Geometry and Topology Seminar

In 2012, Clifford H. Taubes proved a generalized Uhlenbeck's compactness theorem for the PSL(2,C)-bundles on compact 3-dimensional smooth manifolds. This theorem, however, gives us a pair of interesting data to study. This pair of data can be written as (Z, v) where Z is a closed, Hausdorff dimensional 1 subset of M and v is a Z₂-harmonic spinor defined on the complement of Z with its norm extending to Z and vanishing on Z. In this talk, we will consider the case that Z is the image of a C¹-embedding v: S¹ → M. we construct a moduli space consisting of the following data (Z, v) where Z is the image of a C¹-embedding v: S¹ → M and v is a Z_2-harmonic spinor vanishing on Z satisfying |v|_L²1=1. We will prove that this moduli space can be parametrized by the space Met = { all Riemannian metrics on M } as a Lipschitz manifold.