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

SIGMA 13 (2017), 003, 44 pages      arXiv:1308.1005      https://doi.org/10.3842/SIGMA.2017.003

The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs

Batu Güneysu a and Markus J. Pflaum b
a) Institut für Mathematik, Humboldt-Universität, Rudower Chaussee 25, 12489 Berlin, Germany
b) Department of Mathematics, University of Colorado, Boulder CO 80309, USA

Received March 30, 2016, in final form January 05, 2017; Published online January 10, 2017

Abstract
In this paper, we study the formal solution space of a nonlinear PDE in a fiber bundle. To this end, we start with foundational material and introduce the notion of a pfd structure to build up a new concept of profinite dimensional manifolds. We show that the infinite jet space of the fiber bundle is a profinite dimensional manifold in a natural way. The formal solution space of the nonlinear PDE then is a subspace of this jet space, and inherits from it the structure of a profinite dimensional manifold, if the PDE is formally integrable. We apply our concept to scalar PDEs and prove a new criterion for formal integrability of such PDEs. In particular, this result entails that the Euler-Lagrange equation of a relativistic scalar field with a polynomial self-interaction is formally integrable.

Key words: profinite dimensional manifolds; jet bundles; geometric PDEs; formal integrability; scalar fields.

pdf (639 kb)   tex (52 kb)

