Related papers: Fueter sections and $\mathbb{Z}_2$-harmonic 1-form…

This is the second paper in this series. Following the setup of Meng-Taubes, we define the monopole Floer homology for any pair $(Y,\omega)$, where $Y$ is a compact oriented 3-manifold with toroidal boundary and $\omega$ is a suitable…

Let (M,g) be a compact Riemannian three-dimensional manifold with boundary. We prove the compactness of the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface.…

Uhlenbeck's compactness theorem can be used to analyze sequences of connections with anti-self dual curvature on principal SU(2) bundles over oriented 4-dimensional manifolds. The theorems in this paper give an extension of Uhlenbeck's…

We introduce a homology theory whose Euler characteristics counts ASD bundles over four dimensional co-associative submanifolds in (almost) G_2 manifolds. As a TQFT, in relative situations, we have the Fukaya-Floer category of Lagrangians…

Fu and Yau constructed the first smooth family of gauge bundles over a class of non-Kahler, complex 3-folds that are solutions to Strominger's system, the heterotic supersymmetry constraints with nonzero H-flux. In this paper, we begin the…

We consider triholomorphic maps from an almost hyper-Hermitian manifold $\mathcal{M}^{4m}$ into a hyperK\"ahler manifold $\mathcal{N}^{4n}$. This means that $u \in W^{1,2}$ satisfies a quaternionic del-bar equation. We work under the…

This is the third installment in our series of articles (dg-ga/9712005, dg-ga/9710032) on the application of the PU(2) monopole equations to prove Witten's conjecture (hep-th/9411102) concerning the relation between the Donaldson and…

We establish a compensated compactness theorem in the microlocal and geometric analytic framework. For a weakly $L^2_{\rm loc}$-convergent sequence of sections of a vector bundle over a semi-Riemannian manifold whose image under a…

We argue that in type IIB LVS string models, after including the leading order moduli stabilisation effects, the moduli space for the remaining flat directions is compact due the Calabi-Yau K\"ahler cone conditions. In cosmological…

We present the study of type II A flux vacua and their M-theory duals for compactification on a class of Calabi-Yau orientifolds. The Kaehler potential is derived from toroidal compactifications and the superpotential contains a…

Using $\mathbb{Z}_3$ symmetry, we present a topological condition for the existence of the $\mathbb{Z}_2$ harmonic 1-forms over Riemannian manifold. As a corollary, if $L$ is an oriented link on $S^3$ with determinant zero, then there…

Compactifications of type II theories on Calabi-Yau threefolds including electric and magnetic background fluxes are discussed. We derive the bosonic part of the four-dimensional low energy effective action and show that it is a…

In this paper, we establish compactness for various geometric curvature energies including integral Menger curvature, and tangent-point repulsive potentials, defined a priori on the class of compact, embedded $m$-dimensional Lipschitz…

Motivated by the Atiyah-Floer conjecture, we consider $SO(3)$ Santi-self-dual instantons on the product of the real line and a three-manifold with cylindrical end. We prove a Gromov-Uhlenbeck type compactness theorem, namely, any sequence…

This is the third paper of this series. In \cite{Wang20}, we defined the monopole Floer homology for any pair $(Y,\omega)$, where $Y$ is a compact oriented 3-manifold with toroidal boundary and $\omega$ is a suitable closed 2-form viewed as…

Let $\Sigma$ be a compact manifold without boundary whose first homology is nontrivial. Hodge decomposition of the incompressible Euler's equation in terms of 1-forms yields a coupled PDE-ODE system. The $L^2$-orthogonal components are a…

The duality between $E_8\times E_8$ heteritic string on manifold $K3\times T^2$ and Type IIA string compactified on a Calabi-Yau manifold induces a correspondence between vector bundles on $K3\times T^2$ and Calabi-Yau manifolds. Vector…

We investigate the structure of smooth holomorphic foliations with numerically flat tangent bundles on compact K\"ahler manifolds. Extending earlier results on non-uniruled projective manifolds by the second and fourth authors, we show that…

In this paper, by using monotonicity formulas for vector bundle-valued $p$-forms satisfying the conservation law, we first obtain general $L^2$ global rigidity theorems for locally conformally flat (LCF) manifolds with constant scalar…

The aim of this paper is to construct the parabolic version of the Donaldson--Uhlenbeck compactification for the moduli space of parabolic stable bundles on an algenraic surface with parabolic structures along a divisor with normal crossing…