Title: Morse theory & the moduli space of flat connections

Speaker: Prof. Paul Feehan (Rutgers, New Jersey)

Abstract: We shall discuss a variety of results related to Morse (or Morse-Bott) theory for the Yang-Mills energy function on a G-bundle over a closed Riemann manifold, with a long term goal of trying to better understand the meaning of integration over the quotient space of connections. An old result (1985) in a well-known article by Karen Uhlenbeck, asserts that the distance between a connection A and the moduli space of flat connections on a G-bundle over a closed manifold is bounded by a constant times an integral norm of the curvature, F_A, when F_A is suitably small. We shall explain that this result is contradicted by simple examples when the Yang-Mills energy function is not Morse-Bott along the moduli space of flat connections, such as the moduli space of SU(2) connections over a torus. However, a version of Uhlenbeck’s result can be recovered, provided one replaces the norm of the curvature F_A by a suitable power of that quantity, where the power reflects the structure of singularities in the moduli space of flat connections. The proof of our refinement involves gradient flow and Morse theory for the Yang-Mills energy function on the quotient space of connections and a Lojasiewicz distance inequality for the Yang-Mills energy function. In collaboration with Robert and Leslie Sibner in 1990, Uhlenbeck constructed the first examples of non-minimal critical points of the Yang-Mills energy function on SU(2) bundles over the 4-dimensional sphere. We shall describe an approach to constructing arbitrarily high-energy saddle points of the Yang-Mills energy function on SU(2) bundles with any second Chern class over closed Riemannian 4-dimensional manifolds.

Time: Thursday 16 May 2019, 2.30pm

Location: Lecture Room, 1st Floor, School of Theoretical Physics, DIAS, 10 Burlington Road, Dublin 4