Holomorphic Spheres Research Articles

JNR monopoles

Abstract We review the theory of JNR, mass $\frac{1}{2}$ hyperbolic monopoles in particular their spectral curves and rational maps. These are used to establish conditions for a spectral curve to be the spectral curve of a JNR monopole and to show that the rational map of a JNR monopole arises by scattering using results of Atiyah. We show that for JNR monopoles the holomorphic sphere has a remarkably simple form and show that this can be used to give a formula for the energy density at infinity. In conclusion, we illustrate some examples of the energy density at infinity of JNR monopoles.

From Pseudo-Rotations to Holomorphic Curves via Quantum Steenrod Squares

AbstractIn the context of symplectic dynamics, pseudo-rotations are Hamiltonian diffeomorphisms with finite and minimal possible number of periodic orbits. These maps are of interest in both dynamics and symplectic topology. We show that a closed, monotone symplectic manifold, which admits a nondegenerate pseudo-rotation, must have a deformed quantum Steenrod square of the top degree element and hence nontrivial holomorphic spheres. This result (partially) generalizes a recent work by Shelukhin and complements the results by the authors on nonvanishing Gromov–Witten invariants of manifolds admitting pseudo-rotations.

Symplectic cohomology rings of affine varieties in the topological limit

We construct a multiplicative spectral sequence converging to the symplectic cohomology ring of any affine variety X, with first page built out of topological invariants associated to strata of any fixed normal crossings compactification $$(M,{\mathbf {D}})$$ of X. We exhibit a broad class of pairs $$(M,{\mathbf {D}})$$ (characterized by the absence of relative holomorphic spheres or vanishing of certain relative GW invariants) for which the spectral sequence degenerates, and a broad subclass of pairs (similarly characterized) for which the ring structure on symplectic cohomology can also be described topologically. Sample applications include: (a) a complete topological description of the symplectic cohomology ring of the complement, in any projective M, of the union of sufficiently many generic ample divisors whose homology classes span a rank one subspace, (b) complete additive and partial multiplicative computations of degree zero symplectic cohomology rings of many log Calabi-Yau varieties, and (c) a proof in many cases that symplectic cohomology is finitely generated as a ring. A key technical ingredient in our results is a logarithmic version of the PSS morphism, introduced in our earlier work Ganatra and Pomerleano, arXiv:1611.06849.

Monotone Lagrangians in Flag Varieties

Abstract In this paper, we give a formula for the Maslov index of a gradient holomorphic disk, which is a relative version of the Chern number formula of a gradient holomorphic sphere for a Hamiltonian $S^1$-action. Using the formula, we classify all monotone Lagrangian fibers of Gelfand–Cetlin systems on partial flag manifolds.

Holomorphic Spheres and Four-Dimensional Symplectic Pairs

We classify four-dimensional manifolds endowed with symplectic pairs admitting embedded symplectic spheres with nonnegative self-intersection, following the strategy of McDuff’s classification of rational and ruled symplectic four-manifolds.

The diffeomorphism type of symplectic fillings

We show that simply connected contact manifolds that are subcritically Stein fillable have a unique symplectically aspherical filling up to diffeomorphism. Various extensions to manifolds with non-trivial fundamental group are discussed. The proof rests on homological restrictions on symplectic fillings derived from a degree-theoretic analysis of the evaluation map on a suitable moduli space of holomorphic spheres. Applications of this homological result include a proof that compositions of right-handed Dehn twists on Liouville domains are of infinite order in the symplectomorphism group. We also derive uniqueness results for subcritical Stein fillings up to homotopy equivalence and, under some topological assumptions on the contact manifold, up to diffeomorphism or symplectomorphism.

A compactness theorem for Fueter sections

We prove that a sequence of Fueter sections of a bundle of compact hyperkähler manifolds \mathfrak{X} over a 3-manifold M with bounded energy converges (after passing to a subsequence) outside a 1-dimensional closed rectifiable subset S \subset M . The non-compactness along S has two sources: (1) Bubbling-off of holomorphic spheres in the fibres of \mathfrak{X} transverse to a subset \Gamma \subset S , whose tangent directions satisfy strong rigidity properties. (2) The formation of non-removable singularities in a set of \mathcal{H}^1 -measure zero. Our analysis is based on the ideas and techniques that Lin developed for harmonic maps [19]. These methods also apply to Fueter sections on 4-dimensional manifolds; we discuss the corresponding compactness theorem in an appendix. We hope that the work in this paper will provide a first step towards extending the hyperkähler Floer theory developed by Hohloch, Noetzel, and Salamon [15] and Salamon [22] to general target spaces. Moreover, we expect that this work will find applications in gauge theory in higher dimensions.

Bubbles and Onis

In this article we study the gradient flow equation of a variant of the Rabinowitz action functional on very negative line bundles and relate it to periodic orbits on the base of this bundle. On very negative line bundles there are generically no holomorphic spheres.

Polyfolds, cobordisms, and the strong Weinstein conjecture

We prove the strong Weinstein conjecture for closed contact manifolds that appear as the concave boundary of a symplectic cobordism admitting an essential local foliation by holomorphic spheres.

