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Let $(X,\omega)$ be a compact symplectic manifold with a Hamiltonian action of a compact Lie group $G$ and $\mu: X\to \mathfrak g$ be its moment map. In this paper, we study the $L^2$-moduli spaces of symplectic vortices on Riemann surfaces with cylindrical ends. We studied a circle-valued action functional whose gradient flow equation corresponds to the symplectic vortex equations on a cylinder $S^1\times \mathbb R$. Assume that $0$ is a regular value of the moment map $\mu$, we show that the functional is of Bott-Morse type and its critical points of the functional form twisted sectors of the symplectic reduction (the symplecitc orbifold $[\mu^{-1}(0)/G]$). We show that any gradient flow lines approaches its limit point exponentially fast. Fredholm theory and compactness property are then established for the $L^2$-Moduli spaces of symplectic vortices on Riemann surfaces with cylindrical ends., Comment: a few typo are corrected, 41 pages

In this note, we give item-by-item responses to the criticisms raised in [TZ] by Tehrani ad Zinger on our paper [LR]. We illuminate the main ideas and contributions in [LR] in section 2, itemize the responses to issues raised in [TZ] and conclude that we have provided a complete proof of the degeneration formula in our published paper [LR] and its more detailed versions in arXiv. In [TZ], the authors made an effort in comparing the methods and ideas in [LR] vs [IP-1] [IP-2], but their criticisms on [LR] are based on their own lack of sufficient understanding of [LR]., Comment: 20 pages

This is the first part of a trilogy where we apply the theory of virtual manifold/orbifolds developed by the first named author and Tian to study the Gromov-Witten moduli spaces. In this paper, we resolve the main analytic issue arising from the lack of differentiability of $PSL(2, \C)$-action on spaces of $W^{1, p}$-maps from the Riemann sphere to a symplectic manifold $(X, \omega, J)$ with a non-zero homology class $A$. In particular, we establish the slice and tubular neighbourhood theorems for $PSL(2, \C)$-action along smooth maps, and construct a $PSL(2, \C)$-obstruction bundle along $PSL(2, \C)$-orbit of a pseudo-holomorphic map representing a point in the moduli space $\cM_{0, 0}(X, A)$. In Sections 2 and 3 of this paper, we explain an integration theory on virtual orbifolds using proper \'etale groupoids and establish the virtual neighborhood technique for a general orbifold Fredholm system. When the moduli space $\cM_{0, 0}(X, A)$ of pseudo-holomorphic spheres in $(X, \omega, J)$ is compact, applying the virtual neighborhood technique developed in Section 3, we obtain a virtual system for the moduli space $\cM_{0, 0}(X, A)$ of pseudo-holomorphic spheres in $(X, \omega, J)$ and show that the genus zero Gromov-Witten invariant is well-defined.

Kuranishi structures were introduced to symplectic topology by Fukaya and Ono and recently refined by Joyce, in order to extract homological data from compactified moduli spaces of holomorphic maps in cases where geometric regularization approaches such as perturbations of the almost complex structure do not yield a smooth structure on the moduli space. We give a general survey of regularization techniques in symplectic topology, pointing to some general analytic issues, and discussing some specific topological issues of the Kuranishi approach. In the main body of the paper we provide an abstract framework of Kuranishi atlases which separates the analytic and topological issues. Throughout, we focus on the most fundamental issues, which are already present in applying virtual transversality techniques to moduli spaces of holomorphic spheres without nodes or nontrivial isotropy. This is the reinstated 2013 version of this survey and sample construction. A generalized version of the topological theory is now available under 'The topology of Kuranishi atlases' arxiv:1508.01844, with the survey parts and VMC construction updated in 'The fundamental class of smooth Kuranishi atlases with trivial isotropy' arxiv:1508.01560., Comment: Version 2 added material in 2.6 and the appendix. Versions 3 and 4 clarify comments on Siebert's work and changed the title and name of the main object to Kuranishi atlases. Version 5 made further small corrections to citations. Version 6 split off some parts. Versions 7 reverted to Version 5. Version 8 added arxiv numbers in abstract

The moduli space of regular stable maps with values in a complex manifold admits naturally the structure of a complex orbifold. Our proof uses the methods of differential geometry rather than algebraic geometry. It is based on Hardy decompositions and Fredholm intersection theory in the loop space of the target manifold., Comment: 59 pages, 2 figures

