The singularity structureοf the Yang-Mills configuration space

Jürgen Fuchs

Banach Center Publications (1997)

  • Volume: 39, Issue: 1, page 287-299
  • ISSN: 0137-6934

Abstract

top
The geometric description of Yang–Mills theories and their configuration space is reviewed. The presence of singularities in M is explained and some of their properties are described. The singularity structure is analysed in detail for structure group SU(2). This review is based on [28].

How to cite

top

Fuchs, Jürgen. "The singularity structureοf the Yang-Mills configuration space." Banach Center Publications 39.1 (1997): 287-299. <http://eudml.org/doc/208668>.

@article{Fuchs1997,
abstract = {The geometric description of Yang–Mills theories and their configuration space $\mathcal \{M\}$ is reviewed. The presence of singularities in M is explained and some of their properties are described. The singularity structure is analysed in detail for structure group SU(2). This review is based on [28].},
author = {Fuchs, Jürgen},
journal = {Banach Center Publications},
keywords = {gauge transformations; space of connections; singular strata; Yang-Mills functional; connections},
language = {eng},
number = {1},
pages = {287-299},
title = {The singularity structureοf the Yang-Mills configuration space},
url = {http://eudml.org/doc/208668},
volume = {39},
year = {1997},
}

TY - JOUR
AU - Fuchs, Jürgen
TI - The singularity structureοf the Yang-Mills configuration space
JO - Banach Center Publications
PY - 1997
VL - 39
IS - 1
SP - 287
EP - 299
AB - The geometric description of Yang–Mills theories and their configuration space $\mathcal {M}$ is reviewed. The presence of singularities in M is explained and some of their properties are described. The singularity structure is analysed in detail for structure group SU(2). This review is based on [28].
LA - eng
KW - gauge transformations; space of connections; singular strata; Yang-Mills functional; connections
UR - http://eudml.org/doc/208668
ER -

