Partly-local domain-dependent almost complex structures
We fill a gap pointed out by N. Sheridan in the proof of independence of genus zero Gromov-Witten invariants from the choice of divisor in the Cieliebak-Mohnke perturbation scheme.
math.SG↗
arXiv subjects
Guangbo Xu
Publications and source records attributed to Guangbo Xu.
The symplectic approach of gauged linear $σ$-model
Witten's Gauged Linear $σ$-Model (GLSM) unifies the Gromov-Witten theory and the Landau-Ginzburg theory, and provides a global perspective on mirror symmetry. In this article, we summarize a mathematically rigorous construction of the GLSM in the geometric phase using methods from symplectic geometry.
math.SG↗
Local model for the moduli space of affine vortices
We show that the moduli space of regular affine vortices, which are solutions of the symplectic vortex equation over the complex plane, has the structure of a smooth manifold. The construction uses Ziltener's Fredholm theory results [31]. We also extend the result to the case of affine vortices over the upper half plane. These results are necessary ingredients in defining the "open quantum Kirwan map" proposed by Woodward [24].
math.SG↗
Compactness in the adiabatic limit of disk vortices
This paper is the first input towards an open analogue of the quantum Kirwan map. We consider the adiabatic limit of the symplectic vortex equation over the unit disk for a Hamiltonian G-manifold with Lagrangian boundary condition, by blowing up the metric on the disk. We define an appropriate notion of stable solutions in the limit, and prove that any sequence of disk vortices with energy uniformly bounded has a subsequence converging to such a stable object. We also proved several analytical properties of vortices over the upper half plane, which are new type of bubbles appearing in our compactification.
math.SG↗
Gauged Floer homology and spectral invariants
We define a version of spectral invariant in the vortex Floer theory for a $G$-Hamiltonian manifold $M$. This defines potentially new (partial) symplectic quasi-morphism and quasi-states when $M//G$ is not semi-positive. We also establish a relation between vortex Hamiltonian Floer homology and Woodward's quasimap Floer homology by constructing a closed-open string map between them. This yields applications to study non-displaceability problems of subsets in $M//G$
math.SG↗
Analysis of gauged Witten equation
The gauged Witten equation was essentially introduced by Witten in his formulation of gauged linear $σ$-model (GLSM). GLSM is a physics theory which explains the so-called Landau-Ginzburg/Calabi-Yau correspondence. This is the first paper in a series towards a mathematical construction of GLSM. In this paper we study some analytical properties of the gauged Witten equation for a Lagrange multiplier type superpotential. It contains the asymptotic property of finite energy solutions, the linear Fredholm property, the uniform $C^0$-bound, and the compactness of the moduli space of solutions over a fixed smooth $r$-spin curve with uniform energy bound.
math.SG↗
Gauged Hamiltonian Floer homology I: definition of the Floer homology groups
We construct the vortex Floer homology group $VHF (M,μ;H)$ for an aspherical Hamiltonian $G$-manifold $(M, ω)$ with moment map $μ$ and a class of $G$-invariant Hamiltonian loop $H_t$, following the proposal of [3]. This is a substitute for the ordinary Hamiltonian Floer homology of the symplectic quotient of $M$. We achieve the transversality of the moduli space by the classical perturbation argument instead of the virtual technique, so the homology can be defined over ${\mathbb Z}$ or ${\mathbb Z}_2$.
math.SG↗
Morse theory for Lagrange multipliers and adiabatic limits
Given two Morse functions $f, μ$ on a compact manifold $M$, we study the Morse homology for the Lagrange multiplier function on $M \times {\mathbb R}$ which sends $(x, η)$ to $f(x) + ημ(x)$. Take a product metric on $M \times {\mathbb R}$, and rescale its ${\mathbb R}$-component by a factor $λ^2$. We show that generically, for large $λ$, the Morse-Smale-Witten chain complex is isomorphic to the one for $f$ and the metric restricted to ${μ^{-1}(0)}$, with grading shifted by one. On the other hand, let $λ\to 0$, we obtain another chain complex, which is geometrically quite different but has the same homology as the singular homology of $μ^{-1}(0)$ and the isomorphism between them is provided by the homotopy by varying $λ$. Our proofs contain both the implicit function theorem on Banach manifolds and geometric singular perturbation theory.
math.GT↗
Correlation functions of gauged linear $σ$-model
This is the second paper in a series following arXiv:1405.6352, on the construction of a mathematical theory of the gauged linear $σ$-model (GLSM). In this paper, assuming the existence of virtual moduli cycles and their certain properties, we define the correlation function of GLSM for a fixed smooth rigidified $r$-spin curve.
math.SG↗
U(1)-vortices and quantum Kirwan map
We study the symplectic vortex equation over the complex plane, for the target space ${\mathbb C}^N$ ($N\geq 2$) with diagonal U(1)-action. We classify all solutions with finite energy and identify their moduli spaces, which generalizes Taubes' result for N=1. We also studied their compactifications and use them to compute the associated quantum Kirwan maps $κ_Q: H^*_{U(1)}({\mathbb C}^N) \to QH^*({\mathbb P}^{N-1})$.
math.SG↗
Moduli space of twisted holomorphic maps with Lagrangian boundary condition: compactness
Let $(X, ω)$ be a compact symplectic manifold and $L$ be a Lagrangian submanifold. Suppose $(X, L)$ has a Hamiltonian $S^1$ action with moment map $μ$. Take an invariant $ω$-compatible almost complex structure, we consider tuples $(C, P, A, φ)$ where $C$ is a smooth bordered Riemann surface of fixed topological type, $P\to C$ is an $S^1$-principal bundle, $A$ is a connection on $P$ and $φ$ is a section of $P\times_{S^1} X$ satisfying $\ov\partial_A φ=0,\ ι_νF_A+ μ(φ)=c$ with boundary condition $φ(\partial C) \subset P \times_{S^1} L$. Here $F_A$ is the curvature of $A$ and $ν$ is a volume form on $C$ and $c\in i{\mb R}$ is a constant. We compactify the moduli space of isomorphism classes of such objects with energy $\leq K$, where the energy is defined to be the Yang-Mills-Higgs functional $\| F_A\|_{L^2}^2+ \| d_Aφ\|_{L^2}^2+ \| μ(φ)-c \|_{L^2}^2.$ This generalizes the compactness theorem of Mundet-Tian \cite{Mundet_Tian_2009} in the case of closed Riemann surfaces.
math.SG↗
Real embeddings, eta invariant and Chern-Simons current
We present an alternate proof of the Bismut-Zhang localization formula for $η$-invariants without using the analytic techniques developed by Bismut-Lebeau. A Riemann-Roch property for Chern-Simons currents, which is of independent interest, is established in due course.
math.DG↗