Topological field theories and formulae of Casson and Meng-Taubes

A symplectic formula of generalized Casson invariants

Suppose Y is an integer homology 3-sphere, Taubes proved that the number of irreducible critical orbits of the perturbed Chern-Simons functional on Y, counted with signs, is equal to the algebraic intersection number of two character varieties associated with Heegaard splittings when the structure group is SU(2). Taubes' result established a relationship between gauge theory and the Casson invariant. This article proves an analogous identification result for SU(n) generalized Casson invariants. As a special case, we show that the SU(3) Casson invariant of Boden-Herald can be equivalently calcu…

Monopoles and Landau-Ginzburg Models II: Floer Homology

This is the second paper in this series. Following the setup of Meng-Taubes, we define the monopole Floer homology for any pair $(Y,ω)$, where $Y$ is a compact oriented 3-manifold with toroidal boundary and $ω$ is a suitable closed 2-form viewed as a decoration. This construction fits into a (3+1)-topological quantum field theory and generalizes the work of Kronheimer-Mrowka for closed oriented 3-manifolds. By a theorem of Meng-Taubes and Turaev, the Euler characteristic of this Floer homology recovers the Milnor-Turaev torsion invariant of the 3-manifold.

Lescop's Invariant and Gauge Theory

Taubes proved that the Casson invariant of an integral homology 3-sphere equals half the Euler characteristic of its instanton Floer homology. We extend this result to all closed oriented 3-manifolds with positive first Betti number by establishing a similar relationship between the Lescop invariant of the manifold and its instanton Floer homology. The proof uses surgery techniques.

Analytic Torsion, 3d Mirror Symmetry And Supergroup Chern-Simons Theories

We consider topological field theories that compute the Reidemeister-Milnor-Turaev torsion in three dimensions. These are the psl(1|1) and the U(1|1) Chern-Simons theories, coupled to a background complex flat gauge field. We use the 3d mirror symmetry to derive the Meng-Taubes theorem, which relates the torsion and the Seiberg-Witten invariants, for a three-manifold with arbitrary first Betti number. We also present the Hamiltonian quantization of our theories, find the modular transformations of states, and various properties of loop operators. Our results for the U(1|1) theory are in genera…

3-manifold topology and the Donaldson-Witten partition function

We consider Donaldson-Witten theory on four-manifolds of the form $X=Y \times {\bf S}^1$ where $Y$ is a compact three-manifold. We show that there are interesting relations between the four-dimensional Donaldson invariants of $X$ and certain topological invariants of $Y$. In particular, we reinterpret a result of Meng-Taubes relating the Seiberg-Witten invariants to Reidemeister-Milnor torsion. If $b_1(Y)>1$ we show that the partition function reduces to the Casson-Walker-Lescop invariant of $Y$, as expected on formal grounds. In the case $b_1(Y)=1$ there is a correction. Consequently, in the …

关于 Seiberg-Witten 理论中的积分刚性

On integral rigidity in Seiberg-Witten theory

本文提出了一个证明闭4-流形Seiberg-Witten不变量积分刚性的框架，并具体应用于包含非分离三维环面的四维环面，推广了Donaldson定理的Meng-Taubes公式版本。