Counting genus-zero real curves in symplectic manifolds

There are two types of $J$-holomorphic spheres in a symplectic manifold invariant under an anti-symplectic involution: those that have a fixed point locus and those that do not. The former are described by moduli spaces of $J$-holomorphic disks, which are well studied in the literature. In this paper, we first study moduli spaces describing the latter and then combine the two types of moduli spaces to get a well-defined theory of counting real curves of genus 0. We use equivariant localization to show that these invariants (unlike the disk invariants) are essentially the same for the two (standard) involutions on $\mathbb{P}^{4n-1}$.

Symplectic cobordisms and the strong Weinstein conjecture

AbstractWe study holomorphic spheres in certain symplectic cobordisms and derive information about periodic Reeb orbits in the concave end of these cobordisms from the non-compactness of the relevant moduli spaces. We use this to confirm the strong Weinstein conjecture (predicting the existence of null-homologous Reeb links) for various higher-dimensional contact manifolds, including contact type hypersurfaces in subcritical Stein manifolds and in some cotangent bundles. The quantitative character of this result leads to the definition of a symplectic capacity.

On holomorphic curves of constant curvature in the complex grassmann manifold G(2, 5)

In this article, it is proved that there doesn't exist any nonsingular holomorphic sphere in complex Grassmann manifold G(2,5) with constant curvature k = 4/7, 1/2, 4/9. Thus, from [7] it follows that if ϕ :S 2 → G(2,5) is a nonsingular holomorphic curve with constant curvature K, then, K = 4,2,4/3, 1 or 4/5.

Deformations of symplectic vortices

We prove a gluing theorem for a symplectic vortex on a compact complex curve and a collection of holomorphic sphere bubbles. Using the theorem we show that the moduli space of regular strongly stable symplectic vortices on a fixed curve with varying markings has the structure of a stratified-smooth topological orbifold. In addition, we show that the moduli space has a non-canonical C 1-orbifold structure.

Spectral Curves and the Mass of Hyperbolic Monopoles

The moduli spaces of hyperbolic monopoles are naturally fibred by the monopole mass, and this leads to a nontrivial mass dependence of the holomorphic data (spectral curves, rational maps, holomorphic spheres) associated to hyperbolic multi-monopoles. In this paper, we obtain an explicit description of this dependence for general hyperbolic monopoles of magnetic charge two. In addition, we show how to compute the monopole mass of higher charge spectral curves with tetrahedral and octahedral symmetries. Spectral curves of euclidean monopoles are recovered from our results via an infinite-mass limit.

Classification of holomorphic spheres of constant curvature in complex Grassmann manifold G2,5

In this paper, we classify all nonsingular holomorphic spheres in complex Grassmann manifolds G 2,5 with the induced constant curvatures K=4, 2, 4/3,1 and 4/5 into some classes, up to unitary equivalence, in which none of the spheres are congruent; At the same time we also prove that there does not exist the nonsingular holomorphic sphere in G 2,5 with constant curvature 2/3.

Hyperbolic Monopoles and Holomorphic Spheres

We associate to an SU(2) hyperbolic monopole a holomorphicsphere embedded in projective space and use this to uncovervarious features of the monopole.

Wrapped branes with fluxes in 8d gauged supergravity

We study the gravity dual of several wrapped D-brane configurations in presence of 4-form RR fluxes partially piercing the unwrapped directions. We present a systematic approach to obtain these solutions from those without fluxes. We use D=8 gauged supergravity as a starting point to build up these solutions. The configurations include (smeared) M2-branes at the tip of a G_2 cone on S^3 x S^3, D2-D6 branes with the latter wrapping a special Lagrangian 3-cycle of the complex deformed conifold and an holomorphic sphere in its cotangent bundle T^*S^2, D3-branes at the tip of the generalized resolved conifold, and others obtained by means of T duality and KK reduction. We elaborate on the corresponding N=1 and N=2 field theories in 2+1 dimensions.

Holomorphic spheres in loop groups and Bott periodicity

Holomorphic spheres in loop groups and Bott periodicity

A Module Structure on the Symplectic Floer Cohomology

The aim of this paper is to offer an affirmative answer to the Floer conjectures in [2, p. 589] which states that there is a module structure on the Z 2 N -graded symplectic Floer cohomology for monotone symplectic manifolds. By constructing a Z-graded symplectic Floer cohomology as an integral lift of the Z 2 N -graded symplectic Floer cohomology, we gain control of the holomorphic bubbling spheres. This makes a module structure on the Z-graded Floer cohomology. There is a spectral sequence with E 1 *,* given by the Z-graded symplectic Floer cohomology. Such a spectral sequence converges to the Z 2 N -graded symplectic Floer cohomology. Hence we induce a module structure for the Z 2 N -graded symplectic Floer cohomology by the spectral sequence and algebraic topology methods.

Holomorphic spheres in loop groups and Bott periodicity

Holomorphic spheres in loop groups and Bott periodicity