Counting curves on Hirzebruch surfaces: tropical geometry and the Fock space

We study the stationary descendant Gromov–Witten theory of toric surfaces by combining and extending a range of techniques – tropical curves, floor diagrams and Fock spaces. A correspondence theorem is established between tropical curves and descendant invariants on toric surfaces using maximal toric degenerations. An intermediate degeneration is then shown to give rise to floor diagrams, giving a geometric interpretation of this well-known bookkeeping tool in tropical geometry. In the process, we extend floor diagram techniques to include descendants in arbitrary genus. These floor diagrams are then used to connect tropical curve counting to the algebra of operators on the bosonic Fock space, and are showno coincide with the Feynman diagrams of appropriate operators. This extends work of a number of researchers, including Block–Göttsche, Cooper–Pandharipande and Block–Gathmann–Markwig.

Newton–Okounkov Bodies of Flag Varieties and Combinatorial Mutations

A Newton–Okounkov body is a convex body constructed from a projective variety with a globally generated line bundle and with a higher rank valuation on the function field, which gives a systematic method of constructing toric degenerations of projective varieties. Its combinatorial properties heavily depend on the choice of a valuation, and it is a fundamental problem to relate Newton–Okounkov bodies associated with different kinds of valuations. In this paper, we address this problem for flag varieties using the framework of combinatorial mutations, which was introduced in the context of mirror symmetry for Fano manifolds. By applying iterated combinatorial mutations, we connect specific Newton–Okounkov bodies of flag varieties including string polytopes, Nakashima–Zelevinsky polytopes, and Feigin–Fourier–Littelmann–Vinberg polytopes.

Tropical geometry and Newton–Okounkov cones for Grassmannian of planes from compactifications

We construct a family of compactifications of the affine cone of the Grassmannian variety of $2$ -planes. We show that both the tropical variety of the Plücker ideal and familiar valuations associated to the construction of Newton–Okounkov bodies for the Grassmannian variety can be recovered from these compactifications. In this way, we unite various perspectives for constructing toric degenerations of flag varieties.

Toric degenerations of cluster varieties and cluster duality

We introduce the notion of a $Y$-pattern with coefficients and its geometric counterpart: an $\mathcal {X}$-cluster variety with coefficients. We use these constructions to build a flat degeneration of every skew-symmetrizable specially completed $\mathcal {X}$-cluster variety $\widehat {\mathcal {X} }$ to the toric variety associated to its g-fan. Moreover, we show that the fibers of this family are stratified in a natural way, with strata the specially completed $\mathcal {X}$-varieties encoded by $\operatorname {Star}(\tau )$ for each cone $\tau$ of the $\mathbf {g}$-fan. These strata degenerate to the associated toric strata of the central fiber. We further show that the family is cluster dual to $\mathcal {A}_{\mathrm {prin}}$ of Gross, Hacking, Keel and Kontsevich [Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), 497–608], and the fibers cluster dual to $\mathcal {A} _t$. Finally, we give two applications. First, we use our construction to identify the toric degeneration of Grassmannians from Rietsch and Williams [Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians, Duke Math. J. 168 (2019), 3437–3527] with the Gross–Hacking–Keel–Kontsevich degeneration in the case of $\operatorname {Gr}_2(\mathbb {C} ^{5})$. Next, we use it to link cluster duality to Batyrev–Borisov duality of Gorenstein toric Fanos in the context of mirror symmetry.