This note collects a number of standard statements in Riemannian geometry and in Sobolev-space theory that play a prominent role in analytic approaches to symplectic topology. These include relations between connections and complex structures, estimates on exponential-like maps, and dependence of constants in Sobolev and elliptic estimates., Comment: 33 pages, 3 figures

This survey wants to give a short introduction to the transversality problem in symplectic field theory and motivate to approach it using the new Fredholm theory by Hofer, Wysocki and Zehnder. With this it should serve as a lead-in for the user's guide to polyfolds, which will appear soon and is the result of a working group organized by J. Fish, R. Golovko and the author at MSRI Berkeley in fall 2009., Comment: 17 pages, survey

This paper establishes compactness results for the moduli stack of holomorphic curves in suitable exploded manifolds. This result together with the analysis in arXiv:0902.0087 allows the definition of Gromov Witten invariants of these exploded manifolds., Comment: 76 pages. In v2, compactness is proved using more practical to verify assumptions, and there is more focus on $\dbar$-log compatible almost complex structures. arXiv admin note: text overlap with arXiv:0706.3917

Flat coordinates for Frobenius manifolds defined on the orbit space of a Coxeter group W are specified through a certain system of generators of W-invariant polynomials. In this note, starting from basic invariants proposed by M.Mehta, we calculate flat coordinates for the exceptional groups of type E_7 and E_8, leading to a derivation of the potentials for the associated Frobenius structures., Comment: 13 pages

In [LP] the authors defined symplectic "Local Gromov-Witten invariants" associated to spin curves and showed that the GW invariants of a Kahler surface X with p_g>0 are a sum of such local GW invariants. This paper describes how the local GW invariants arise from an obstruction bundle (in the sense of Taubes) over the space of stable maps into curves. Together with the results of [LP], this reduces the calculation of the GW invariants of complex surfaces to computations in the GW theory of curves.

It was pointed out by Y. Eliashberg in his ICM 2006 plenary talk that the rich algebraic formalism of symplectic field theory leads to a natural appearance of quantum and classical integrable systems, at least in the case when the contact manifold is the prequantization space of a symplectic manifold. In this paper we generalize the definition of gravitational descendants in SFT from circle bundles in the Morse-Bott case to general contact manifolds. After we have shown that for the basic examples of holomorphic curves in SFT, that is, branched covers of cylinders over closed Reeb orbits, the gravitational descendants have a geometric interpretation in terms of branching conditions, we compute the corresponding sequences of Poisson-commuting functions when the contact manifold is the unit cotangent bundle of a Riemannian manifold., Comment: 44 pages, no figures

For any non-degenerate, quasi-homogeneous hypersurface singularity W and an admissible group of diagonal symmetries G, Fan, Jarvis, and Ruan have constructed a cohomological field theory which is a candidate for the mathematical structure behind the Landau-Ginzburg A-model. When using the orbifold Milnor ring of a singularity W as a B-model, and the Frobenius algebra H_{W,G} constructed by Fan, Jarvis, and Ruan, as an A-model, the following conjecture is obtained: For a quasi-homogeneous singularity W and a group G of symmetries of W, there is a dual singularity W^T such that the orbifold A-model of W/G is isomorphic to the B-model of W^T. I will show that this conjecture holds for a two-dimensional invertible loop potential $W$ with its maximal group of diagonal symmetries $G_{W}$., Comment: 10 pages

In this article, we study the Berglund--H\"ubsch transpose construction W^T for invertible quasihomogeneous potential W. We introduce the dual group G^T and establish the state space isomorphism between the Fan-Jarvis-Ruan-Witten A-model of W/G and the orbifold Milnor ring B-model of W^T/G^T. Furthermore, we prove a mirror symmetry theorem at the level of Frobenius algebra structure for G^max. Then, we interpret Arnol'd strange duality of exceptional singularities W as mirror symmetry between W/J and its strange dual W^SD., Comment: 32 pages

