Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1404.1898

On Symplectic Sum Formulas in Gromov-Witten Theory

Mohammad F. Tehrani·Aleksey Zinger
View original source ↗Read source document ↗Export metadata ↓

Abstract

This manuscript describes in detail the symplectic sum formulas in Gromov-Witten theory and related topological and analytic issues. In particular, we analyze and compare two analytic approaches to these formulas. The Ionel-Parker formula contains two unique features, rim tori refinements of relative invariants and the so-called S-matrix, which have been a mystery in GW-theory over the past decade. We explain why the latter, which appears due to imprecise reasoning, should not be present and how the former should be interpreted. While the gluing argument in the IP work attempts to address all of the issues relevant to certain "semi-positive" cases, it contains several highly technical, but crucial, mistakes, which invalidate it and thus the whole paper almost completely. The SFT type idea behind the Li-Ruan approach has the potential of avoiding many issues with the degeneration of the metric on the target occurring in the IP approach. Unfortunately, the LR paper is vague about the key notions and aspects of the setup, including the definition of relative stable maps, and does not contain even a description of the local structure of the relative moduli space or an attempt at a complete proof of any major statement. The only technical arguments in this paper concern fairly minor points and are either incorrect or unnecessary. Neither of the two papers even considers gluing stable maps with extra rubber structure, which is necessary for defining the relevant invariants outside of a relatively narrow collection of "semi-positive" cases. In this manuscript, we re-formulate the (numerical) symplectic sum formula, describe the issues arising in both approaches, and explain how the Li-Ruan SFT type idea can be used to address them.

Explore related subjects

math.SGmath.AG

Keep this discovery

BibTeXRIS

Mohammad F. Tehrani, Aleksey Zinger. 2014-12-28. On Symplectic Sum Formulas in Gromov-Witten Theory. https://arxiv.org/abs/1404.1898

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

KEEP EXPLORING

Related papers

arXiv2026

Extended Future Tube Conjecture for Unipotent Subgroups

Let $\Omega$ be the Lorentz future cone in $\mathbb{R}^{d+1}$ with respect to the Lorentz product and let $T^M$ be the $M$-fold product of the future tube $T=\mathbb{R}^{d+1}+i\Omega$. The Lorentz group $\mathrm{SO}_0(1,d)$ acts diagonally on $T^M$, and its complexification $\mathrm{SO}(1,d)^\mathbb{C}$ acts on $\mathbb{C}^{(d+1)\times M}$. We prove that the domain $G^\mathbb{C}\cdot T^M$ is a Stein manifold for any connected unipotent subgroup $G$ of $\mathrm{SO}_0(1,d)$.

math.SG↗
arXiv2026

The First Correction Term in the Asymptotic Expansion of Bohr--Sommerfeld Lagrangian States

Let $\Lambda$ be a compact Bohr--Sommerfeld Lagrangian submanifold of a compact K\"ahler manifold equipped with a holomorphic prequantum line bundle. We study the asymptotic expansion of the Lagrangian states associated with $\Lambda$. In particular, we compute explicitly the first nontrivial correction term and show that it is expressed in terms of geometric invariants of the ambient K\"ahler manifold and the Lagrangian submanifold, including their scalar curvatures, the second fundamental form, and the mean curvature. As a consequence, we obtain the corresponding second-order asymptotic formula for the $L^2$-norm of the Lagrangian states.

math.SG↗
arXiv2026

Classification of Legendrian doubles and suspensions

We define a construction of Legendrians inside contact manifolds that arise by doubling an exact Lagrangian filling in the page of an open book decomposition. This can be seen as a generalization of a previous construction by Courte and Ekholm to arbitrary open books. These Legendrians, called Legendrian doubles, are shown to admit regular flexible exact Lagrangian fillings, and they are thus classified up to Legendrian isotopy by classical data. Finally, we show that the Legendrian suspension construction, as defined by Arikan and the author in previous work,-this is a Legendrian contained inside a page of an open book that is obtained by using Seidel's suspension of Lefschetz fibrations- is a Legendrian double.

math.SG↗