Browsing by Author "Ranganathan, Dhruv"

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    Counting curves on toric surfaces tropical geometry & the Fock space
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Cavalieri, Renzo; Johnson, Paul D.; Markwig, Hannah; Ranganathan, Dhruv
    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 shown to coincide with the Feynman diagrams of appropriate operators. This extends work of a number of researchers, including Block–Göttche, Cooper–Pandharipande, and Block–Gathmann–Markwig.
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    A graphical interface for the Gromov-Witten theory of curves
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2016) Cavalieri, Renzo; Johnson, Paul; Markwig, Hannah; Ranganathan, Dhruv
    We explore the explicit relationship between the descendant Gromov–Witten theory of target curves, operators on Fock spaces, and tropical curve counting. We prove a classical/tropical correspondence theorem for descendant invariants and give an algorithm that establishes a tropical Gromov–Witten/Hurwitz equivalence. Tropical curve counting is related to an algebra of operators on the Fock space by means of bosonification. In this manner, tropical geometry provides a convenient “graphical user interface” for Okounkov and Pandharipande’s celebrated GW/H correspondence. An important goal of this paper is to spell out the connections between these various perspectives for target dimension 1, as a first step in studying the analogous relationship between logarithmic descendant theory, tropical curve counting, and Fock space formalisms in higher dimensions.