Abstract
According to Atiyah-Bott [ABA] the moduli space of flat connections on a compact oriented 2-manifold with prescribed holonomies around the boundary is a finite-dimensional symplectic manifold, possibly singular. A standard approach [W1W2] to computing invariants (symplectic volumes, Riemann-Roch numbers, etc.) of the moduli space is to study the “factorization” of invariants under gluing of 2-manifolds along boundary components. Given such a factorization result, any choice of a “pants decomposition” of the 2-manifold reduces the computation of invariants to the three-holed sphere.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. F. Atiyah, The geometry and physics of knots. Cambridge University Press, Cambridge, 1990
M. F. Atiyah, R. Bott, The Yang-Mills equations over Riemann surfacesPhil. Trans. R. Soc. London 308(1982), 523–615.
N. Berline, E. Getzler, M. Vergne, Heat kernels and Dirac operatorsGrundlehren der Mathematischen Wissenschaften 298. Springer-Verlag, Berlin, 1992.
T. Bröcker, T. tom Dieck Representations of compact Lie groups.Graduate Texts in Mathematics, 98. Springer-Verlag, New York, 1985.
S. Chang, private communication.
S. K. Donaldson, Boundary value problems for Yang-Mills fieldsJ. Geom. Phys. 8(1992), 89–122.
S. K. Donaldson, Gluing techniques in the cohomology of moduli spaces. Topological methods in modern mathematics (Stony Brook, NY, 1991), 137–170, Publish or Perish, Houston, TX, 1993.
V. Ginzburg, V. Guillemin, Y. Karshon, Cobordism theory and localization formulas for Hamiltonian group actionsInt. Math. Res. Notices 5(1996), 221–234.
V. Ginzburg, V. Guillemin, Y. Karshon, Cobordism techniques in symplectic geometry, (in preparation.)
V. Guillemin, S. SternbergSymplectic Techniques in PhysicsCambridge University Press, 1990.
V. Guillemin, E. Lerman, S. SternbergSymplectic Fibrations and Multiplicity DiagramsCambridge University Press, Cambridge, 1996.
V. Guillemin, E. Prato, Heckman, B. Kostant, and Steinberg for-mulas for symplectic manifolds, Adv. Math. 82(1990), 160–179.
L. HörmanderThe Analysis of Linear Partial Differential Oper- ators I2nd ed. Springer-Verlag, 1990.
J. Huebschmann, Symplectic and Poisson structures of certain moduli spaces, IDuke Math. J. 80(1995), 737–756.
L. Jeffrey, Extended moduli spaces of flat connections on Riemann surfacesMath. Ann. 298(1994), 667–692.
L. Jeffrey, J. Weitsman, Toric structures on the moduli space of flat connections on a Riemann surface. II.Inductive decomposition of the moduli spaceMath. Ann. 307(1997), 93–108.
L. Jeffrey, J. Weitsman, Symplectic geometry of the moduli space of flat connections on a Riemann surface, inductive decompositions and vanishing theorems, preprint, December 1996, revised August 1997.
Y. Karshon, Moment maps and non-compact cobordisms, preprint (1997), dg-ga/9701006.
K. Liu, Heat kernel and moduli spaceMath. Res. Lett.3 (1996), 743–76.
K. Liu, Heat kernel and moduli spaces II, preprint, dg-ga/9612001.
E. Meinrenken, C. Woodward, Hom. Honian loop group actions and Verlinde factorization. To appear inJ. Diff. Geom.
E. Meinrenken, C. Woodward, Fusion of Hamiltonian loop group manifolds and cobordism. To appear inMath. Zeit.
A. Pressley, G. SegalLoop GroupsOxford University Press, Ox-ford, 1988.
G. Segal, Lecture notes, Oxford, 1988.
S. Sternberg, Minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang-Mills fieldProc. Nat. Acad. Sci. U.S.A. 74(1977), 5253–5254.
R. Sjamaar, R. Lerman, Stratified symplectic spaces and reduc-tion, Ann. of Math. 134(1991), 375–422.
M. Thaddeus, Stable pairs, linear systems and the Verlinde formulaInvent. Math. 117(1994), 317–353.
E. Witten, On quantum gauge theories in two dimensionsComm. Math. Phys. 141(1991), 153–209.
E. Witten, Two-dimensional gauge theories revisitedJ. Geom. Phys. 9(1992), 303–368.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Science+Business Media New York
About this chapter
Cite this chapter
Meinrenken, E., Woodward, C. (1999). Moduli Spaces of Flat Connections on 2-Manifolds, Cobordism, and Witten’s Volume Formulas. In: Brylinski, JL., Brylinski, R., Nistor, V., Tsygan, B., Xu, P. (eds) Advances in Geometry. Progress in Mathematics, vol 172. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1770-1_12
Download citation
DOI: https://doi.org/10.1007/978-1-4612-1770-1_12
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-7274-8
Online ISBN: 978-1-4612-1770-1
eBook Packages: Springer Book Archive