The quantum differential equations can be regarded as examples of equations with certain universal properties which are of wider interest beyond quantum cohomology itself. We present this point of view as part of a framework which accommodates the KdV equation and other well known integrable systems. In the case of quantum cohomology, the theory is remarkably effective in packaging geometric information; we illustrate this with reference to simple examples of Gromov-Witten invariants, variations of Hodge structure, the Reconstruction Theorem, and the Crepant Resolution Conjecture. Based on lectures given at the summer school "Geometric and Topological Methods for Quantum Field Theory", Villa de Leyva, 2007., Comment: 26 pages

We consider the construction of Frobenius manifolds associated to projective special geometry and analyse the dependence on choices involved. In particular, we prove that the underlying F-manifold is canonical. We then apply this construction to integrable systems of Hitchin type., Comment: 43 pages

We establish a product formula for Gromov-Witten invariants for closed, connected, relatively semi-positive Hamiltonian fibrations over any symplectic base. Furthermore, we show that the fibration projection induces a locally trivial (orbi-)fibration map from the moduli space of pseudo-holomorphic maps with marked points in the total space of the Hamiltonian fibration to the corresponding moduli space of pseudo-holomorphic maps with marked points in the base. We use this induced map to recover the product formula by means of integration. Finally, we give applications to c-splitting and symplectic uniruledness., Comment: 59 pages, no figure, changes in the presention, minor corrections, added proofs in the gluing section.

We consider the symplectic vortex equations for a linear Hamiltonian torus action. We show that the associated genus zero moduli space itself is homotopic (in the sense of a homotopy of regular G-moduli problems) to a toric manifold with combinatorial data directly obtained from the original torus action. This allows to view the wall crossing formula of Cieliebak and Salamon for the computation of vortex invariants as a consequence of a generalized Jeffrey-Kirwan localization formula for integrals over symplectic quotients., Comment: 84 pages

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 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., Comment: 44 pages, 1 figure. Title changed

We prove a duality principle for a special class of submanifolds in pseudo-Euclidean spaces. This class of submanifolds with potential of normals is introduced in this paper. We prove also, for example, that an arbitrary Frobenius manifold can be realized as a certain flat submanifold of this very natural class., Comment: 3 pages

We give examples of families of Frobenius type structures on the punctured plane and we study their limits at the boundary. We then discuss the existence of a limit Frobenius manifold. We also give an example of a logarithmic Frobenius manifold., Comment: 21 pages; please note that this preprint has been superseded by arXiv:0909.4063

Let $(M,\omega)$ be a closed symplectic manifold and $\textup{Ham}(M,\omega)$ the group of Hamiltonian diffeomorphisms of $(M,\omega)$. Then the Seidel homomorphism is a map from the fundamental group of $\textup{Ham}(M,\omega)$ to the quantum homology ring $QH_*(M;\Lambda)$. Using this homomorphism we give a sufficient condition for when a nontrivial loop $\psi$ in $\textup{Ham}(M,\omega)$ determines a nontrivial loop $\psi\times\textup{id}_N$ in $\textup{Ham}(M\times N,\omega\oplus\eta)$, where $(N,\eta)$ is a closed symplectic manifold such that $\pi_2(N)=0$., Comment: 13 pages

We prove that the orbifold quantum ring is preserved under singular symplectic flops. Hence we verify Ruan's conjecture for this case., Comment: 34pages

In this paper we study certain algebraic properties of the quantum homology algebra for the class of symplectic toric Fano manifolds. In particular, we examine the semi-simplicity of the quantum homology algebra, and the more general property of containing a field as a direct summand. Our main result provides an easily-verified sufficient condition for these properties which is independent of the symplectic form. Moreover, we answer two questions of Entov and Polterovich negatively by providing examples of toric Fano manifolds with non semi-simple quantum homology algebra, and others in which the Calabi quasi-morphism is non-unique., Comment: Revised version. Two corollaries have been added

In this paper, we gave some explicit relations between absolute and relative Gromov-Witten invariants. We proved that a symplectic manifold is symplectic rationally connected if it contains a positive divisor symplectomorphic to $P^n$.