References

  1. Abbati M.C., Manià A., On differential structure for projective limits of manifolds, J. Geom. Phys. 29 (1999), 35-63, math-ph/9804007.
  2. Anderson I.M., The variational bicomplex, Technical report, Department of Mathematics, Utah State University, 1989.
  3. Artin M., Algebraic approximation of structures over complete local rings, Inst. Hautes Études Sci. Publ. Math. (1969), 23-58.
  4. Ashtekar A., Lewandowski J., Differential geometry on the space of connections via graphs and projective limits, J. Geom. Phys. 17 (1995), 191-230.
  5. Atiyah M.F., Hitchin N.J., Singer I.M., Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), 425-461.
  6. Barnich G., Brackets in the jet-bundle approach to field theory, in Secondary Calculus and Cohomological Physics (Moscow, 1997), Contemp. Math., Vol. 219, Amer. Math. Soc., Providence, RI, 1998, 17-27, hep-th/9709164.
  7. Bernstein I.N., Rosenfel'd B.I., Homogeneous spaces of infinite-dimensional Lie algebras and the characteristic classes of foliations, Russ. Math. Surv. 28 (1973), no. 4, 107-142.
  8. Bickel H., Lie-projective groups, J. Lie Theory 5 (1995), 15-24.
  9. Bocharov A.V., Chetverikov V.N., Duzhin S.V., Khor'kova N.G., Krasil'shchik I.S., Samokhin A.V., Torkhov Yu.N., Verbovetsky A.M., Vinogradov A.M., Symmetries and conservation laws for differential equations of mathematical physics, Translations of Mathematical Monographs, Vol. 182, Amer. Math. Soc., Providence, RI, 1999.
  10. Bourbaki N., Algebra I, Chapters 1-3, Elements of Mathematics, Springer-Verlag, Berlin - Heidelberg, 1989.
  11. Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.L., Griffiths P.A., Exterior differential systems, Mathematical Sciences Research Institute Publications, Vol. 18, Springer-Verlag, New York, 1991.
  12. Chetverikov V.N., On the structure of integrable $\mathscr C$-fields, Differential Geom. Appl. 1 (1991), 309-325.
  13. Constantine G.M., Savits T.H., A multivariate Faa di Bruno formula with applications, Trans. Amer. Math. Soc. 348 (1996), 503-520.
  14. Dodson C.T.J., Galanis G., Vassiliou E., Geometry in a Fréchet context. A projective limit approach, London Mathematical Society Lecture Note Series, Vol. 428, Cambridge University Press, Cambridge, 2016.
  15. Donaldson S.K., An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983), 279-315.
  16. Dütsch M., Fredenhagen K., Perturbative algebraic field theory, and deformation quantization, in Mathematical Physics in Mathematics and Physics (Siena, 2000), Fields Inst. Commun., Vol. 30, Amer. Math. Soc., Providence, RI, 2001, 151-160, hep-th/0101079.
  17. Ehrenpreis L., Guillemin V.W., Sternberg S., On Spencer's estimate for $\delta $-Poincaré, Ann. of Math. 82 (1965), 128-138.
  18. Eilenberg S., Steenrod N., Foundations of algebraic topology, Princeton University Press, Princeton, New Jersey, 1952.
  19. Finster F., Tolksdorf J., Bosonic loop diagrams as perturbative solutions of the classical field equations in $\phi^4$-theory, J. Math. Phys. 53 (2012), 052305, 32 pages, arXiv:1201.5497.
  20. Freed D.S., Uhlenbeck K.K., Instantons and four-manifolds, Mathematical Sciences Research Institute Publications, Vol. 1, Springer-Verlag, New York, 1984.
  21. Gharesifard B., Lewis A.D., Mansouri A.-R., A geometric framework for stabilization by energy shaping: sufficient conditions for existence of solutions, Commun. Inf. Syst. 8 (2008), 353-398.
  22. Giachetta G., Mangiarotti L., Gauge invariance and formal integrability of the Yang-Mills-Higgs equations, Internat. J. Theoret. Phys. 35 (1996), 1405-1422.
  23. Giachetta G., Mangiarotti L., Sardanashvily G., Advanced classical field theory, World Sci. Publ., Hackensack, NJ, 2009.
  24. Goldschmidt H., Integrability criteria for systems of nonlinear partial differential equations, J. Differential Geometry 1 (1967), 269-307.
  25. Grothendieck A., Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), 140 pages.
  26. Hofmann K.H., Morris S.A., Projective limits of finite-dimensional Lie groups, Proc. London Math. Soc. 87 (2003), 647-676.
  27. Krasil'shchik I.S., Lychagin V.V., Vinogradov A.M., Geometry of jet spaces and nonlinear partial differential equations, Advanced Studies in Contemporary Mathematics, Vol. 1, Gordon and Breach Science Publishers, New York, 1986.
  28. Kruglikov B., Involutivity of field equations, J. Math. Phys. 51 (2010), 032502, 16, arXiv:0902.1685.
  29. Kruglikov B., Lychagin V., Mayer brackets and solvability of PDEs. II, Trans. Amer. Math. Soc. 358 (2006), 1077-1103.
  30. Kruglikov B., Lychagin V., Geometry of differential equations, in Handbook of Global Analysis, Elsevier Sci. B.V., Amsterdam, 2008, 725-771.
  31. Kuranishi M., On E. Cartan's prolongation theorem of exterior differential systems, Amer. J. Math. 79 (1957), 1-47.
  32. Lewis A.D., Geometric partial differential equations, definitions and properties, unpublished lecture notes.
  33. Marsden J.E., Montgomery R., Morrison P.J., Thompson W.B., Covariant Poisson brackets for classical fields, Ann. Physics 169 (1986), 29-47.
  34. Pflaum M.J., On the deformation quantization of symplectic orbispaces, Differential Geom. Appl. 19 (2003), 343-368, math-ph/0208020.
  35. Pommaret J.-F., Systems of partial differential equations and Lie pseudogroups, Mathematics and its Applications, Vol. 14, Gordon & Breach Science Publishers, New York, 1978.
  36. Pommaret J.-F., Lie pseudogroups and mechanics, Mathematics and its Applications, Vol. 16, Gordon & Breach Science Publishers, New York, 1988.
  37. Pommaret J.-F., Partial differential equations and group theory. New perspectives for applications, Mathematics and its Applications, Vol. 293, Kluwer Academic Publishers Group, Dordrecht, 1994.
  38. Rejzner K., Perturbative algebraic quantum field theory. An introduction for mathematicians, Mathematical Physics Studies, Springer, Cham, 2016.
  39. Reyes E.G., Jet bundles, symmetries, Darboux transforms, in Algebraic Aspects of Darboux Transformations, Quantum Integrable Systems and Supersymmetric Quantum Mechanics, Contemp. Math., Vol. 563, Amer. Math. Soc., Providence, RI, 2012, 137-164.
  40. Sardanashvily G., Fibre bundles, jet manifolds and Lagrangian theory, arXiv:0908.1886.
  41. Saunders D.J., The geometry of jet bundles, London Mathematical Society Lecture Note Series, Vol. 142, Cambridge University Press, Cambridge, 1989.
  42. Seiler W.M., Involution. The formal theory of differential equations and its applications in computer algebra, Algorithms and Computation in Mathematics, Vol. 24, Springer-Verlag, Berlin, 2010.
  43. Spencer D.C., Deformation of structures on manifolds defined by transitive, continuous pseudogroups. II. Deformations of structure, Ann. of Math. 76 (1962), 399-445.
  44. Spencer D.C., Overdetermined systems of linear partial differential equations, Bull. Amer. Math. Soc. 75 (1969), 179-239.
  45. Stasheff J., The (secret?) homological algebra of the Batalin-Vilkovisky approach, in Secondary Calculus and Cohomological Physics (Moscow, 1997), Contemp. Math., Vol. 219, Amer. Math. Soc., Providence, RI, 1998, 195-210, hep-th/9712157.
  46. Taubes C.H., Self-dual connections on $4$-manifolds with indefinite intersection matrix, J. Differential Geom. 19 (1984), 517-560.
  47. Trèves F., Topological vector spaces, distributions and kernels, Academic Press, New York - London, 1967.
  48. Vinogradov A.M., Cohomological analysis of partial differential equations and secondary calculus, Translations of Mathematical Monographs, Vol. 204, Amer. Math. Soc., Providence, RI, 2001.
  49. Vitagliano L., Secondary calculus and the covariant phase space, J. Geom. Phys. 59 (2009), 426-447, arXiv:0809.4164.
  50. Zuckerman G.J., Action principles and global geometry, in Mathematical Aspects of String Theory (San Diego, Calif., 1986), Adv. Ser. Math. Phys., Vol. 1, World Sci. Publ., Singapore, 1987, 259-284.

Previous article  Next article   Contents of Volume 13 (2017)