A Quantum Kirwan Map, I: Fredholm Theory

Fabian Ziltener

Home/Archive/arXiv

arXiv · 0905.4047

A Quantum Kirwan Map, I: Fredholm Theory

Fabian Ziltener

View original source ↗Read source document ↗Export metadata ↓

Abstract

Consider a Hamiltonian action of a compact connected Lie group $G$ on an aspherical symplectic manifold $(M,ω)$. Under some assumptions on $(M,ω)$ and the action, D. A. Salamon conjectured that counting gauge equivalence classes of symplectic vortices on the plane $R^2$ gives rise to a quantum deformation $Qκ_G$ of the Kirwan map. This article is the first of three, whose goal is to define $Qκ_G$ rigorously. Its main result is that the vertical differential of the vortex equations over $R^2$ (at the level of gauge equivalence) is a Fredholm operator of a specified index. Potentially, the map $Qκ_G$ can be used to compute the quantum cohomology of many symplectic quotients. Conjecturally it also gives rise to quantum generalizations of non-abelian localization and abelianization (see [Woodward-Ziltener]).

Explore related subjects

math.SG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Fabian Ziltener. 2012-09-27. A Quantum Kirwan Map, I: Fredholm Theory. https://arxiv.org/abs/0905.4047

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

An algebraic generalization of Giroux's criterion

We compute the contact homology algebra of a neighborhood $(\mathbb{R}_τ \times W, ξ)$ of a convex hypersurface $W$ and determine when this algebra is zero or non-zero. Thus we provide a tool for inferring the tightness of such $ξ$ for $W$ of any even dimension. In more detail, consider the augmentations $ε^{\pm}$ of chain-level contact homology algebras of the dividing set $(Γ, ξ_Γ)$ determined by the positive and negative regions of $W$. We compute $CH(\mathbb{R}_τ \times W, ξ)$ as the derived tensor product of the $ε^{\pm}$. Consequently the vanishing or non-vanishing of $CH(\mathbb{R}_τ \times W, ξ)$ is determined by the induced morphisms $Hε^{\pm}$ from $CH(Γ, ξ_Γ)$ to the coefficient ring.

math.SG↗

arXiv2026

Hofer-Like Geometry Revisited

We prove that the inclusion of the Hamiltonian group $\Ham(M,ω)$ into the identity component $G_ω(M)$ of the symplectic diffeomorphism group is a bi-Lipschitz embedding with respect to the Hofer norm and the Hofer-like norm, and we identify geometric conditions under which this embedding is isometric: settling a conjecture of Banyaga. This conjecture was proved by Buss and Leclercq; our proof provides explicit equivalence constants. We also detail and simplify Banyaga's original proof of the non-degeneracy of the Hofer-like norm. We then extend the analysis to all of $G_ω(M)$: for $ϕ$ with flux class $γ$, the Hofer-like norm is given by a semidirect-product formula, the infimum over the harmonic locus $\Harm(γ)$ plus a Hofer residue. This yields a geometric condition for the two norms to agree on the Hamiltonian group. In particular, this geometric condition holds on all closed surfaces of genus $g\ge 2$.

math.SG↗

arXiv2026

Kodaira fibres and wrapped Floer cohomology

Let $F$ be a singular fibre of a relatively minimal complex elliptic fibration with smooth total space, and let $Ω$ be a nonvanishing holomorphic two-form near $F$. We show that a small neighbourhood of $F$ is a Weinstein domain for $\mathrm{Re}\,Ω$ whose completion is a Legendrian surgery, with cocores obtained by completing holomorphic disks transverse to the components of $F$. For every coefficient field and every multiplicative bulk class, we compute the wrapped Floer cohomology of these cocores and prove that it is concentrated in degree zero. The cocores generate, so the bulk-deformed wrapped Fukaya category is equivalent to the category of perfect modules over an explicit algebra: a multiplicative preprojective algebra of affine type for the normal crossing fibres, and a quiver algebra with relations for types $II$, $III$ and $IV$. Applications include formality of the affine plumbing dg algebras, mirror equivalences with resolved affine surfaces at the trivial bulk class, and with quotient stacks of algebraic tori at root-of-unity bulk classes for the four star-shaped fibres.

math.SG↗