun. Math. Phys. 126, (1989) 409.
A. Szenes. Hilbert polynomials of moduli spaces of rank 2 vector bundles I. Harvard preprint (1991).
M. Thaddeus. Conformal field theory and the cohomology of the moduli space of stable bundles. J. Diff. Geom., 35, 131 (1992).
E. Verlinde. Fusion rules and modular transformations in 2d conformal field theory. Nucl. Phys. B300, (1988) 351.
J. Weitsman. Real polarization of the moduli space of flat connections on a Riemann Surface. Commun. Math. Phys. 145, (1992) 425.
E. Witten. On quantum gauge theories in two dimensions, Commun. Math. Phys. 140, (1991) 153.
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Jeffrey, L.C., Weitsman, J. (1993). Torus Actions, Moment Maps, and the Symplectic Geometry of the Moduli Space of Flat Connections on a Two-Manifold. In: Osborn, H. (eds) Low-Dimensional Topology and Quantum Field Theory. NATO ASI Series, vol 315. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-1612-9_28
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DOI: https://doi.org/10.1007/978-1-4899-1612-9_28
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