Generalisations of Rozansky-Witten invariants

Topological quantum field theory and hyperkähler geometry

Rozansky and Witten proposed a 3-dimensional sigma-model whose target space is a hyperkähler manifold. They conjectured that this theory has an associated TQFT, with Hilbert spaces given by certain cohomology groups of the hyperkähler manifold. On the other hand, there is a certain modified TQFT constructed by Murakami and Ohtsuki using the universal quantum invariant. We explain how the Rozansky-Witten TQFT can be obtained from the latter by applying a `hyperkähler weight system'.

$PU(N)$ monopoles, higher rank instantons, and the monopole invariants

A famous conjecture in gauge theory mathematics, attributed to Witten, suggests that the polynomial invariants of Donaldson are expressible in terms of the Seiberg-Witten invariants if the underlying four-manifold is of simple type. Mathematicians have sought a proof of the conjecture by means of a `cobordism program' involving $PU(2)$ monopoles. A higher rank version of the Donaldson invariants was recently introduced by Kronheimer. Before being defined, the physicists Mariño and Moore had already suggested that there should be a generalisation of Witten's conjecture to this type of invariant…

On higher rank instantons & the monopole cobordism program

Witten's conjecture suggests that the polynomial invariants of Donaldson are expressible in terms of the Seiberg-Witten invariants if the underlying four-manifold is of simple type. A higher rank version of the Donaldson invariants was introduced by Kronheimer. Before even having been defined, the physicists Mariño and Moore had already suggested that there should be a generalisation of Witten's conjecture to this type of invariants. We study a generalisation of the classical cobordism program to the higher rank situation and obtain vanishing results which gives evidence that the generalisatio…

On the Generalized Casson Invariant

The path integral generalization of the Casson invariant as developed by Rozansky and Witten is investigated. The path integral for various three manifolds is explicitly evaluated. A new class of topological observables is introduced that may allow for more effective invariants. Finally it is shown how the dimensional reduction of these theories corresponds to a generalization of the topological B sigma model.

A new weight system on chord diagrams via hyperkähler geometry

A weight system on graph homology was constructed by Rozansky and Witten using a compact hyperkähler manifold. A variation of this construction utilizing holomorphic vector bundles over the manifold gives a weight system on chord diagrams. We investigate these weights from the hyperkähler geometry point of view.

Generalised Seiberg-Witten equations and almost-Hermitian geometry

In this article, we study a generalisation of the Seiberg-Witten equations, replacing the spinor representation with a hyperKahler manifold equipped with certain symmetries. Central to this is the construction of a (non-linear) Dirac operator acting on the sections of the non-linear fibre-bundle. For hyperKahler manifolds admitting a hyperKahler potential, we derive a transformation formula for the Dirac operator under the conformal change of metric on the base manifold. As an application, we show that when the hyperKahler manifold is of dimension four, then away from a singular set, the equat…

What to expect from $U(n)$ Seiberg-Witten monopoles for $n > 1$

We study generalisations to the structure groups U(n) of the familiar (abelian) Seiberg-Witten monopole equations on a four-manifold $X$ and their moduli spaces. For $n=1$ one obtains the classical monopole equations. For $n > 1$ our results indicate that there should not be any non-trivial gauge-theoretical invariants which are obtained by the scheme `evaluation of cohomology classes on the fundamental cycle of the moduli space'. For, if $b_2^+$ is positive the moduli space should be `cobordant' to the empty space because we can deform the equations so as the moduli space of the deformed equa…

The families Seiberg-Witten invariants of smooth families of Kähler surfaces

We consider a generalisation of the Seiberg-Witten invariant to the families Seiberg-Witten invariants of a smooth family of 4-manifolds with fibres diffeomorphic to a 4-manifold $X$. Of particular interest is the special case when the family has a smoothly varying Kähler structure. We obtain a general computation of the invariants when $b_1=0$ in terms of characteristic classes of some vector bundles corresponding to the cohomology groups of holomorphic line bundles of the family. Finally, we apply the formula to some examples of Kähler families where some more further explicit computations c…

稳定曲率不变量的图算术

A Graphical Calculus for Stable Curvature Invariants

本文提出了一种图算术用于计算稳定曲率不变量，通过描述费菲安多项式和归一化矩多项式，推导了关于紧致爱因斯坦流形的曲率恒等式。

四流形上的一个瞬子不变量族及其与Khovanov同调的关系

A Family of Instanton-Invariants for Four-Manifolds and Their Relation to Khovanov Homology

本文基于物理论证，提出四流形上的一个参数族Haydys-Witten瞬子Floer同调群，通过维度约简和边界条件，建立与Khovanov同调的关系。

Skew Invariant Theory of Symplectic Groups, Pluri-Hodge Groups and 3-Manifold Invariants

This article deals with a number of topics which are, somewhat surprisingly, related. Firstly, the fundamental theorem of skew invariant theory for the symplectic group giving the generators and relations of symplectic invariants is established. The relations are the so called P_n relations which appear in the study of certain 3-manifold invariants. Next a class of cohomology groups are introduced, called Pluri-Hodge groups (somewhat in keeping with the notion of pluri-canonical groups). These are Dolbeault groups on a complex manifold X with values in tensor powers of sheaves of holomorphic f…

Constraints on families of smooth 4-manifolds from Bauer-Furuta invariants

We obtain constraints on the topology of families of smooth $4$-manifolds arising from a finite dimensional approximation of the families Seiberg-Witten monopole map. Amongst other results these constraints include a families generalisation of Donaldson's diagonalisation theorem and Furuta's $10/8$ theorem. As an application we construct examples of continuous $\mathbb{Z}_p$-actions for any odd prime $p$, which can not be realised smoothly. As a second application we show that the inclusion of the group of diffeomorphisms into the group of homeomorphisms is not a weak homotopy equivalence for …