Related papers: Equivariant gluing theory on regular instanton mod…
Recently Andrei Teleman considered instanton moduli spaces over negative definite four-manifolds $X$ with $b_2(X) \geq 1$. If $b_2(X)$ is divisible by four and $b_1(X) =1$ a gauge-theoretic invariant can be defined; it is a count of flat…
In this article we introduce a method to construct $\rm{G}_2$-instantons on $\rm{G}_2$-manifolds arising from Joyce's generalised Kummer construction. The method is based on gluing ASD instantons over ALE spaces to flat bundles on…
We solve a conjecture of Morgan and Szabo (Embedded genus 2 surfaces in four-manifolds, Preprint) about the relationship of the basic classes of two four-manifolds $X_i$ of simple type with $b_1=0$, $b^+>1$, such that there are embedded…
We develop the gluing theory of contact instantons in the context of open strings and in the context of closed strings \emph{with vanishing charge}, for example in the symplectization context. This is one of the key ingredients for the…
We study the moduli spaces of self-dual instantons on CP^2 in a simple group G. When G is a classical group, these instanton solutions can be realised using ADHM-like constructions which can be naturally embedded into certain three…
We consider the moduli space ${\cal M}(G)$ of $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a semisimple complex Lie group, and study the action of a finite group $\Gamma$ on ${\cal M}(G)$ induced by a holomorphic…
We generalize the recently proposed noncommutative ADHM construction to the case of $\Gamma$-equivariant instantons over $\R^4$, with $\Gamma$ a Kleinian group. We show that a certain form of the inhomogeneous ADHM equations describes…
We prove a gluing formula for the families Seiberg-Witten invariants of families of $4$-manifolds obtained by fibrewise connected sum. Our formula expresses the families Seiberg-Witten invariants of such a connected sum family in terms of…
We review and elaborate on certain aspects of the connections between instanton counting in maximally supersymmetric gauge theories and the computation of enumerative invariants of smooth varieties. We study in detail three instances of…
We explain a construction of $G_2$-instantons on manifolds obtained by resolving $G_2$-orbifolds. This includes the case of $G_2$-instantons on resolutions of $T^7/\Gamma$ as a special case. The ingredients needed are a $G_2$-instanton on…
We introduce a method to construct $G_2$-instantons over compact $G_2$-manifolds arising as the twisted connected sum of a matching pair of building blocks [Kov03,KL11,CHNP12]. Our construction is based on gluing $G_2$-instantons obtained…
We construct noncommutative Donaldson-Thomas invariants associated with abelian orbifold singularities by analysing the instanton contributions to a six-dimensional topological gauge theory. The noncommutative deformation of this gauge…
This paper is one step toward infinite energy gauge theory and the geometry of infinite dimensional moduli spaces. We generalize a gluing construction in the usual Yang-Mills gauge theory to an ``infinite energy'' situation. We show that we…
We investigate Yang--Mills instanton theory over four dimensional asymptotically locally flat (ALF) geometries, including gravitational instantons of this type, by exploiting the existence of a natural smooth compactification of these…
The problem of equivariant rigidity is the $\Gamma$-homeomorphism classification of $\Gamma$-actions on manifolds with compact quotient and with contractible fixed sets for all finite subgroups of $\Gamma$. In other words, this is the…
Taubes' gluing theorems establish the existence of ASD connections on closed, oriented 4-manifolds. We extend these gluing results to the mASD connections of Morgan-Mrowka-Ruberman on oriented 4-manifolds with cylindrical ends. As a…
We associate several invariants to a knot in an integer homology 3-sphere using $SU(2)$ singular instanton gauge theory. There is a space of framed singular connections for such a knot, equipped with a circle action and an equivariant…
We propose a method to construct G_2-instantons over a compact twisted connected sum G_2-manifold, applying a gluing result of S\'a Earp and Walpuski to instantons over a pair of 7-manifolds with a tubular end (see arXiv:1310.7933). In our…
We review a method to construct $\rm{G}_2$--instantons over compact $\rm{G}_2$--manifolds arising as the twisted connected sum of a matching pair of Calabi-Yau $3$-folds with cylindrical end, based on the series of articles [SE15, SEW15,…
We provide and study an equivariant theory of group (co)homology of a group G with coefficients in a gamma-equivariant G-module A, when a separate group "gamma" acts on G and A, generalizing the classical Eilenberg-MacLane (co)homology of…