Descendant log Gromov-Witten invariants for toric varieties and tropical curves

Mandel, Travis; Ruddat, Helge

doi:10.48550/arxiv.1612.02402

Cited by 14 publications

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“…The tropical intersection-theoretic description of multiplicities. In genus 0, the abovementioned correspondence between tropical and algebraic Gromov-Witten counts was proved in [Ran17,Gro18] in terms of tropical intersection theory, a quite different approach from that of [NS06,MR]. In particular, this indicates that the multiplicity Mult(Γ) of ( 7) can be expressed in terms of the tropical intersection theory developed in [AR10,Rau16].…”

Section: 3mentioning

confidence: 99%

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Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves

Mandel,

Ruddat

2019

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We define a symmetric monoidal category Trop2Cob which, roughly, has degrees of tropical curves as its objects and types of tropical curves as its morphisms. A symmetric monoidal functor with domain Trop2Cob is what we call a (2D) tropical quantum field theory (TrQFT). We generalize the correspondence of 2D TQFT's with commutative Frobenius algebras to TrQFTs and construct a particular TrQFT which computes multiplicities of tropical curves (and thus log Gromov-Witten invariants) from local computations akin to Mikhalkin's formula in two dimensions. For genus zero curves, this yields a splitting theorem as well as a practical expression of multiplicities in terms of iterated Schouten-Nijenhuis brackets (and products and higher brackets) of polyvector fields on the dual/mirror algebraic torus. Contents 1. Introduction 1 2. Review of tropical curves and their multiplicities 7 3. Tropical quantum field theory 12 4. A splitting formula for genus 0 20 5. Multiplicities from brackets of polyvector fields 23 References 32

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Section: 3mentioning

confidence: 99%

“…With this convention, the notions of type and degree are easily extended to curves Γ with no vertices, but to simplify the exposition, we assume for the rest of this section that Γ [0] = ∅. See [MR,Rmk. 4.17] for some details on this case.…”

Section: Review Of Tropical Curves and Their Multiplicitiesmentioning

confidence: 99%

Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves

Mandel,

Ruddat

2019

Preprint

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“…For tropical curves in R n , n ≥ 3, the first tropical enumerative invariant was found by T. Nishinou and B. Siebert [17]. It then was largely extended by T. Mandel and H. Ruddat [12] by allowing ψ-constraints, complicated boundary conditions, and curves of positive genera. The weights of tropical curves in these enumerative problems involve factors depending on the entire curves, not only on their vertices, edges or other small fragments.…”

Section: Introductionmentioning

confidence: 99%

“…The author thanks the MPIM for support and excellent working conditions. Special thanks are due to T. Mandel and H. Ruddat, who attracted my attention to their works [12,13] and explained me important details of their approach both to numerical and refined tropical enumerative invariants. I am also grateful to T. Blomme, M. Polyak, and the unknown referee for valuable remarks which helped me to correct mistakes and improve the presentation.…”

Section: Introductionmentioning

confidence: 99%

On refined count of rational tropical curves

Shustin¹

2018

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We address the problem of existence of refined (i.e., depending on a formal parameter) tropical enumerative invariants, and we present two new examples of a refined count of rational marked tropical curves. One of the new invariants counts plane rational tropical curves with an unmarked vertex of arbitrary valency. It was motivated by the tropical enumeration of plane cuspidal tropical curves given by Y. Ganor and the author, which naturally led to consideration of plane tropical curves with an unmarked fourvalent vertex. Another refined invariant counts rational tropical curves of a given degree in the Euclidean space of arbitrary dimension matching specific constraints, which make the spacial refined invariant similar to known planar invariants.

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“…Corollary 3.8 and Theorem 3.9. The original motivation is that these tropical curve counts can be related to holomorphic curve counts [Mik05,NS06,Gro18,MRa], and this should lead to proofs of the expected mirror symmetry correspondences. Indeed, upcoming work of the author…”

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confidence: 99%

Scattering diagrams, theta functions, and refined tropical curve counts

Mandel

2015

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Working over various monoid-graded Lie algebras and in arbitrary dimension, we express scattering diagrams and theta functions in terms of counts of tropical curves/disks, weighted by multiplicities given in terms of iterated Lie brackets. Over the tropical vertex group, already important in the Gross-Siebert mirror symmetry program, our tropical curve counts give descendant log Gromov-Witten invariants. Upcoming work will use this to prove the Gross-Hacking-Keel Frobenius structure conjecture for cluster varieties. The non-degeneracy of the trace-pairing for this conjecture is also proven here. Working over the quantum torus algebra yields theta functions for quantum cluster varieties, and our tropical description sets up for geometric interpretations of these. As an immediate application, we prove the quantum Frobenius conjecture of [FG09]. We also prove a refined version of the [CPS] result on consistency of theta functions.

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Descendant log Gromov-Witten invariants for toric varieties and tropical curves

Cited by 14 publications

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Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves

Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves

On refined count of rational tropical curves

Scattering diagrams, theta functions, and refined tropical curve counts

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