The tropical vertex

By Mark Gross, R. Pandharipande, Bernd S Siebert. 2009

189 citations recorded by the source corpus.

Elements of the tropical vertex group are formal families of symplectomorphisms of the 2-dimensional algebraic torus. We prove ordered product factorizations in the tropical vertex group are equivalent to calculations of certain genus 0 relative Gromov-Witten invariants of toric surfaces. The relative invariants which arise have full tangency to a toric divisor at a single unspecified point. The method uses scattering diagrams, tropical curve counts, degeneration formulas, and exact multiple cover calculations in orbifold Gromov-Witten theory.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d33ed978b686811b30772671cf77e5283d557954

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Citations in the corpus

Quivers, curves, and the tropical vertex (2009, methodology): “t nontrivial cases to consider are the commutators of the elements Sℓ1 = θ (1 ,0) (1+tx)ℓ1 and Tℓ2 = θ (0 ,1) (1+ty)ℓ2 where ℓ 1,ℓ 2 > 0. By an elementary result of [10] reviewed in Section 1.3 of [6], there exists a unique factorization (1.2) T−1 ℓ2 ◦Sℓ1 ◦Tℓ2 ◦S −1 ℓ1 = Y→ θ (a,b),f a,b where th”
Theta functions on varieties with effective anti-canonical class (2016, methodology): “hy the wall structures provides a well-deﬁned deformation relies on the reduction to this situation via the enumerative interpretation of the inductive process of wall insertion for toric surfaces in [GPS]. A combination of the two methods will be used in [GHKS] in the general K3 case. 4 MARKGROSS,P”
Descendant log Gromov-Witten invariants for toric varieties and tropical curves (2016, background): “1,...,fuk. The coordinates zvj on TV(Σ) can be viewede as homogeneous coordinates on TV(Σ). In terms of these coordinates, any ϕ∈ Mtt 0,1,s(Y †,∆) can (similarly to in the proof of Proposition 3.2 in [GPS10]) be expressed as zvj → ϕ j := cj(X+ajY)wj if j= 1,...,s cj if j= s+1,...,k, where Xand Y ”
Mirror symmetry for log Calabi-Yau surfaces I (2011, methodology): “ by imposing the kβ-fold tangency condition. The moduli space carries a virtual fundamental class. Furthermore, we have Lemma 3.2. M(Y˜o/Co,β) is proper over k. Proof. This follows as in the proof of [GPS09], Theorem 4.2. In brief, let Rbe a valuation ring with quotient ﬁeld K, with S= SpecR, T = Sp”
Logarithmic Gromov–Witten theory with expansions (2019, background): “This is the situation in each of the papers [14, 17, 29, 47, 55].”
Canonical bases for cluster algebras (2014, methodology): “ be obtained from D′by taking the asymptotic scattering diagram of D′, i.e., just translate each line of D′so it passes through the origin and each ray of D′so its endpoint is the origin. See §1.4 of [GPS] for more details. If after performing this translation, we obtain a number of rays with the sa”
Poisson automorphisms and quiver moduli (2008, methodology): “1 to a factorization formula of [3] in terms of Gromov-Witten theory.”
Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants (2010, background): “elements Yy (a,b)∈Z2 >0−{0} Y |k|6const(a,b) (−1)abqk/2xayb;q c(a,b;k) ∞ with c(a,b;k) ∈Z, is closed under the product. The corresponding group can be called the quantum tropical vertex group (cf. [25]) since for a quantum admissible series Fthe automorphism Ad(F) in the limit q1/2 →1 gives rise ”