Schwarz showed that when a closed symplectic manifold (M,\om) is symplectically aspherical (i.e. the symplectic form and the first Chern class vanish on \pi_2(M)) then the spectral invariants, which are initially defined on the universal cover of the Hamiltonian group, descend to the Hamiltonian group Ham (M,\om). In this note we describe less stringent conditions on the Chern class and quantum homology of M under which the (asymptotic) spectral invariants descend to Ham (M,\om). For example, they descend if the quantum multiplication of M is undeformed and H_2(M) has rank >1, or if the minimal Chern number is at least n+1 (where \dim M=2n) and the even cohomology of M is generated by divisors. The proofs are based on certain calculations of genus zero Gromov--Witten invariants. As an application, we show that the Hamiltonian group of the one point blow up of T^4 admits a Calabi quasimorphism. Moreover, whenever the (asymptotic) spectral invariants descend it is easy to see that Ham (M,\om) has infinite diameter in the Hofer norm. Hence our results establish the infinite diameter of Ham in many new cases. We also show that the area pseudonorm -- a geometric version of the Hofer norm -- is nontrivial on the (compactly supported) Hamiltonian group for all noncompact manifolds as well as for a large class of closed manifolds., Comment: 34 pages, no figures; to appear in Commentarii Math. Helv; v4 corrects a small error in Prop 2.3

The Drinfeld - Sokolov construction associates a hierarchy of bihamiltonian integrable systems with every untwisted affine Lie algebra. We compute the complete set of invariants of the related bihamiltonian structures with respect to the group of Miura type transformations., Comment: 73 pages, no figures

We give a short review of Frobenius manifolds and algebraic integrability and study their intersection. The simplest case is the relation between the Frobenius manifold of simple singularities, which is almost dual to the integrable open Toda chain. New types of manifolds called extra special Kaehler and special F-manifolds are introduced which capture the intersection., Comment: 18 pages, 1 figure, to appear in Encyclopedia of Peyresq (2007)

We review and streamline our previous results and the results of Y.Ostrover on the existence of Calabi quasi-morphisms and symplectic quasi-states on symplectic manifolds with semi-simple quantum homology. As an illustration, we discuss the case of symplectic toric Fano 4-manifolds. We present also new results due to D.McDuff: she observed that for the existence of quasi-morphisms/quasi-states it suffices to assume that the quantum homology contains a field as a direct summand, and she showed that this weaker condition holds true for one point blow-ups of non-uniruled symplectic manifolds., Comment: A minor change: clarified the recipe for computing the quantum homology of a symplectic toric Fano manifold

In this note we introduce parameterized Gromov-Witten invariants for symplectic fiber bundles and study the topology of the symplectomorphism group. We also give sample applications showing the non-triviality of certain homotopy groups of some symplectomorphism groups., Comment: 23 pages, final version

We use Donaldson hypersurfaces to construct pseudo-cycles which define Gromov-Witten invariants for any symplectic manifold which agree with the invariants in the cases where transversality could be achieved by perturbing the almost complex structure., Comment: 53 pages, final version

The present paper explores how the spectral sequence introduced in a previous work (and obtained by taking moduli spaces of any dimension into account in the Floer construction), interacts with the presence of bubbling. As consequences are obtained some relations between binary Gromov-Witten invariants and relative Ganea-Hopf invariants, a criterion for detecting the monodromy of bubbling as well as algebraic criteria for the detection of periodic orbits., Comment: 27 pages

We prove that the associativity equations of two-dimensional topological quantum field theories are very natural reductions of the fundamental nonlinear equations of the theory of submanifolds in pseudo-Euclidean spaces and give a natural class of potential flat torsionless submanifolds. We show that all potential flat torsionless submanifolds in pseudo-Euclidean spaces bear natural structures of Frobenius algebras on their tangent spaces. These Frobenius structures are generated by the corresponding flat first fundamental form and the set of the second fundamental forms of the submanifolds (in fact, the structural constants are given by the set of the Weingarten operators of the submanifolds). We prove in this paper that each N-dimensional Frobenius manifold can locally be represented as a potential flat torsionless submanifold in a 2N-dimensional pseudo-Euclidean space. By our construction this submanifold is uniquely determined up to motions. Moreover, in this paper we consider a nonlinear system, which is a natural generalization of the associativity equations, namely, the system describing all flat torsionless submanifolds in pseudo-Euclidean spaces, and prove that this system is integrable by the inverse scattering method., Comment: 10 pages, Proceedings of the Workshop "Nonlinear Physics. Theory and Experiment. IV. Gallipoli (Lecce), Italy, June 22 - July 1, 2006

