# Morse theory for the Yang-Mills energy function near flat connections

# Morse theory for the Yang-Mills energy function near flat connections

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. An old result in the literature 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 useful version of the full 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.