We define rational numbers counting holomorphic disks bounding a complex lagrangian submanifold on a hyperkhaler manifold of real dimension four. We provide a simple a direct proof of Kontsevich-Soibelman Wall Crossing Formula for these rational invariants.

math.SG↗

arXiv2011

Frame ambiguity in Open Gromov-Witten invariants

We consider Open Gromov-Witten invariants for noncompact Calabi-Yau in the case the Lagrangian has the topology of $\R^2 \times S^1$. The definition of the invariant involves the choice of a frame for the Lagrangian, in accord with string theory. Our result applies to the examples arising from Large $N$-duality. In particular it leads to knot and link invariants counting holomorphic curves.

math.SG↗

arXiv2011

Obstruction classes and Open Gromov-Witten invariants

We define genus zero open Gromov-Witten invariants for Calabi-Yau three folds and relatively spin Lagrangian submanifold of Maslov index zero.

math.SG↗

arXiv2011

Open Gromov-Witten theory on Calabi-Yau three-folds I

We propose a general theory of the Open Gromov-Witten invariant on Calabi-Yau three-folds. We introduce the moduli space of multi-curves and show how it leads to invariants. Our construction is based on an idea of Witten. In the special case that each connected component of the Lagrangian submanifold has the rational homology of a sphere we define rational numbers $F_{g,h}$ for each genus $g$ and $h$ boundary components.

math.SG↗

arXiv2010

Master Equation and Perturbative Chern-Simons theory

We extend the Chern-Simons perturbative invariant of Axelrod and Singer to non-acyclic connections. We construct a solution of the quantum master equation on the space of functions on the cohomology of the connection. We prove that this solution is well defined up to master homotopy. We discuss also invariants of links.

math.DG↗

arXiv2009

Open Gromov-Witten theory on Calabi-Yau three-folds II

We propose a general theory of the Open Gromov-Witten invariant on Calabi-Yau three-folds. In this paper we construct the Open Gromov-Witten potential. The evaluation of the potential on its critical points leads to numerical invariants.

math.SG↗

arXiv2008

A simple proof of a theorem of Fukaya and Oh

We study the moduli space of pseudo pointed holomorphic disks with boundaries mapped in the zero section of the cotangent bundle of a manifold. We define perturbations of the equation for which it is possible to describe explicitly all the solutions of the problem in terms of Morse graphs on the manifold. In particular, this proves that the $A_\infty$ structure of the zero section of the cotangent bundle is equivalent to the Morse $A_\infty$ structure of the base manifold.

math.SG↗