The stack of Yang-Mills fields on Lorentzian manifolds

Marco Benini, Alexander Schenkel, Urs Schreiber
April 05, 2017
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 [S. Hollander, Israel 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 $\mathrm{B}G_\mathrm{con}$.

Yang-Mills theory, globally hyperbolic Lorentzian manifolds, Cauchy problem, stacks, presheaves of groupoids, homotopical algebra, model categories