Abstract
We provide an abstract definition and an explicit construction of the stack of non-Abelian Yang–Mills fields on globally hyperbolic Lorentzian manifolds. We also formulate a stacky version of the Yang–Mills Cauchy problem and show that its well-posedness is equivalent to a whole family of parametrized PDE problems. Our work is based on the homotopy theoretical approach to stacks proposed in Hollander (Isr. J. Math. 163:93–124, 2008), which we shall extend by further constructions that are relevant for our purposes. In particular, we will clarify the concretification of mapping stacks to classifying stacks such as BGcon.
Article PDF
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Barwick C.: On left and right model categories and left and right Bousfield localizations. Homol. Homotopy Appl. 12, 245–320 (2010)
Bär, C., Ginoux, N., Pfäffle, F.: Wave Equations on Lorentzian Manifolds and Quantization. Eur. Math. Soc. Zürich (2007). arXiv:0806.1036 [math.DG]
Beem J.K., Ehrlich P.E., Easley K.L.: Global Lorentzian Geometry. Marcel Dekker, New York (1996)
Benini, M., Hanisch, F., Schenkel, A.: Locally covariant Poisson algebra for non-linear scalar fields in the Cahiers topos. (In preparation)
Benini M., Schenkel A.: Poisson algebras for non-linear field theories in the Cahiers topos. Ann. Henri Poincaré 18, 1435 (2017) arXiv:1602.00708 [math-ph]
Benini M., Schenkel A.: Quantum field theories on categories fibered in groupoids. Commun. Math. Phys. 356(1), 19 (2017) arXiv:1610.06071 [math-ph]
Benini M., Schenkel A., Szabo R.J.: Homotopy colimits and global observables in Abelian gauge theory. Lett. Math. Phys. 105(9), 1193 (2015) arXiv:1503.08839 [math-ph]
Brunetti R., Fredenhagen K., Verch R.: The generally covariant locality principle: a new paradigm for local quantum field theory. Commun. Math. Phys. 237, 31 (2003) arXiv:math-ph/0112041
Choquet-Bruhat Y.: Global existence theorems for hyperbolic harmonic maps. Ann. Inst. H. Poincaré Phys. Théor. 46(1), 97–111 (1987)
Choquet-Bruhat Y.: Yang–Mills–Higgs fields in three space time dimensions. Mémoires de la Société Mathématique de France 46, 73–97 (1991)
Chrusciel P.T., Shatah J.: Global existence of solutions of the Yang–Mills equations on globally hyperbolic four dimensional Lorentzian manifolds. Asian J. Math. 1, 530 (1997)
Collini, G.: Fedosov quantization and perturbative quantum field theory. Ph.D. thesis, Universität Leipzig (2016). arXiv:1603.09626 [math-ph]
Deligne P., Mumford D.: The Irreducibility of the Space of Curves of Given Genus. Publ. Math. IHES 36, 75–110 (1969)
Dugger D.: Universal homotopy theories. Adv. Math. 164(1), 144–176 (2001)
Dwyer, W.G., Spalinski, J.: Homotopy theories and model categories. In: Handbook of Algebraic Topology, vol. 73. North-Holland, Amsterdam (1995)
Eggertsson, R.: Stacks in gauge theory. B.Sc. thesis Utrecht (2014). https://ncatlab.org/schreiber/files/Eggertsson2014.pdf
Fewster C.J., Verch R.: Dynamical locality and covariance: what makes a physical theory the same in all spacetimes?. Ann. Henri Poincaré 13(7), 1613 (2012) arXiv:1106.4785v3 [math-ph]
Fiorenza D., Rogers C.L., Schreiber U.: Higher U(1)-gerbe connections in geometric prequantization. Rev. Math. Phys. 28(06), 1650012 (2016) arXiv:1304.0236 [math-ph]
Fiorenza, D., Sati, H., Schreiber, U.: A higher stacky perspective on Chern–Simons theory. In: Mathematical Aspects of Quantum Field Theories. Mathematical Physics Studies, pp. 153–211. Springer (2015) arXiv:1301.2580 [hep-th]
Fiorenza D., Schreiber U., Stasheff J.: Čech cocycles for differential characteristic classes: an \({\infty}\)-Lie theoretic construction. Adv. Theor. Math. Phys. 16(1), 149 (2012) arXiv:1011.4735 [math.AT]
Giraud J.: Cohomologie Non-abelienne. Grundlehren der mathematischen Wissenschaften. Springer, Berlin (1971)
Hepworth R.A.: Vector fields and flows on differentiable stacks. Theory Appl. Categ. 22(21), 542–587 (2009) arXiv:0810.0979 [math.DG]
Hirschhorn P.S.: Model Categories and Their Localizations. Mathematical Surveys and Monographs, vol. 99. Amer. Math. Soc., Providence (2003)
Hollander S.: A homotopy theory for stacks. Isr. J. Math. 163, 93–124 (2008) arXiv:math.AT/0110247
Hollander S.: Characterizing algebraic stacks. Proc. Am. Math. Soc. 136(4), 1465–1476 (2008) arXiv:0708.2705 [math.AT]
Hollander S.: Descent for quasi-coherent sheaves on stacks. Algebr. Geom. Topol. 7, 411–437 (2007) arXiv:0708.2475 [math.AT]
Hovey M.: Model Categories. Mathematical Surveys and Monographs, vol. 63. Amer. Math. Soc., Providence (1999)
Khavkine, I., Schreiber, U.: Synthetic geometry of differential equations: I. Jets and comonad structure. arXiv:1701.06238 [math.DG]
Lurie J.: Higher Topos Theory. Annals of Mathematics Studies, vol. 170. Princeton University Press, Princeton (2009)
Mac Lane S., Moerdijk I.: Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Universitext. Springer, Berlin (1994)
O’Neill B.: Semi-Riemannian Geometry. Academic Press, New York (1983)
Rejzner K.: Perturbative Algebraic Quantum Field Theory: An Introduction for Mathematicians. Mathematical Physics Studies. Springer, Berlin (2016)
Rezk, C.: Toposes and homotopy toposes. http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf
Schreiber U.: Differential cohomology in a cohesive infinity-topos. Current version available at https://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos. arXiv:1310.7930 [math-ph]
Segal G.: Classifying spaces and spectral sequences. Publ. Math. IHES 34, 105 (1968)
Strickland N.P.: K(N)-local duality for finite groups and groupoids. Topology 39, 733–772 (2000)
Toën B., Vezzosi G.: Homotopical algebraic geometry I: topos theory. Adv. Math. 193(2), 257–372 (2005) arXiv:math.AG/0404373
Zuckerman, G.J.: Action principles and global geometry. In: Mathematical Aspects of String Theory. Advanced Series in Mathematical Physics, vol. 1. World Scientific, Singapore (1987)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. Schweigert
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Benini, M., Schenkel, A. & Schreiber, U. The Stack of Yang–Mills Fields on Lorentzian Manifolds. Commun. Math. Phys. 359, 765–820 (2018). https://doi.org/10.1007/s00220-018-3120-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-018-3120-1