Moduli Spaces of Flat Connections on 2-Manifolds, Cobordism, and Witten’s Volume Formulas

  • E. Meinrenken
  • C. Woodward
Part of the Progress in Mathematics book series (PM, volume 172)


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.


Manifold Stratification Nite 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A]
    M. F. Atiyah, The geometry and physics of knots. Cambridge University Press, Cambridge, 1990MATHCrossRefGoogle Scholar
  2. [AB]
    M. F. Atiyah, R. Bott, The Yang-Mills equations over Riemann surfacesPhil. Trans. R. Soc. London 308(1982), 523–615.MathSciNetGoogle Scholar
  3. [BGV]
    N. Berline, E. Getzler, M. Vergne, Heat kernels and Dirac operatorsGrundlehren der Mathematischen Wissenschaften 298. Springer-Verlag, Berlin, 1992.MATHCrossRefGoogle Scholar
  4. [BrD]
    T. Bröcker, T. tom Dieck Representations of compact Lie groups.Graduate Texts in Mathematics, 98. Springer-Verlag, New York, 1985.MATHGoogle Scholar
  5. [C]
    S. Chang, private communication.Google Scholar
  6. [D1]
    S. K. Donaldson, Boundary value problems for Yang-Mills fieldsJ. Geom. Phys. 8(1992), 89–122.MathSciNetMATHCrossRefGoogle Scholar
  7. [D2]
    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.Google Scholar
  8. [GGK1]
    V. Ginzburg, V. Guillemin, Y. Karshon, Cobordism theory and localization formulas for Hamiltonian group actionsInt. Math. Res. Notices 5(1996), 221–234.MathSciNetCrossRefGoogle Scholar
  9. [GGK2]
    V. Ginzburg, V. Guillemin, Y. Karshon, Cobordism techniques in symplectic geometry, (in preparation.)Google Scholar
  10. [GS]
    V. Guillemin, S. SternbergSymplectic Techniques in PhysicsCambridge University Press, 1990.MATHGoogle Scholar
  11. [GLS]
    V. Guillemin, E. Lerman, S. SternbergSymplectic Fibrations and Multiplicity DiagramsCambridge University Press, Cambridge, 1996.MATHCrossRefGoogle Scholar
  12. [GP]
    V. Guillemin, E. Prato, Heckman, B. Kostant, and Steinberg for-mulas for symplectic manifolds, Adv. Math. 82(1990), 160–179.MathSciNetMATHGoogle Scholar
  13. [Ho]
    L. HörmanderThe Analysis of Linear Partial Differential Oper- ators I2nd ed. Springer-Verlag, 1990.CrossRefGoogle Scholar
  14. [H]
    J. Huebschmann, Symplectic and Poisson structures of certain moduli spaces, IDuke Math. J. 80(1995), 737–756.MathSciNetMATHCrossRefGoogle Scholar
  15. [J]
    L. Jeffrey, Extended moduli spaces of flat connections on Riemann surfacesMath. Ann. 298(1994), 667–692.MathSciNetMATHCrossRefGoogle Scholar
  16. [JW1]
    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.MathSciNetMATHCrossRefGoogle Scholar
  17. [JW2]
    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.Google Scholar
  18. [K]
    Y. Karshon, Moment maps and non-compact cobordisms, preprint (1997), dg-ga/9701006.Google Scholar
  19. [L1]
    K. Liu, Heat kernel and moduli spaceMath. Res. Lett.3 (1996), 743–76.MathSciNetMATHGoogle Scholar
  20. [L2]
    K. Liu, Heat kernel and moduli spaces II, preprint, dg-ga/9612001.Google Scholar
  21. [MW1]
    E. Meinrenken, C. Woodward, Hom. Honian loop group actions and Verlinde factorization. To appear inJ. Diff. Geom. Google Scholar
  22. [MW2]
    E. Meinrenken, C. Woodward, Fusion of Hamiltonian loop group manifolds and cobordism. To appear inMath. Zeit. Google Scholar
  23. [PS]
    A. Pressley, G. SegalLoop GroupsOxford University Press, Ox-ford, 1988.MATHGoogle Scholar
  24. [Se]
    G. Segal, Lecture notes, Oxford, 1988.Google Scholar
  25. [S]
    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.MathSciNetMATHCrossRefGoogle Scholar
  26. [SL]
    R. Sjamaar, R. Lerman, Stratified symplectic spaces and reduc-tion, Ann. of Math. 134(1991), 375–422.MathSciNetMATHCrossRefGoogle Scholar
  27. [T]
    M. Thaddeus, Stable pairs, linear systems and the Verlinde formulaInvent. Math. 117(1994), 317–353.MathSciNetMATHCrossRefGoogle Scholar
  28. [W1]
    E. Witten, On quantum gauge theories in two dimensionsComm. Math. Phys. 141(1991), 153–209.MathSciNetMATHCrossRefGoogle Scholar
  29. [W2]
    E. Witten, Two-dimensional gauge theories revisitedJ. Geom. Phys. 9(1992), 303–368.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • E. Meinrenken
    • 1
  • C. Woodward
    • 2
  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.Department of MathematicsRutgers UniversityUSA

Personalised recommendations