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This study aimed to investigate the impact of the replanting of basswood Ganoderma lingzhi on the soil bacterial community and reveal the obstacle phenomenon of replanting basswood G. lingzhi. In this study, the soil bacterial community of wild 20 cm (N0a) and 40 cm deep soil (N0b), cultivated once (N1a, N1b) and twice (N2a, N2b), were investigated by Illumina MiSeq sequencing. The predominant bacterial phyla at the phylum classification level were Acidobacteria, Chloracidobacteria, Nitrospira, Spartobacteria, Gemmatimonadetes, Acidobacteria-6. Still, only the relative abundance of Chloracidobacteria and Acidobacteriia increased after two years of replanting of basswood G. lingzhi. At the genus level, the dominant genus included many unclassified bacteria. Among the known genera, the best genus was DA101, which showed a decreasing trend after two years of replanting. Network analysis showed that more connections of bacterial communities were observed in soil samples of the group "a," indicating that the replanting of basswood G. lingzhi can affect the relationship between soil bacterial communities at depths. The phylogenetic investigation of communities by reconstruction of unobserved states (PICRUSt) analysis showed that the gene metabolism function of soil bacteria was quite different after one year of replanting basswood G. lingzhi. The replanting of basswood G. lingzhi changed the composition and function of the soil bacterial community, and also affected the bacterial community diversity in the soil at different depths.

The gauged linear sigma model (GLSM) is a two-dimensional supersymmetric quantum field theory introduced by Witten. It has played important roles in many studies related to string theory and mirror symmetry by both mathematicians and physicists. In previous work, the authors have completed a mathematically rigorous construction of the correlation functions of the GLSM in geometric phase. In this article, we first review the construction. Then we explain how to use the adiabatic limit technique to prove a relation between the GLSM correlation function and the Gromov–Witten invariants. Such a relation provides an enumerative interpretation of the mirror map for certain Calabi–Yau manifolds such as the quintic.

In this paper we prove a wall-crossing formula, a crucial ingredient needed to prove that the correlation function of gauged linear sigma model is independent of the choice of perturbations., This article is split from 1602.07638. Submitted and revised version. 48 pages

This is the second paper in a series following Tian and Xu (2015), 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.

The gauged Witten equation was essentially introduced by Witten in his formulation of the gauged linear σ-model (GLSM), 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 which we set up a proper framework for studying the gauged Witten equation and its perturbations. We also prove several analytical properties of solutions and moduli spaces of the perturbed gauged Witten equation. We prove that solutions have nice asymptotic behavior on cylindrical ends of the domain. Under a good perturbation scheme, the energies of solutions are shown to be uniformly bounded by a constant depending only on the topological type. We prove that the linearization of the perturbed gauged Witten equation is Fredholm, and we calculate its Fredholm index. Finally, we define a notion of stable solutions and prove a compactness theorem for the moduli space of solutions over a fixed domain curve.

We construct the vortex Floer homology group $VHF (M,\mu;H)$ for an aspherical Hamiltonian $G$-manifold $(M, \omega)$ with moment map $\mu$ 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 ${\mathbb Z}_2$.

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 $M//G$, 48 pages. Comments welcome!

Given two Morse functions $f, \mu$ on a compact manifold $M$, we study the Morse homology for the Lagrange multiplier function on $M \times {\mathbb R}$ which sends $(x, \eta)$ to $f(x) + \eta \mu(x)$. Take a product metric on $M \times {\mathbb R}$, and rescale its ${\mathbb R}$-component by a factor $\lambda^2$. We show that generically, for large $\lambda$, the Morse-Smale-Witten chain complex is isomorphic to the one for $f$ and the metric restricted to ${\mu^{-1}(0)}$, with grading shifted by one. On the other hand, let $\lambda\to 0$, we obtain another chain complex, which is geometrically quite different but has the same homology as the singular homology of $\mu^{-1}(0)$ and the isomorphism between them is provided by the homotopy by varying $\lambda$. Our proofs contain both the implicit function theorem on Banach manifolds and geometric singular perturbation theory., Comment: v3. 39 pages, published version, 4 figures. v2: 38 pages, 4 figures. Removed the last section of the previous version, improved literature

Let ( X , ω ) be a compact symplectic manifold and L be a compact Lagrangian submanifold. Suppose that ( X , L ) has a Hamiltonian S 1 action with moment map μ . Take an S 1 -invariant, ω -compatible almost complex structure, and we consider tuples ( C , P , A , φ ) where C is a smooth bordered Riemann surface of fixed topological type, P → C is an S 1 -principal bundle, A is a connection on P and φ is a section of P × S 1 X satisfying ∂ ¯ A φ = 0 , ι ν F A + μ ( φ ) = c with boundary condition φ ( ∂ C ) ⊂ P × S 1 L . Here F A is the curvature of A and ν is a volume form on C and c ∈ i R is a constant. We compactify the moduli space of isomorphism classes of such objects with energy ≤ K , where the energy is defined to be the Yang–Mills–Higgs functional ‖ F A ‖ L 2 2 + ‖ d A φ ‖ L 2 2 + ‖ μ ( φ ) − c ‖ L 2 2 . This generalizes the compactness theorem of Mundet–Tian (2009) [17] in the case of closed Riemann surfaces.

We construct virtual cycles on moduli spaces of perturbed gauged Witten equation over a fixed smooth r -spin curve, under the framework of [TX15]. Together with the wall-crossing formula proved in the companion paper [TX19], it completes the construction of the correlation function for the gauged linear sigma model announced in [TX16] as well as the proof of its invariance., Comment: The original version is split into two parts as requested by the journal. The proof of the wall-crossing formula is given in the companion paper [TX19]. 60 pages

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 [A Quantum Kirwan Map: Bubbling and Fredholm Theory, Memiors of the American Mathematical Society, Vol. 230 (American Mathematical Society, Providence, RI, 2014), pp. 1–129]. 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 [Gauged Floer theory for toric moment fibers, Geom. Funct. Anal. 21 (2011) 680–749].

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., 59 pages, revised version after referee report

We study the symplectic vortex equation over the complex plane, for the target space [Formula: see text] ([Formula: see text]) with diagonal [Formula: see text]-action. Using adiabatic limit argument, we classify all solutions with finite energy and identify their moduli spaces, which generalizes Taubes’ result for [Formula: see text].