相关论文: An SO(3)-monopole cobordism formula relating Donal…

We introduce the notion of symmetric obstruction theory and study symmetric obstruction theories which are compatible with C*-actions. We prove that the contribution of an isolated fixed point under a C*-action to equivariant…

We consider a twisted version of the four-dimensional N=4 supersymmetric Yang-Mills theory with gauge groups SU(2) and SO(3), and bare masses for two of its chiral multiplets, thereby breaking N=4 down to N=2. Using the wall-crossing…

We introduce the foliated anti-self dual equation for higher dimensional smooth manifolds with codimension-4 Riemannian foliations. Several fundamental results are established, towards the defining of a Donaldson type invariant for such…

We formulate a very general conjecture relating the analytical invariants of a normal surface singularity to the Seiberg-Witten invariants of its link provided that the link is a rational homology sphere. As supporting evidence, we…

This note contains two remarks about the application of the d-invariant in Heegaard Floer homology and Donaldson's diagonalization theorem to knot theory. The first is the equivalence of two obstructions they give to a 2-bridge knot being…

In this thesis, we prove several results concerning field-theoretic invariants of knots and 3-manifolds. In Chapter 2, for any knot $K$ in a closed, oriented 3-manifold $M$, we use $SU(2)$ representation spaces and the Lagrangian field…

In this paper, we give a systematic study of Seiberg-Witten theory on closed oriented manifold $M$ with codimension-$3$ oriented Riemannian foliation $F$. Under a certain topological condition, we construct the basic Seiberg-Witten…

The adjunction inequality is a key tool for bounding the genus of smoothly embedded surfaces in 4-manifolds. Using gauge-theoretic invariants, many versions of this inequality have been established for both closed surfaces and surfaces with…

We establish constraints on the topology of smooth Lefschetz fibrations with $4$-dimensional fibers, by studying the family Bauer-Furuta invariant. To compute this invariant, we analyze the framed bordism class of 1-dimensional…

This article analyzes the interplay between symplectic geometry in dimension four and the invariants for smooth four-manifolds constructed using holomorphic triangles introduced in math.SG/0110169. Specifically, we establish a non-vanishing…

We introduce a Floer homotopy version of the contact invariant introduced by Kronheimer-Mrowka-Ozv\'ath-Szab\'o. Moreover, we prove a gluing formula relating our invariant with the first author's Bauer-Furuta type invariant, which refines…

In this paper, we investigate the Seiberg-Witten gauge theory for Seifert fibered spaces. The monopoles over these three-manifolds, for a particular choice of metric and perturbation, are completely described. Gradient flow lines between…

We use the construction of unfolded Seiberg-Witten Floer spectra of general 3-manifolds defined in our previous paper to extend the notion of relative Bauer-Furuta invariants to general 4-manifolds with boundary. One of the main purposes of…

We introduce a new class of representations of the cohomological Hall algebras of Kontsevich and Soibelman, which we call cohomological Hall modules, or CoHM for short. These representations are constructed from self-dual representations of…

We investigate rational homology cobordisms of 3-manifolds with non-zero first Betti number. This is motivated by the natural generalization of the slice-ribbon conjecture to multicomponent links. In particular we consider the problem of…

We prove that $\mathrm{SO}(3)$ modular functors in genus $0$ have geometric origin and support integral variations of Hodge structures for any odd level $r$ and $r$-th root of unity $\zeta_r\in\mathbb{C}$. We identify the TQFT intersection…

We study higher rank Donaldson-Thomas invariants of a Calabi-Yau 3-fold using Joyce-Song's wall-crossing formula. We construct quivers whose counting invariants coincide with the Donaldson-Thomas invariants. As a corollary, we prove the…

We investigate the breaking of SU(3) into its subgroups from the viewpoints of explicit and spontaneous breaking. A one-to-one link between these two approaches is given by the complex spherical harmonics, which form a complete set of…

The goal of this paper is to give a new proof of a theorem of Meng and Taubes that identifies the Seiberg-Witten invariants of 3-manifolds with Milnor torsion. The point of view here will be that of topological quantum field theory. In…

We establish the exact triangle in Seiberg-Witten-Floer theory relating the monopoloe homologies of any two closed 3-manifolds which are obtained from each other by $\pm 1$-surgery. We also show that the sum of the modified version of the…