Tropical refined curve counting from higher genera and lambda classes

Abstract

Block and Göttsche have defined a q -number refinement of counts of tropical curves in . Under the change of variables , we show that the result is a generating series of higher genus log Gromov–Witten invariants with insertion of a lambda class. This gives a geometric interpretation of the Block-Göttsche invariants and makes their deformation invariance manifest.

Quantum mirrors of log Calabi–Yau surfaces and higher-genus curve counting

“…in the context of cluster varieties [FG09a], [FG09b], was expected, and quantum broken lines have been studied by Mandel [Man15]. The key novelty of the present paper, building on the previous work [Bou17], [Bou18] of the author, is the connection between these algebraic/combinatorial q-deformations and the geometric deformation given by higher genus log Gromov-Witten theory.…”

“…Then β.E j = p j < 0 and so every stable log map f ∶ C → Y m of class β has a component dominating E j . If d ≠ −R ⩾0 m j , then we can do an analogue of the cycle argument of Proposition 11 of [Bou17] and Lemma 15 of [Bou18]. Knowing the asymptotic behavior of the tropical map to the tropicalization B of Y m , imposed by the tangency condition ℓ β m d , and using repetitively the balancing condition, we get that C needs to contain a cycle of components mapping surjectively to ∂Y m .…”

“…Vanishing properties of the lambda class (see e.g. Lemma 8 of [Bou17]) then imply that N Ym ∂Ym g,β = 0. If d = −R ⩾0 m j for some j, then the same argument implies the vanishing of N Ym ∂Ym g,β , except if β is a multiple of some E j , which is not the case by the assumption β ∈ G.…”

“…The main result of the present paper is a construction of a deformation quantization of X → S. Our construction follows the lines of Gross-Hacking-Keel [GHK15a] except that, rather than to use only genus zero log Gromov-Witten invariants, we use higher genus log Gromov-Witten invariants, the genus parameter playing the role of the quantization parameter ̵ h on the mirror side. We construct a quantum version of a scattering diagram and we prove its consistency using the main result of [Bou18], which itself relies on the connection with refined tropical geometry [Bou17]. Once the consistency of the quantum scattering diagram is guaranteed, some quantum version of the broken lines are well-defined and can be used to construct a deformation quantization of H 0 (X , O X ).…”

Donaldson–Thomas invariants from tropical disks

“…On the other hand, the quantum tropical curve counts in rank 2 are Block-Göttsche invariants [2], which have been related to higher-genus Gromov-Witten invariants in [3,4] and to real curve counts in [39]. Upcoming work of the second author will extend the correspondence of [39] to higher-dimensions, although an extension to tropical disks is still more distant.…”