Abstract
Long time existence and uniqueness of solutions to the Yang-Mills heat equation is proven over a compact 3-manifold with smooth boundary. The initial data is taken to be a Lie algebra valued connection form in the Sobolev space H 1. Three kinds of boundary conditions are explored, Dirichlet type, Neumann type and Marini boundary conditions. The last is a nonlinear boundary condition, specified by setting the normal component of the curvature to zero on the boundary. The Yang-Mills heat equation is a weakly parabolic nonlinear equation. We use gauge symmetry breaking to convert it to a parabolic equation and then gauge transform the solution of the parabolic equation back to a solution of the original equation. Apriori estimates are developed by first establishing a gauge invariant version of the Gaffney-Friedrichs inequality. A gauge invariant regularization procedure for solutions is also established. Uniqueness holds upon imposition of boundary conditions on only two of the three components of the connection form because of weak parabolicity. This work is motivated by possible applications to quantum field theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alvarez J., Eydenberg M.S., Obiedat H.: The action of operator semigroups on the topological dual of the Beurling-Björck space. J. Math. Anal. Appl. 339(1), 405–418 (2008)
Arnaudon M., Bauer R.O., Thalmaier A.: A probabilistic approach to the Yang-Mills heat equation. J. Math. Pures Appl. (9) 81(2), 143–166 (2002)
Atiyah M. F., Bott R.: The Yang-Mills equations over Riemann surfaces. Philos. Trans. Roy. Soc. London Ser. A 308(1505), 523–615 (1983)
Bourguignon J.-P., Lawson H.B. Jr: Stability and isolation phenomena for Yang-Mills fields. Commun. Math. Phys. 79(2), 189–230 (1981)
Butzer, P.L., Berens, H.: Semi-groups of operators and approximation. Die Grundlehren der mathematischen Wissenschaften, Band 145, New York: Springer-Verlag New York Inc., 1967
Chen, Y.M., Shen, C.L.: Evolution of Yang-Mills connections. In: Differential geometry (Shanghai, 1991), River Edge, NJ: World Sci. Publ., 1993, pp. 33–41
Conner P.E.: The Neumann’s problem for differential forms on Riemannian manifolds. Mem. Amer. Math. Soc. 1956(20), 56 (1956)
DeTurck D.M.: Deforming metrics in the direction of their Ricci tensors. J. Diff. Geom. 18(1), 157–162 (1983)
Donaldson S.K.: Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. London Math. Soc. (3) 50(1), 1–26 (1985)
Donaldson S.K.: Boundary value problems for Yang-Mills fields. J. Geom. Phys. 8(1–4), 89–122 (1992)
Donaldson, S.K., Kronheimer, P.B.: The geometry of four-manifolds. Oxford Mathematical Monographs, New York: The Clarendon Press/Oxford University Press, 1990
Friedrichs K.O.: Differential forms on Riemannian manifolds. Comm. Pure Appl. Math. 8, 551–590 (1955)
Gaffney M.P.: The harmonic operator for exterior differential forms. Proc. Nat. Acad. Sci. U. S. A. 37, 48–50 (1951)
Gallot, S., Hulin, D., Lafontaine, J.: Riemannian geometry. Third ed., Universitext, Berlin: Springer- Verlag, 2004
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, Classics in Mathematics. Berlin: Springer-Verlag, 2001, reprint of the 1998 edition
Ginibre J., Velo G.: Global existence of coupled Yang-Mills and scalar fields in (2 + 1)-dimensional space-time. Phys. Lett. B 99(5), 405–410 (1981)
Ginibre, J., Velo, G.: The Cauchy problem for coupled Yang-Mills and scalar fields in the temporal gauge. Commun. Math. Phys. 82(1), 1–28 (1981/82)
Glimm, J., Jaffe, A.: Quantum physics. Second ed., New York: Springer-Verlag, 1987
Gross L.: Convergence of U(1)3 lattice gauge theory to its continuum limit. Commun. Math. Phys. 92(2), 137–162 (1983)
Gryc W.E.: On the holonomy of the Coulomb connection over manifolds with boundary. J. Math. Phys. 49(6), 062904 (2008)
Hassell A.: The Yang-Mills-Higgs heat flow on R 3. J. Funct. Anal. 111(2), 431–448 (1993)
Hong M.-C.: Heat flow for the Yang-Mills-Higgs field and the Hermitian Yang-Mills-Higgs metric. Ann. Global Anal. Geom. 20(1), 23–46 (2001)
Hong M.-C., Tian G.: Asymptotical behaviour of the Yang-Mills flow and singular Yang-Mills connections. Math. Ann. 330(3), 441–472 (2004)
Hong M.-C., Tian G.: Global existence of the m-equivariant Yang-Mills flow in four dimensional spaces. Commun. Anal. Geom. 12(1–2), 183–211 (2004)
Kogut J.B., Suskind L.: Hamiltonian formulation of Wilson’s lattice gauge theories. Phys. Rev. D 11, 395–408 (1975)
LePage G.P. et al.: Accurate determinations of α s from realistic lattice qcd. Phys. Rev. Lett. 95, 052002-1–052002-4 (2005)
Lions J.L.: Sur les espaces d’interpolation; dualité. Math. Scand. 9, 147–177 (1961)
