We discuss the statistical mechanics of a gas of gauged vortices in the canonical formalism. At critical self-coupling, and for low temperatures, it has been argued that the configuration space for vortex dynamics in each topological class of the abelian Higgs model approximately truncates to a finite-dimensional moduli space with a Kähler structure. For the case where the vortices live on a 2-...

We present a unified eight-dimensional approach to instanton equations on several seven-dimensional manifolds associated to a six-dimensional homogeneous nearly Kähler manifold. The cone over the sinecone on a nearly Kähler manifold has holonomy group Spin(7) and can be foliated by submanifolds with either holonomy group G2, a nearly parallel G2-structure or a cocalibrated G2-structure. We show...

We identify and examine a generalization of topological sigma models suitable for coupling to topological open strings. The targets are Kähler manifolds with a real structure, i.e. with an involution acting as a complex conjugation, compatible with the Kähler metric. These models satisfy axioms of what might be called “equivariant topological quantum field theory,” generalizing the axioms of to...

M-theory suggests the study of 11-dimensional space-times compactified on some 7-manifolds. From its intimate relation to superstrings, one possible class of such 7-manifolds are those that have Calabi-Yau threefolds as boundary. In this article, we construct a special class of such 7-manifolds, named as K3-Thurston (K3T) 7manifolds. The factor from the K3 part of the deformation space of these...

When computing an average of positive definite (PD) matrices, the preservation of additional matrix structure is desirable for interpretations in applications. An interesting and widely present structure is that of PD Toeplitz matrices, which we endow with a geometry originating in signal processing theory. As an averaging operation, we consider the barycenter, or minimizer of the sum of square...

In this expository article we review the problem of finding Einstein metrics on compact Kähler manifolds and Sasaki manifolds. In the former half of this article we see that, in the Kähler case, the problem fits better with the notion of stability in Geometric Invariant Theory if we extend the problem to that of finding extremal Kähler metrics or constant scalar curvature Kähler (cscK) metrics....