Browsing by Author "Stern, Mark A"

The central idea of this dissertation is to interpret certain invariants constructed from Laplace spectral data on a compact Riemannian manifold as regularized integrals of closed differential forms on the space of Riemannian metrics, or more generally on a space of metrics on a vector bundle. We apply this idea to both the Ray-Singer analytic torsion

and the eta invariant, explaining their dependence on the metric used to define them with a Stokes' theorem argument. We also introduce analytic multi-torsion, a generalization of analytic torsion, in the context of certain manifolds with local product structure; we prove that it is metric independent in a suitable sense.

This dissertation examines two different subjects within the study of instantons: the construction of Nahm transforms for instantons invariant under certain group actions; and a generalization of the proof that Yang--Mills minimizers are instantons.

The first Nahm transform examined is the ADHM construction for $S^1$-invariant instantons on $S^4$, which correspond to singular monopoles on $\RR^3$. In this case, there is a decomposition of the ADHM data in terms of $S^1$-subrepresentations of $\ker \Dir$. The moduli spaces of $S^1$-invariant $SU(2)$-instantons are given up to charge 3, and examples of ADHM data for instantons of charge $4$ are also provided.

The second Nahm transform considered is for instantons on a certain flat quotient of $\RR^4$ with nonabelian fundamental group. Equivalently, one can consider these to be $\ZZ_2$-invariant instantons on $T^4$, and the Nahm transform yields instantons invariant under a crystallographic action.

In our study of minimal Yang--Mills connections, we extend results of Bourguignon--Lawson--Simons and Stern, who showed that connections that minimize $\|F_\nabla\|^2$ on homogeneous manifolds must be instantons or have instanton subbundles. We extend the previous arguments by considering variations constructed using conformal vector fields, and also allow these vector fields to be incomplete. We prove a minimality result over a half-cylinder.

In this work, we apply $L^2$-index theory to compute the index of a non-Fredholm elliptic operator. The operator arises in Type I' string theory, and the index is found to be non-zero, thus implying existence of bound states.

Motivated by questions arising in the study of harmonic maps and Yang Mills theory, we study new techniques for producing optimal monotonicity relations for geometric partial differential equations. We apply these results to sharpen epsilon regularity results. As a sample application, we analyze energy minimizing maps from compact manifolds to the space of hermitian matrices, where the energy of the map includes the usual kinetic term and a singular potential designed to force the image of the map to lie in a set homotopic to a Grassmannian.