Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 048, 35 pages      arXiv:1407.4741      https://doi.org/10.3842/SIGMA.2015.048

General Boundary Formulation for $n$-Dimensional Classical Abelian Theory with Corners

Homero G. Díaz-Marín ab
a) Escuela Nacional de Ingenierí a y Ciencias, Instituto Tecnológico y de Estudios Superiores de Monterrey, C.P. 58350 Morelia, México
b) Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, C.P. 58190 Morelia, México

Received October 30, 2014, in final form June 04, 2015; Published online June 24, 2015

Abstract
We propose a general reduction procedure for classical field theories provided with abelian gauge symmetries in a Lagrangian setting. These ideas come from an axiomatic presentation of the general boundary formulation (GBF) of field theories, mostly inspired by topological quantum field theories (TQFT). We construct abelian Yang-Mills theories using this framework. We treat the case for space-time manifolds with smooth boundary components as well as the case of manifolds with corners. This treatment is the GBF analogue of extended TQFTs. The aim for developing this classical formalism is to accomplish, in a future work, geometric quantization at least for the abelian case.

Key words: gauge fields; action; manifolds with corners.

pdf (526 kb)   tex (41 kb)

References

  1. Al-Zamil Q.S.A., Montaldi J., Witten-Hodge theory for manifolds with boundary and equivariant cohomology, Differential Geom. Appl. 30 (2012), 179-194, arXiv:1004.2687.
  2. Ancona V., Gaveau B., Okada M., Harmonic forms and cohomology of singular stratified spaces, Bull. Sci. Math. 131 (2007), 422-456.
  3. Arbieto A., Matheus C., A pasting lemma and some applications for conservative systems, Ergodic Theory Dynam. Systems 27 (2007), 1399-1417, math.DS/0601433.
  4. Atiyah M., Topological quantum field theories, Inst. Hautes Études Sci. Publ. Math. (1988), 175-186.
  5. Cattaneo A.S., Mnev P., Reshetikhin N., Classical and quantum Lagrangian field theories with boundary, PoS Proc. Sci. (2011), PoS(CORFU2011), 044, 25 pages, arXiv:1207.0239.
  6. Cattaneo A.S., Mnev P., Reshetikhin N., Semiclassical quantization of classical field theories, in Mathematical Aspects of Quantum Field Theories, Editors D. Calaque, T. Strobl, Mathematical Physics Studies, Springer, Berlin, 2015, 275-324, arXiv:1311.2490.
  7. Dacorogna B., Moser J., On a partial differential equation involving the Jacobian determinant, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), 1-26.
  8. Driver B.K., Hall B.C., Yang-Mills theory and the Segal-Bargmann transform, Comm. Math. Phys. 201 (1999), 249-290, hep-th/9808193.
  9. Duff G.F.D., Differential forms in manifolds with boundary, Ann. of Math. 56 (1952), 115-127.
  10. Duff G.F.D., Spencer D.C., Harmonic tensors on Riemannian manifolds with boundary, Ann. of Math. 56 (1952), 128-156.
  11. Gelca R., Topological quantum field theory with corners based on the Kauffman bracket, Comment. Math. Helv. 72 (1997), 216-243, q-alg/9603002.
  12. Giachetta G., Mangiattori L., Sardanashvily G., Advanced classical field theory, World Sci. Publ., Singapore, 2009.
  13. Iliev B.Z., Handbook of normal frames and coordinates, Progress in Mathematical Physics, Vol. 42, Birkhäuser Verlag, Basel, 2006.
  14. Kijowski J., Tulczyjew W.M., A symplectic framework for field theories, Lecture Notes in Phys., Vol. 107, Springer-Verlag, Berlin - New York, 1979.
  15. Landsman N.P., Mathematical topics between classical and quantum mechanics, Springer Monographs in Mathematics, Springer-Verlag, New York, 1998.
  16. Lauda A.D., Pfeiffer H., Open-closed strings: two-dimensional extended TQFTs and Frobenius algebras, Topology Appl. 155 (2008), 623-666, math.AT/0510664.
  17. Milnor J.W., Topology from the differentiable viewpoint, Based on notes by David W. Weaver, The University Press of Virginia, Charlottesville, Va., 1965.
  18. Moser J., On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286-294.
  19. Oeckl R., General boundary quantum field theory: foundations and probability interpretation, Adv. Theor. Math. Phys. 12 (2008), 319-352, hep-th/0509122.
  20. Oeckl R., Two-dimensional quantum Yang-Mills theory with corners, J. Phys. A: Math. Theor. 41 (2008), 135401, 20 pages, hep-th/0608218.
  21. Oeckl R., Affine holomorphic quantization, J. Geom. Phys. 62 (2012), 1373-1396, arXiv:1104.5527.
  22. Oeckl R., Holomorphic quantization of linear field theory in the general boundary formulation, SIGMA 8 (2012), 050, 31 pages, arXiv:1009.5615.
  23. Sati H., Duality and cohomology in $M$-theory with boundary, J. Geom. Phys. 62 (2012), 1284-1297, arXiv:1012.4495.
  24. Schwarz G., Hodge decomposition - a method for solving boundary value problems, Lecture Notes in Math., Vol. 1607, Springer-Verlag, Berlin, 1995.
  25. Segal G., The definition of conformal field theory, in Topology, Geometry and Quantum Field Theory, Cambridge University Press, Cambridge, 2004, 421-577.
  26. Weinstein A., Symplectic categories, Port. Math. 67 (2010), 261-278, arXiv:0911.4133.
  27. Witten E., Two-dimensional gauge theories revisited, J. Geom. Phys. 9 (1992), 303-368, hep-th/9204083.
  28. Woodhouse N.M.J., Geometric quantization, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1992.

Previous article  Next article   Contents of Volume 11 (2015)