A Fock space approach to Severi degrees

Cooper, Yaim; Pandharipande, Rahul

doi:10.1112/plms.12017

“…Cooper and Pandharipande pioneered a Fock space approach to the Severi degrees of P 1 × P 1 and P 2 by using degeneration techniques [17]. Block and Göttsche generalised their work to a broader class of surfaces (the h-transverse surfaces, see for instance [3] and [9]), and to refined curve counts, via quantum commutators on the Fock space side [6].…”

Section: •2 Context and Motivationmentioning

confidence: 99%

“…In this section we build on work of Cooper and Pandharipande [17] and Block and Göttsche [6] and express relative descendant Gromov-Witten invariants of Hirzebruch surfaces as matrix elements for an operator on a Fock space. The results of this section continue to hold if one replaces the stationary logarithmic descendants of the previous section.…”

Section: Floor Diagrams Via the Operator Theorymentioning

confidence: 99%

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

Cavalieri¹,

Johnson²,

Markwig³

et al. 2021

Math. Proc. Camb. Phil. Soc.

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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.

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“…Cooper and Pandharipande [17] remarked that the combinatorics of the degeneration formula in relative Gromov-Witten theory for P 1 × P 1 and P 2 can be nicely encoded into an operator formalism in Fock space. This approach has been recently generalized to Hirzebruch surfaces by Cooper [16].…”

Section: Refined Fock Spacesmentioning

confidence: 99%

“…Cooper and Pandharipande [17] remarked that the combinatorics of the degeneration formula in relative Gromov-Witten theory for P 1 × P 1 and P 2 can be nicely encoded into an operator formalism in Fock space. This approach has been recently generalized to Hirzebruch surfaces by Cooper [16]. Block and Göttsche [4] generalized this remark to h-transverse toric surfaces by recognizing that the floor diagrams were the Feynman diagrams of the operator formalism in Fock space.…”

Section: Refined Fock Spacesmentioning

confidence: 99%

Refined floor diagrams from higher genera and lambda classes

Bousseau

¹

2021

Sel. Math. New Ser.

2

0

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We show that, after the change of variables $$q=e^{iu}$$ q = e iu , refined floor diagrams for $${\mathbb {P}}^2$$ P 2 and Hirzebruch surfaces compute generating series of higher genus relative Gromov–Witten invariants with insertion of a lambda class. The proof uses an inductive application of the degeneration formula in relative Gromov–Witten theory and an explicit result in relative Gromov–Witten theory of $${\mathbb {P}}^1$$ P 1 . Combining this result with the similar looking refined tropical correspondence theorem for log Gromov–Witten invariants, we obtain a non-trivial relation between relative and log Gromov–Witten invariants for $${\mathbb {P}}^2$$ P 2 and Hirzebruch surfaces. We also prove that the Block–Göttsche invariants of $${\mathbb {F}}_0$$ F 0 and $${\mathbb {F}}_2$$ F 2 are related by the Abramovich–Bertram formula.

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“…Another treatment of effective configurations has been proposed in [CP12]. A real version of the WDVV equations for rational 4-symplectic manifolds have been proposed by Solomon [Sol].…”

Section: Introductionmentioning

confidence: 99%