References

top
  1. [1] A. Yu. Alekseev and V. Schomerus, Representation theory of Chern–Simons observables, preprint q-alg/9503016. 
  2. [2] L. Alvarez-Gaumé and P. Ginsparg, The topological meaning of non-abelian anomalies, Nuclear Phys. B 243 (1984), 449. Zbl0579.58038
  3. [3] J. M. Arms, The structure of the solution set for the Yang–Mills equations, Math. Proc. Cambridge Philos. Soc. 90 (1981), 361. 
  4. [4] J. M. Arms, J. E. Marsden and V. Moncrief, Symmetry and bifurcations of momentum mappings, Comm. Math. Phys. 78 (1981), 455. Zbl0486.58008
  5. [5] A. Ashtekar and J. Lewandowski, Differential geometry on the space of connections via graphs and projective limits, preprint hep-th/9412073. 
  6. [6] A. Ashtekar, D. Marol f and J. Mourão, Integration on the space of connections modulo gauge transformations, preprint gr-qc/9403042. 
  7. [7] M. Asorey, F. Falceto, J. L. López and G. Luzón, Nodes, monopoles and confinement in 2+1-dimensional gauge theories, Phys. Lett. B 349 (1995), 125. Zbl0840.58058
  8. [8] M. Asorey and P. K. Mitter, Regularized, continuum Yang–Mills process and Feynman–Kac functional integral, Comm. Math. Phys. 80 (1981), 43. Zbl0476.58008
  9. [9] M. Asorey and P. K. Mitter, Cohomology of the Yang–Mills gauge orbit space and dimensional reduction, Ann. Inst. H. Poincaré Phys. Théor. A45 (1986), 61. Zbl0596.55003
  10. [10] M. Atiyah and J. Jones, Topological aspects of Yang–Mills theory, Comm. Math. Phys. 61 (1978), 97. Zbl0387.55009
  11. [11] M. Audin, The Topology of Torus Actions on Symplectic Manifolds, Birkhäuser, Basel 1991. 
  12. [12] P. van Baal, More (thoughts on) Gribov copies, Nuclear Phys. B 369 (1992), 259. 
  13. [13] P. van Baal and B. van den Heuvel, Zooming in on the SU(2) fundamental domain, Nuclear Phys. B 417 (1994), 215. Zbl1007.81511
  14. [14] O. Babelon and C. M. Viallet, The geometrical interpretation of the Faddeev–Popov determinant, Phys. Lett. B 85 (1979), 246. 
  15. [15] O. Babelon and C. M. Viallet, On the Riemannian geometry of the configuration space of gauge theories, Comm. Math. Phys. 81 (1981), 515. Zbl0495.58003
  16. [16] J. C. Baez, Generalized measures in gauge theory, Lett. Math. Phys. 31 (1994), 213. Zbl0798.58009
  17. [17] R. Bott, Morse theory and the Yang–Mills equations, in: Differential geometrical methods in mathematical physics, Lecture Notes in Math. 836, Springer, Berlin, 1980, p. 269. Zbl0486.58009
  18. [18] J. P. Brasselet and M. Ferrarotti, Regular differential forms on stratified spaces, preprint Pisa, Sept. 1992. Zbl0837.55003
  19. [19] A. Chodos, Canonical quantization of non-Abelian gauge theories in the axial gauge, Phys. Rev. D (3) 17 (1978), 2624. 
  20. [20] M. Daniel and C. M. Viallet, The gauge fixing problem around classical solutions of the Yang–Mills theory, Phys. Lett. B 76 (1978), 458. 
  21. [21] G. Dell’Antonio and D. Zwanziger, Every gauge orbit passes inside the Gribov horizon, Comm. Math. Phys. 138 (1991), 291. Zbl0726.53067
  22. [22] S. K. Donaldson, Polynomial invariants for smooth four-manifolds, Topology 29 (1990), 257. Zbl0715.57007
  23. [23] S. K. Donaldson and P. B. Kronheimer, The Geometry of Four-Manifolds, Clarendon Press, Oxford 1990. Zbl0820.57002
  24. [24] J. Eichhorn, T. Friedrich, Seiberg–Witten theory, this volume. 
  25. [25] M. Ferrarotti, Some results about integration on regular stratified sets, Ann. Mat. Pura Appl. (4) CL (1988), 263. Zbl0691.58017
  26. [26] A. E. Fischer, Resolving the singularities in the space of Riemannian geometries, J. Math. Phys. 27 (1986), 718. Zbl0604.58007
  27. [27] D. S. Freed, On determinant line bundles, in: Mathematical Aspects of String Theory, S. T. Yau, ed., World Scientific, Singapore, 1987, p. 189. 
  28. [28] J. Fuchs, M. G. Schmidt and C. Schweigert, On the configuration space of gauge theories, Phys. B 426 (1994), 107. Zbl1049.81556
  29. [29] H. B. Gao and H. Römer, Some features of blown-up non-linear σ-models, Classical Quantum Gravity 12 (1995), 17. Zbl0817.53043
  30. [30] M. J. Gotay, Reduction of homogeneous Yang–Mills fields, J. Geom. Phys. 6 (1989), 349. Zbl0692.53041
  31. [31] V. Gribov, Quantization of nonabelian gauge theories, Nuclear Phys. B 139 (1978), 1. 
  32. [32] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley Interscience, New York 1978. Zbl0408.14001
  33. [33] A. Heil, A. Kersch, N. Papadopoulos, B. Reifenhäuser and F. Scheck, Anomalies from nonfree action of the gauge group, Ann. Physics 200 (1990), 206. Zbl0719.53064
  34. [34] A. Heil, A. Kersch, N. Papadopoulos, B. Reifenhäuser and F. Scheck, Structure of the space of reducible connections for Yang–Mills theories, J. Geom. Phys. 7 (1990), 489. Zbl0759.53019
  35. [35] W. Kondracki and J. S. Rogulski, On the stratification of the orbit space for the action of automorphisms on connections, Dissertationes Math. (Rozprawy Mat.) CCL (1986), 1. Zbl0614.57025
  36. [36] W. Kondracki and P. Sadowski, Geometric structure on the orbit space of gauge connections, J. Geom. Phys. 3 (1986), 421. Zbl0624.53055
  37. [37] E. Langmann, M. Salmhofer and A. Kovner, Consistent axial-like gauge fixing on hypertori, Modern Phys. Lett. A 9 (1994), 2913. Zbl1020.58503
  38. [38] P. K. Mitter and C. M. Viallet, On the bundle of connections and the gauge orbit manifold in Yang–Mills theory, Comm. Math. Phys. 79 (1981), 457. Zbl0474.58004
  39. [39] V. Moncrief, Reduction of the Yang–Mills equations, in: Differential geometrical methods in mathematical physics, Lecture Notes in Math. 836, Springer, Berlin, 1980, p. 276. Zbl0489.58006
  40. [40] M. Narasimhan and T. Ramadas, Geometry of SU(2) gauge fields, Comm. Math. Phys. 67 (1979), 121. Zbl0418.53029
  41. [41] L. Rozansky, A contribution of the trivial connection to the Jones polynomial and Witten’s invariant of 3d manifolds I. and II., preprints hep-th/9401069 and hep-th/9403021. 
  42. [42] N. Seiberg, Exact results on the space of vacua of four-dimensional SUSY gauge theories, Phys. Rev. D (3) 49 (1994), 6857. 
  43. [43] M. A. Semenov-Tyan-Shanskiĭ and V. A. Franke, A variational principle for the Lorentz condition and restriction of the domain of path integration in non-abelian gauge theory (Russian), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 120 (1982), 159; transl. J. Soviet. Math. 34 (1986), 1999. 
  44. [44] I. M. Singer, Some remarks on the Gribov ambiguity, Comm. Math. Phys. 60 (1978), 7. Zbl0379.53009
  45. [45] I. M. Singer, The geometry of the orbit space for nonabelian gauge theories, Phys. Scripta T 24 (1981), 817. Zbl1063.81623
  46. [46] K. K. Uhlenbeck, Removable singularities in Yang–Mills fields, Comm. Math. Phys. 83 (1982), 11. Zbl0491.58032
  47. [47] E. Witten, Monopoles and four manifolds, Math. Res. Lett. 1 (1994), 769. Zbl0867.57029
  48. [48] C. N. Yang and R. M. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev. (2) 96 (1954), 191. Zbl06538052
  49. [49] D. Zwanziger, Non-perturbative modification of the Faddeev–Popov formula and banishment of the naive vacuum, Nuclear Phys. B 209 (1982), 336. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.