24 research outputs found
Refined floor diagrams from higher genera and lambda classes
We show that, after the change of variables , refined floor
diagrams for and Hirzebruch surfaces compute generating series
of higher genus relative Gromov-Witten invariants with insertion of a lambda
class. The proof uses an inductive application of the degeneration formula in
relative Gromov-Witten theory and an explicit result in relative Gromov-Witten
theory of . Combining this result with the similar looking
refined tropical correspondence theorem for log Gromov-Witten invariants, we
obtain some non-trivial relation between relative and log Gromov-Witten
invariants for and Hirzebruch surfaces. We also prove that the
Block-G\"ottsche invariants of and are related by
the Abramovich-Bertram formula.Comment: 44 pages, 8 figures, revised version, exposition greatly improved,
main results unchanged, published in Selecta Mathematic
The quantum tropical vertex
Gross-Pandharipande-Siebert have shown that the 2-dimensional
Kontsevich-Soibelman scattering diagrams compute certain genus zero log
Gromov-Witten invariants of log Calabi-Yau surfaces. We show that the
-refined 2-dimensional Kontsevich-Soibelman scattering diagrams compute,
after the change of variables , generating series of certain
higher genus log Gromov-Witten invariants of log Calabi-Yau surfaces.
This result provides a mathematically rigorous realization of the physical
derivation of the refined wall-crossing formula from topological string theory
proposed by Cecotti-Vafa, and in particular can be seen as a non-trivial
mathematical check of the connection suggested by Witten between higher genus
open A-model and Chern-Simons theory.
We also prove some new BPS integrality results and propose some other BPS
integrality conjectures.Comment: v2: 68 pages, revised version (minor mistake in Section 5 corrected),
published in Geometry and Topolog
On an example of quiver DT/relative GW correspondence
We explain and generalize a recent result of Reineke-Weist by showing how to
reduce it to the Gromov-Witten/Kronecker correspondence by a degeneration and
blow-up. We also refine the result by working with all genera on the
Gromov-Witten side and with refined Donaldson-Thomas invariants on the quiver
side.Comment: 37 pages, comments welcom
Quivers and curves in higher dimension
We prove a correspondence between Donaldson-Thomas invariants of quivers with
potential having trivial attractor invariants and genus zero punctured
Gromov-Witten invariants of holomorphic symplectic cluster varieties. The proof
relies on the comparison of the stability scattering diagram, describing the
wall-crossing behavior of Donaldson-Thomas invariants, with a scattering
diagram capturing punctured Gromov-Witten invariants via tropical geometry.Comment: 38 pages, 3 figures. Comments welcome
Fock-Goncharov dual cluster varieties and Gross-Siebert mirrors
Cluster varieties come in pairs: for any cluster variety there
is an associated Fock-Goncharov dual cluster variety. On the
other hand, in the context of mirror symmetry, associated with any log
Calabi-Yau variety is its mirror dual, which can be constructed using the
enumerative geometry of rational curves in the framework of the Gross-Siebert
program. In this paper we bridge the theory of cluster varieties with the
algebro-geometric framework of Gross-Siebert mirror symmetry. Particularly, we
show that the mirror to the cluster variety is a degeneration of
the Fock-Goncharov dual cluster variety and vice versa. To do
this, we investigate how the cluster scattering diagram of
Gross-Hacking-Keel-Kontsevich compares with the canonical scattering diagram
defined by Gross-Siebert to construct mirror duals in arbitrary dimensions.
Consequently, we derive an enumerative interpretation of the cluster scattering
diagram. Along the way, we prove the Frobenius structure conjecture for a class
of log Calabi-Yau varieties obtained as blow-ups of toric varieties.Comment: 51 pages, revised version published in Journal f\"ur die reine und
angewandte Mathematik (Crelles Journal