Holomorphic disk invariants with boundary in the real Lagrangian of a quintic 3-fold are calculated by localization and proven mirror transforms. A careful discussion of the underlying virtual intersection theory is included. The generating function for the disk invariants is shown to satisfy an extension of the Picard-Fuchs differential equations associated to the mirror quintic. The Ooguri-Vafa multiple cover formula is used to define virtually enumerative disk invariants. The results may also be viewed as providing a virtual enumeration of real rational curves on the quintic., Comment: 52 pages, 5 figures. Added background on open mirror symmetry in Section 0.4, and added details especially in Lemma 7 and Lemma 18

On a compact K\"{a}hler manifold $X$ with a holomorphic 2-form $\a$, there is an almost complex structure associated with $\a$. We show how this implies vanishing theorems for the Gromov-Witten invariants of $X$. This extends the approach, used in \cite{lp} for K\"{a}hler surfaces, to higher dimensions.

We prove a structure theorem for the Gromov-Witten invariants of compact Kahler surfaces with geometric genus $p_g>0$. Under the technical assumption that there is a canonical divisor that is a disjoint union of smooth components, the theorem shows that the GW invariants are universal functions determined by the genus of this canonical divisor components and the holomorphic Euler characteristic of the surface. We compute special cases of these universal functions., Comment: 27 pages

We show that moduli spaces of stable maps admits virtual orbifold structure. The symplectic version of virtual localization formula is obtained., Comment: 62 pages

We study the degeneration and the gluing of Kuranishi structures in Gromov-Witten theory under a symplectic cut. This leads us to a degeneration axiom and a gluing axiom for open Gromov-Witten invariants. They provide then a route to the construction of virtual fundamental chains via specialization. Comments on the equivalence of the degeneration formula of closed Gromov-Witten invariants by Li-Ruan/Li versus Ionel-Parker are given in the Appendix., Comment: 102+2 pages

This paper is concerned with the rational symplectic field theory in the Floer case. For this observe that in the general geometric setup for symplectic field theory the contact manifolds can be replaced by mapping tori of symplectic manifolds with symplectomorphisms. While the cylindrical contact homology is given by the Floer homologies of powers of the symplectomorphism, the other algebraic invariants of symplectic field theory provide natural generalizations of symplectic Floer homology. For symplectically aspherical manifolds and Hamiltonian symplectomorphisms we study the moduli spaces of rational curves and prove a transversality result, which does not need the polyfold theory by Hofer, Wysocki and Zehnder. Besides that our result shows that one does not get nontrivial operations on Floer homology from symplectic field theory, we use it to compute the full contact homology of the corresponding Hamiltonian mapping torus., Comment: 40 pages, revised version to appear in Commentarii Mathematici Helvetici

We define a new family of open Gromov-Witten type invariants based on intersection theory on the moduli space of pseudoholomorphic curves of arbitrary genus with boundary in a Lagrangian submanifold. We assume the Lagrangian submanifold arises as the fixed points of an anti-symplectic involution and has dimension 2 or 3. In the strongly semi-positive genus 0 case, the new invariants coincide with Welschinger's invariant counts of real pseudoholomorphic curves. Furthermore, we calculate the new invariant for the real quintic threefold in genus 0 and degree 1 to be 30., Comment: 79 pages

First, we provide another proof that the signed count of the real $J$-holomorphic spheres (or $J$-holomorphic discs) passing through a generic real configuration of $k$ points is independent of the choice of the real configuration and the choice of $J$, if the dimension of the Lagrangian submanifold $L$ (fixed points set of the involution) is two or three, and also if we assume $L$ is orientable and relatively spin, and $M$ is strongly semi-positive. This theorem was first proved by Welschinger in a more general setting, and we provide more natural approach using the degree of evaluation maps from the moduli spaces of $J$-holomorphic discs. Then, we define the invariant count of discs intersecting cycles of a symplectic manifold at fixed interior marked points, and intersecting real points at the boundary under certain assumptions. The last result is new and was not proved by Welshinger's method., Comment: 18 pages, 2 figures, typo corrected

The goal of this paper is to give an efficient computation of the 3-point Gromov-Witten invariants of Fano hypersurfaces, starting from the Picard-Fuchs equation. This simplifies and to some extent explains the original computations of Jinzenji. The method involves solving a gauge-theoretic differential equation, and our main result is that this equation has a unique solution., Comment: 23 pages