Conformal Yang-Mills renormalisation and higher Yang-Mills energies

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• Rod Gover (University of Auckland)
• Wednesday 11 September 2024, 11:30-12:30
• Seminar Room 1, Newton Institute.

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TWTW01 - Twistors in Geometry & Physics

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 four. 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. The 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 a connection Q-curvature, 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.  

This talk is part of the Isaac Newton Institute Seminar Series series.

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