Lüscher, M.: Properties and uses of the Wilson flow in lattice QCD. J. High Energy Phys. no. 8, 071, 18. (2010)
Lüscher M.: Trivializing maps, the Wilson flow and the HMC algorithm. Commun. Math. Phys. 293(3), 899–919 (2010)
Lüscher, M., Weisz, P.: Perturbative analysis of the gradient flow in non-abelian gauge theories. J. High Energy Phys. no. 2, 051, i, 22 (2011)
Marini A.: Dirichlet and Neumann boundary value problems for Yang-Mills connections. Comm. Pure Appl. Math. 45(8), 1015–1050 (1992)
Marini, A.: Elliptic boundary value problems for connections: a non-linear Hodge theory. Mat. Contemp. 2, 195–205 (1992), Workshop on the Geometry and Topology of Gauge Fields (Campinas, 1991)
Marini A.: The generalized Neumann problem for Yang-Mills connections. Comm. Part. Diff. Eqs. 24(3-4), 665–681 (1999)
Matsuzawa T.: A calculus approach to hyperfunctions. I. Nagoya Math. J. 108, 53–66 (1987)
Matsuzawa T.: A calculus approach to hyperfunctions II. Trans. Amer. Math. Soc. 313(2), 619–654 (1989)
Matsuzawa T.: A calculus approach to hyperfunctions. III. Nagoya Math. J. 118, 133–153 (1990)
Mendez-Hernandez P.J., Murata M.: Semismall perturbations, semi-intrinsic ultracontractivity, and integral representations of nonnegative solutions for parabolic equations. J. Funct. Anal. 257(6), 1799–1827 (2009)
Mitrea D., Mitrea M., Taylor M.: Layer potentials, the Hodge Laplacian, and global boundary problems in nonsmooth Riemannian manifolds. Mem. Amer. Math. Soc. 150(713), x+120 (2001)
Mitrea M.: Dirichlet integrals and Gaffney-Friedrichs inequalities in convex domains. Forum Math. 13(4), 531–567 (2001)
Morrey C.B. Jr: A variational method in the theory of harmonic integrals. II. Amer. J. Math. 78, 137–170 (1956)
Morrey C.B. Jr, Eells J. Jr: A variational method in the theory of harmonic integrals. I. Ann. of Math. (2) 63, 91–128 (1956)
Narasimhan M.S., Ramadas T.R.: Geometry of SU(2) gauge fields. Commun. Math. Phys. 67(2), 121–136 (1979)
Pulemotov A.: The Li-Yau-Hamilton estimate and the Yang-Mills heat equation on manifolds with boundary. J. Funct. Anal. 255(10), 2933–2965 (2008)
Råde J.: On the Yang-Mills heat equation in two and three dimensions. J. Reine Angew. Math. 431, 123–163 (1992)
Ray D.B., Singer I.M.: R-torsion and the Laplacian on Riemannian manifolds. Adv. in Math. 7, 145–210 (1971)
Sadun, L.A.: Continuum regularized Yang-Mills theory. Ph. D. Thesis, Univ. of California, Berkeley, 1987, 67+ pages
Saranen J.: On an inequality of Friedrichs. Math. Scand. 51(2), 310–322 (1982)
Seiler, E.: Gauge theories as a problem of constructive quantum field theory and statistical mechanics. Lecture Notes in Physics, Vol. 159, Berlin: Springer-Verlag, 1982
Sengupta, A.: Gauge theory on compact surfaces. Mem. Amer. Math. Soc. 126(600) (1997)
Sengupta, A.N.: Gauge theory in two dimensions: topological, geometric and probabilistic aspects. In: Stochastic analysis in mathematical physics, Hackensack, NJ: World Sci. Publ., 2008, pp. 109–129
Singer I.M.: Some remarks on the Gribov ambiguity. Commun. Math. Phys. 60(1), 7–12 (1978)
Singer I.M.: The geometry of the orbit space for nonabelian gauge theories. Phys. Scripta 24(5), 817–820 (1981)
Streater, R.F., Wightman, A.S.: PCT, spin and statistics, and all that. Princeton Landmarks in Physics, Princeton, NJ: Princeton University Press, 2000, Corrected third printing of the 1978 edition
Taibleson M.H.: On the theory of Lipschitz spaces of distributions on Euclidean n-space. I. Principal properties. J. Math. Mech. 13, 407–479 (1964)
Taibleson M.H.: On the theory of Lipschitz spaces of distributions on Euclidean n-space. II. Translation invariant operators, duality, and interpolation. J. Math. Mech. 14, 821–839 (1965)
Taibleson M.H.: On the theory of Lipschitz spaces of distributions on Euclidean n-space. III. Smoothness and integrability of Fourier tansforms, smoothness of convolution kernels. J. Math. Mech. 15, 973–981 (1966)
Taylor, M.E.: Partial differential equations, Texts in Applied Mathematics, Vol. 23, New York: Springer-Verlag, 1996
Taylor, M.E.: Partial differential equations. III. Applied Mathematical Sciences, Vol. 117, New York: Springer-Verlag, 1997, Corrected reprint of the 1996 original
Wilson K.G.: Confinement of quarks. Phys. Rev. D 10, 2445–2459 (1974)
Zwanziger D.: Covariant quantization of gauge fields without Gribov ambiguity. Nucl. Phys. B 192(1), 259–269 (1981)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M. Salmhofer
Rights and permissions
About this article
Cite this article
Charalambous, N., Gross, L. The Yang-Mills Heat Semigroup on Three-Manifolds with Boundary. Commun. Math. Phys. 317, 727–785 (2013). https://doi.org/10.1007/s00220-012-1558-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-012-1558-0