相关论文: Casson--type invariants in dimension four

In this self-contained book, following Edward Witten, Maxim Kontsevich, Greg Kuperberg and Dylan Thurston, we define an invariant Z of framed links in rational homology 3-spheres, and we study its properties. The invariant Z, which is often…

A family of locally equivalent models is considered. They can be taken as a generalization to $d+1$ dimensions of the Topological Massive and ``Self-dual'' models in 2+1 dimensions. The corresponding 3+1 models are analized in detail. It is…

The Seiberg-Witten analysis of the low-energy effective action of d=4 N=2 SYM theories reveals the relation between the Donaldson and Seiberg-Witten (SW) monopole invariants. Here we apply analogous reasoning to d=3 N=4 theories and propose…

We use Khovanov-Rozansky gl(N) link homology to define invariants of oriented smooth 4-manifolds, as skein modules constructed from certain 4-categories with well-behaved duals. The technical heart of this construction is a proof of the…

In quantum geometry, we consider a set of loops, a compact orientable surface and a solid compact spatial region, all inside $\mathbb{R} \times \mathbb{R}^3 \equiv \mathbb{R}^4$, which forms a triple. We want to define an ambient isotopic…

The SU(3)-Casson invariant for integral homology 3-spheres as studied by Boden-Herald possesses a 'spectral flow obstruction' to being an integer valued invariant which depends only on the non-degenerate (perturbed) moduli space of flat…

We study four dimensional $N=2$ supersymmetric gauge theories on $R^3 \times S^1$ with a circle of radius $R$. They interpolate between four dimensional gauge theories ($R=\infty$) and $N=4$ supersymmetric gauge theories in three dimensions…

A theory of finite type invariants for arbitrary compact oriented 3-manifolds is proposed, and illustrated through many examples arising from both classical and quantum topology. The theory is seen to be highly non-trivial even for…

We propose a novel type of duality that connects a sequence of well-known theories with even-multiplicity scalar amplitudes: it relates the Yang-Mills theory coupled to a specific scalar matter sector to the nonlinear sigma model on a…

We explore various aspects of 2-form topological gauge theories in (3+1)d. These theories can be constructed as sigma models with target space the second classifying space $B^2G$ of the symmetry group $G$, and they are classified by…

Using Heegaard Floer homology, we construct a numerical invariant for any smooth, oriented $4$-manifold $X$ with the homology of $S^1 \times S^3$. Specifically, we show that for any smoothly embedded $3$-manifold $Y$ representing a…

We study low-dimensional problems in topology and geometry via a study of contact and Cauchy-Riemann ($CR$) structures. A contact structure is called spherical if it admits a compatible spherical $CR$ structure. We will talk about spherical…

We study the Goussarov-Habiro finite type invariants theory for framed string links in homology balls. Their degree 1 invariants are computed: they are given by Milnor's triple linking numbers, the mod 2 reduction of the Sato-Levine…

Around 2007, A. Chang, J. Qing, and P. Yang proved a conformal gap theorem for Bach-flat metrics with round sphere as the model case. In this article, we extend this result to prove conformally invariant gap theorems for Bach-flat…

We give a new definition of a universal finite type invariant of three-dimensional oriented rational homology spheres which counts configurations of trivalent graphs in such manifolds. Kontsevich introduced this invariant following Witten's…

This article surveys invariants of four-manifolds and their relation to Donaldson-Witten theory, and other topologically twisted Yang-Mills theories. The article is written for the second edition of the Encyclopedia of Mathematical Physics,…

We consider the link invariants defined by the quantum Chern-Simons field theory with compact gauge group U(1) in a closed oriented 3-manifold M. The relation of the abelian link invariants with the homology group of the complement of the…

An invariant is introduced for negative definite plumbed $3$-manifolds equipped with a spin$^c$-structure. It unifies and extends two theories with rather different origins and structures. One theory is lattice cohomology, motivated by the…

Bott--Cattaneo's theory defines the integral invariants for a framed rational homology 3-sphere equipped with an acyclic orthogonal local system, in terms of graph cocycles without self-loops. The 2-loop term of their invariants is…

We extend the definition of L\"uscher's lattice topological charge to the case of $4$d $SU(N)$ gauge fields coupled with $\mathbb{Z}_N$ $2$-form gauge fields. This result is achieved while maintaining the locality, the $SU(N)$ gauge…