publications

preprint:

2024

1. Refined Gromov-Witten invariants

   Andrea Brini, and Yannik Schuler

   arXiv:2410.00118 [math.AG], 2024

   Abs arXiv

   We study the enumerative geometry of stable maps to Calabi-Yau 5-folds Z with a group action preserving the Calabi-Yau form. In the central case Z=X×C2, where X is a Calabi-Yau 3-fold with a group action scaling the holomorphic volume form non-trivially, we conjecture that the disconnected equivariant Gromov-Witten generating series of Z returns the Nekrasov-Okounkov equivariant K-theoretic PT partition function of X and, under suitable rigidity conditions, its refined BPS index. We show that in the unrefined limit the conjecture reproduces known statements about the higher genus Gromov-Witten theory of X; we prove it for X the resolved conifold; and we establish a refined cycle-level local/relative correspondence for local del Pezzo surfaces, implying the Nekrasov-Shatashvili limit of the conjecture when X is the local projective plane. We further establish B-model physics predictions of Huang-Klemm for refined higher genus mirror symmetry for local P2. In particular, we prove that our refined Gromov-Witten generating series obey extended holomorphic anomaly equations, are quasi-modular functions of Γ1(3), have leading asymptotics at the conifold point given by the logarithm of the Barnes double-Gamma function, and satisfy a version of the higher genus Crepant Resolution Correspondence with the refined orbifold Gromov-Witten theory of [C3/μ3]. This refines results, and partially proves conjectures, of Lho-Pandharipande, Coates-Iritani, and Bousseau-Fan-Guo-Wu.

2. The log-open correspondence for two-component Looijenga pairs

   Yannik Schuler

   arXiv: 2404.15412 [math.AG], 2024

   Abs arXiv

   A two-component Looijenga pair is a rational smooth projective surface with an anticanonical divisor consisting of two transversally intersecting curves. We establish an all-genus correspondence between the logarithmic Gromov-Witten theory of a two-component Looijenga pair and open Gromov-Witten theory of a toric Calabi-Yau threefold geometrically engineered from the surface geometry. This settles a conjecture of Bousseau, Brini and van Garrel in the case of two boundary components. We also explain how the correspondence implies BPS integrality for the logarithmic invariants and provides a new means for computing them via the topological vertex method.

2023

1. Gromov–Witten theory of bicyclic pairs

   Michel van Garrel, Navid Nabijou, and Yannik Schuler

   arXiv: 2310.06058 [math.AG], 2023

   Abs arXiv

   A bicyclic pair is a smooth surface equipped with a pair of smooth divisors intersecting in two reduced points. Resolutions of self-nodal curves constitute an important special case. We investigate the logarithmic Gromov-Witten theory of bicyclic pairs. We establish all-genus correspondences with local Gromov-Witten theory and open Gromov-Witten theory, and a genus zero correspondence with orbifold Gromov-Witten theory. For self-nodal curves in P(1,1,r) we obtain a closed formula for the genus zero invariants and relate these to the invariants of local curves. We also establish a conceptual relationship to the invariants obtained by smoothing the self-nodal curve. The technical heart of the paper is a qualitatively new analysis of the degeneration formula for stable logarithmic maps, complemented by torus localisation and scattering techniques.

published:

2023

1. On quasi-tame Looijenga pairs

   Andrea Brini, and Yannik Schuler

   Commun. Num. Theor. Phys., 2023

   Abs arXiv doi

   We prove a conjecture of Bousseau, van Garrel and the first-named author relating, under suitable positivity conditions, the higher genus maximal contact log Gromov-Witten invariants of Looijenga pairs to other curve counting invariants of Gromov-Witten/Gopakumar-Vafa type. The proof consists of a closed-form q-hypergeometric resummation of the quantum tropical vertex calculation of the log invariants in presence of infinite scattering. The resulting identity of q-series appears to be new and of independent combinatorial interest.

2. Higher Airy structures and topological recursion for singular spectral curves

   Gaëtan Borot, Reinier Kramer, and Yannik Schüler

   Ann. Inst. Henri Poincaré Comb. Phys. Interact., 2023

   Abs arXiv doi

   We give elements towards the classification of quantum Airy structures based on the W(gl_r)-algebras at self-dual level based on twisted modules of the Heisenberg VOA of gl_r​ for twists by arbitrary elements of the Weyl group S_r. In particular, we construct a large class of such quantum Airy structures. We show that the system of linear ODEs forming the quantum Airy structure and determining uniquely its partition function is equivalent to a topological recursion à la Chekhov–Eynard–Orantin on singular spectral curves. In particular, our work extends the definition of the Bouchard–Eynard topological recursion (valid for smooth curves) to a large class of singular curves and indicates impossibilities to extend naively the definition to other types of singularities. We also discuss relations to intersection theory on moduli spaces of curves, giving a general ELSV-type representation for the topological recursion amplitudes on smooth curves, and formulate precise conjectures for application in open r-spin intersection theory.

theses:

2024

1. Topics in Gromov-Witten theory

   Yannik Schuler

   Apr 2024

   PhD thesis

   Abs PDF

   This thesis explores different aspects of Gromov–Witten theory and is divided into two parts. The first investigates conjectures of Bousseau, Brini and van Garrel relating three a priori very different curve counts: Logarithmic Gromov–Witten theory of Looijenga pairs (certain logarithmic Calabi–Yau surfaces), open Gromov–Witten theory of toric Calabi–Yau threefolds and local Gromov–Witten theory of higher dimensional Calabi–Yau varieties. We concentrate on the case where the logarithmic boundary of the initial surface geometry has two components. First we establish the logarithmic-open correspondence in an explicit example where the Looijenga pair is a del Pezzo surface of degree six. The proof relies on a direct calculation using quantum scattering diagrams and involves an intricate identity of q-hypergeometric functions. After this case study we proceed with a more general, geometric approach and ultimately establish the logarithmic-local and all-genus logarithmic-open correspondence for all Looijenga pairs with two boundary components. The proof of the correspondences involves a delicate application the degeneration formula and torus localisation. In the second part we propose a mathematical interpretation of the so called refined topological string on a Calabi–Yau threefold in terms of equivariant Gromov–Witten theory of an extended Calabi–Yau fivefold geometry. We perform initial checks which indicate that our proposal meets several expectations formulated in the physics literature. We state a refined BPS integrality conjecture and provide evidence in case the threefold is the resolved conifold or a local del Pezzo surface. In the latter case we do so by identifying the Nekrasov–Shatashvili limit with the relative Gromov–Witten theory of the surface relative a smooth anticanonical curve.