We study the complex symplectic structure of the quiver varieties corresponding to the moduli spaces of SU(2) instantons on both commutative and non-commutative R^4. We identify global Darboux coordinates and quadratic Hamiltonians on classical phase spaces for which these quiver varieties are natural completions. We also show that the group of non-commutative symplectomorphisms of the corresponding path algebra acts transitively on the moduli spaces of non-commutative instantons.
This paper should be viewed as a step towards extending known results for Calogero-Moser spaces to the instanton moduli spaces.
We prove that Manolescu and Woodward's Symplectic Instanton homology, and its twisted versions are natural, and define maps associated to four dimensional cobordisms within this theory. This allows one to define representations of the mapping class group and the fundamental group of a 3-manifold, and to have a geometric interpretation of the maps appearing in the long exact sequence for symplectic instanton homology, together with vanishing criterions.
We define a twisted version of Manolescu and Woodward's Symplectic Instanton homology, prove that this invariant fits into the framework of Wehrheim and Woodward's Floer Field theory, and describe its behaviour for connected sum and Dehn surgery.
Fix a compact 4-dimensional manifold with self-dual 2nd Betti number one and with a given symplectic form. This article proves the following: The Frechet space of tamed almost complex structures as defined by the given symplectic form has an open and dense subset whose complex structures are compatible with respect to a symplectic form that is cohomologous to the given one. The theorem is proved by constructing the new symplectic form by integrating over a space of currents that are defined by pseudo-holomorphic curves.
Since its inception, Floer homology has been an important tool in low-dimensional topology. Floer theoretic invariants of $3$-manifolds tend to be either gauge theoretic or symplecto-geometric in nature, and there is a general philosophy that each gauge theoretic Floer homology should have a corresponding symplectic Floer homology and vice-versa. In this article, we construct a Lagrangian Floer invariant for any closed, oriented $3$-manifold $Y$ (called the symplectic instanton homology of $Y$ and denoted $\mathrm{SI}(Y)$) which is conjecturally equivalent to a Floer homology defined using a c…
There have been a number of constructions of Lagrangian Floer homology invariants for $3$-manifolds defined in terms of symplectic character varieties arising from Heegaard splittings. With the aim of establishing an Atiyah-Floer counterpart of Kronheimer and Mrowka's singular instanton homology, we generalize one of these, due to H. Horton, to produce a Lagrangian Floer invariant of a knot or link $K \subset Y$ in a closed, oriented $3$-manifold, which we call symplectic instanton knot homology ($\mathrm{SIK}$). We use a multi-pointed Heegaard diagram to parametrize the gluing together of a p…
Symplectic instanton homology is an invariant for closed oriented three-manifolds, defined by Manolescu and Woodward, which conjecturally corresponds to a symplectic version of a variant of Floer's instanton homology. In this thesis we study the behaviour of this invariant under connected sum, Dehn surgery, and four-dimensional cobordisms. We prove a Künneth-type formula for the connected sum : let $Y$ and $Y'$ be two closed oriented three-manifolds, we show that the symplectic instanton homology of their connected sum is isomorphic to the direct sum of the tensor product of their symplectic i…
In this paper, we study the algebraic symplectic geometry of the singular moduli spaces of Higgs bundles of degree $0$ and rank $n$ on a compact Riemann surface $X$ of genus $g$. In particular, we prove that such moduli spaces are symplectic singularities, in the sense of Beauville [Bea00], and admit a projective symplectic resolution if and only if $g=1$ or $(g, n)=(2,2)$. These results are an application of a recent paper by Bellamy and Schedler [BS16] via the so-called Isosingularity Theorem.
In arXiv:1611.09927, we constructed a well-defined Lagrangian Floer invariant for any closed, oriented $3$-manifold $Y$ via the symplectic geometry of so-called traceless $\mathrm{SU}(2)$-character varieties. This invariant, $\mathrm{SI}(Y)$, which we refer to as the symplectic instanton homology of $Y$, was also shown to satisfy an exact triangle for Dehn surgeries on knots which is typical of Floer-theoretic invariants of $3$-manifolds. In this article, we demonstrate further structural properties of this symplectic instanton homology. For example, Floer theories are expected to roughly sati…
A mathematical theory of D-string world-sheet instantons, II: Moduli stack of $Z$-(semi)stable morphisms from Azumaya nodal curves with a fundamental module to a projective Calabi-Yau 3-fold
In this paper we study the symplectic and Poisson geometry of moduli spaces of flat connections over quilted surfaces. These are surfaces where the structure group varies from region to region in the surface, and where a reduction (or relation) of structure occurs along the boundaries of the regions. Our main theoretical tool is a new form moment-map reduction in the context of Dirac geometry. This reduction framework allows us to use very general relations of structure groups, and to investigate both the symplectic and Poisson geometry of the resulting moduli spaces from a unified perspective…
We study quiver gauge theories on the round and squashed seven-spheres, and orbifolds thereof. They arise by imposing $G$-equivariance on the homogeneous space $G/H=\mathrm{SU}(4)/\mathrm{SU}(3)$ endowed with its Sasaki-Einstein structure, and $G/H=\mathrm{Sp}(2)/\mathrm{Sp}(1)$ as a 3-Sasakian manifold. In both cases we describe the equivariance conditions and the resulting quivers. We further study the moduli spaces of instantons on the metric cones over these spaces by using the known description for Hermitian Yang-Mills instantons on Calabi-Yau cones. It is shown that the moduli space of i…
