Narrow Results

This paper aims at explaining some incarnations of the idea of topological recursion: in two-dimensional quantum field theories (2d TQFTs), in cohomological field theories (CohFT), and in the computation of volumes of the moduli space of curves. It gives an introduction to the formalism of quantum Airy structures on which the topological recursion is based, which is seen at work in the above topics.

For a smooth, irreducible projective surface S over ℂ, the number of r-nodal curves in an ample linear system (where is a line bundle on S) can be expressed using the rth Bell polynomial P_r in universal functions a_i, 1 ≤ i ≤ r, of (S, ), which are ℤ-linear polynomials in the four Chern numbers of S and . We use this result to establish a proof of the classical shape conjectures of Di Francesco–Itzykson and Göttsche governing node polynomials in the case of ℙ². We also give a recursive procedure which provides the -term of the polynomials a_i.

We compute the purely real Welschinger invariants, both original and modified, for all real del Pezzo surfaces of degree ≥ 2. We show that under some conditions, for such a surface X and a real nef and big divisor class D ∈ Pic(X), through any generic collection of - DK_X - 1 real points lying on a connected component of the real part ℝX of X one can trace a real rational curve C ∈ |D|. This is derived from the positivity of appropriate Welschinger invariants. We furthermore show that these invariants are asymptotically equivalent, in the logarithmic scale, to the corresponding genus zero Gromov–Witten invariants. Our approach consists in a conversion of Shoval–Shustin recursive formulas counting complex curves on the plane blown up at seven points and of Vakil's extension of the Abramovich–Bertram formula for Gromov–Witten invariants into formulas computing real enumerative invariants.

We study the problem of computing Gopakumar–Vafa (GV) invariants for multiparameter families of symmetric Calabi–Yau threefolds admitting flops to diffeomorphic manifolds. There are infinite Coxeter groups, generated by permutations and flops, that act as symmetries on the GV-invariants of these manifolds. We describe how these groups are related to symmetries in GLSMs, and the existence of multiple mirrors. Some representation theory of these Coxeter groups is also discussed. The symmetries provide an infinite number of relations between the GV-invariants for each fixed genus. This remarkable fact is of assistance in obtaining higher-genus invariants via the BCOV recursion. This paper is based on joint work with Philip Candelas and Xenia de la Ossa.

Je rappelle les divers problèmes de géométrie énumérative réelle desquels j'ai pu extraire des invariants à valeurs entières, fournissant un pendant réel aux invariants de Gromov-Witten. Je discute l'optimalité des bornes inférieures fournies par ces invariants ainsi que certaines de leurs propriétés arithmétiques. Je présente enfin davantage de résultats garantissant la présence ou l'absence de disques pseudo-holomorphes à bord dans une sous-variété lagrangienne d'une variété symplectique donnée.