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

Abstract

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.

Refined floor diagrams from higher genera and lambda classes

“…This relation could probably be obtained directly using a degeneration argument in a log context, but it is interesting that tropical geometry gives an alternative argument. In the unrefined case, similar remarks are made in [15,16].…”

“…4. The unrefined, that is, g = g ,n , version of this degeneration argument can be found for example in the proof of Theorem 4.9 of [15], or, in the Fock space language, in §2.5 of [16]. We adapt this degeneration argument to the refined, that is, g ≥ g ,n , case using the vanishing results proved in Sect.…”

Floor diagrams and enumerative invariants of line bundles over an elliptic curve

“…Both approaches got generalized to various settings with new different kinds of constraints, for instance with the insertion of ψ-constraints. See [36], [37], [38], [27] or [17] for generalization or other proofs of the correspondence theorem.…”

“…They enable concrete, and to some extent efficient, computations of tropical invariants. The use of floor diagrams has since been generalized to other situations, for instance by F. Block and L. Göttsche [4] to compute enumerative invariants of toric surfaces constructed from an h-transverse polygon, and by R. Cavalieri, P. Johnson, H. Markwig and D. Ranganathan [16] [17] to compute descendant Gromov-Witten invariants of Hirzebruch surfaces.…”

“…This approach was pioneered by Y. Cooper and R. Pandharipande in [18], where they relate Severi degrees of CP 1 × CP 1 and CP 2 to matrix elements of exponential of some operator on a Fock space. This approach was generalized in [4] to a wider family of toric surfaces, and in [17] for Hirzebruch surfaces with the insertion of descendant conditions.…”