Séance Séminaire

Séminaire Gaston Darboux

Conformal Yang-Mills renormalisation and higher Yang-Mills energies

Given a gauge connection on a Riemannian 4-manifold, the norm squared of its curvature gives a Lagrangian density whose integral is the Yang-Mills action/energy -- the variation of which gives the celebrated Yang-Mills equations. An important feature of both this energy and the equations is their conformal invariance in dimension 4. A natural question is whether there are analogous objects in higher dimensions. We prove that there are such conformally invariant objects on even dimensional manifolds equipped with a connection. One proof, in dimension six, uses a type of Q-curvature that one can associate to connections, and we investigate applications of the result to conformal gravity type equations. Another proof uses a Poincare-Einstein manifold in one higher dimension and a suitable Dirichlet problem for the interior Yang-Mills equations on this structure. The higher Yang-Mills equations arise from an obstruction to smoothly solving the asymptotic problem, while the higher energy is a log term (the so-called anomaly term) in the asymptotic expansion of the divergent interior energy. More arises including links to the non-local renormalised Yang-Mills energy, and a related higher non-linear Dirichlet-Neumann operator. This is joint work with Emanuele Latini, Andrew Waldron, and Yongbing Zhang