arXiv subjects

Tim Gräfnitz

Publications and source records attributed to Tim Gräfnitz.

5 recordsLinked to original sources

arXiv2025

Enumerative Geometry of Quantum Periods

We interpret the $q$-refined theta function $\vartheta_1$ of a log Calabi-Yau surface $(\mathbb{P},E)$ as a natural $q$-refinement of the open mirror map, defined by quantum periods of mirror curves for outer Aganagic-Vafa branes on the local Calabi-Yau $K_{\mathbb{P}}$. The series coefficients are all-genus logarithmic two-point invariants, directly extending the relation found in [GRZ]. Yet we find an explicit discrepancy at higher genus in the relation to open Gromov-Witten invariants of the Aganagic-Vafa brane. Using a degeneration argument, we express the difference in terms of relative invariants of an elliptic curve. With $π: \widehat{\mathbb{P}} \rightarrow \mathbb{P}$ the toric blow up of a point, we use the Topological Vertex [AKMV] to show a correspondence between open invariants of $K_{\mathbb{P}}$ and closed invariants of $K_{\widehat{\mathbb{P}}}$ generalizing a variant of [CLLT][LLW] to arbitrary genus and winding. We also equate winding-1, open-BPS invariants with closed Gopakumar-Vafa invariants.

math.AG↗

arXiv2025

Smoothings from zero mutable Laurent polynomials via log resolutions and divisorial extractions

A conjecture by Corti, Filip and Petracci, inspired by mirror symmetry, states that smoothing types of affine Gorenstein toric 3-folds correspond to zero mutable Laurent polynomials. We propose a method to prove this conjecture via log crepant log resolutions constructed from compatible collections of divisorial extractions. For affine cones over weighted projective planes we prove for several infinite families of zero mutable Laurent polynomials that they indeed describe curves that admit a compatible collection of divisorial extractions. The construction of log crepant log resolutions and smoothings will be worked out in joint work with Alessio Corti and Helge Ruddat.

math.AG↗

arXiv2025

Gromov-Witten Invariants and Mirror Symmetry For Non-Fano Varieties Via Tropical Disks

Under mirror symmetry a non-Fano variety $X$ corresponds to an instanton corrected Hori-Vafa potential $W$. The classical period of $W$ equals the regularized quantum period of $X$, which is a generating function for descendant Gromov-Witten invariants. These periods define closed mirror maps relating complex with symplectic parameters and open mirror maps relating coordinates on the mirror curves. We interpret the corrections to $W$ by broken lines in a scattering diagram, so that $W$ is the primitive theta function $\vartheta_1$. We show that, after wall crossing to infinity and application of the closed mirror map, $W=\vartheta_1$ is equal to the open mirror map. By tropical correspondence, $\vartheta_1$ is a generating function for $2$-marked logarithmic Gromov-Witten invariants, which are algebraic analogues of counts of Maslov index $2$ disks. This generalizes the predictions of mirror symmetry to the non-Fano case.

math.AG↗

arXiv2024

The proper Landau-Ginzburg potential is the open mirror map

The mirror dual of a smooth toric Fano surface $X$ equipped with an anticanonical divisor $E$ is a Landau-Ginzburg model with superpotential, W. Carl-Pumperla-Siebert give a definition of the the superpotential in terms of tropical disks using a toric degeneration of the pair $(X,E)$. When $E$ is smooth, the superpotential is proper. We show that this proper superpotential equals the open mirror map for outer Aganagic-Vafa branes in the canonical bundle $K_X$, in framing zero. As a consequence, the proper Landau-Ginzburg potential is a solution to the Lerche-Mayr Picard-Fuchs equation. Along the way, we prove a generalization of a result about relative Gromov-Witten invariants by Cadman-Chen to arbitrary genus using the multiplication rule of quantum theta functions. In addition, we generalize a theorem of Hu that relates Gromov-Witten invariants of a surface under a blow-up from the absolute to the relative case. One of the two proofs that we give introduces birational modifications of a scattering diagram. We also demonstrate how the Hori-Vafa superpotential is related to the proper superpotential by mutations from a toric chamber to the unbounded chamber of the scattering diagram.

math.AG↗

arXiv2023

Scattering diagrams: polynomiality and the dense region

We use deformations and mutations of scattering diagrams to show that the coefficients of a scattering diagram with initial functions $f1 = (1+tx)^μ$ and $f2 = (1+ty)^ν$ are polynomial in $μ$, $ν$ and non-trivial in a certain dense region. We discuss consequences for Gromov-Witten invariants and quiver representations.

math.AG↗