The closed topological vertex via the Cremona transform

Abstract

ABSTRACT. We compute the local Gromov-Witten invariants of the "closed vertex", that is, a configuration of three P 1 's meeting in a single triple point in a Calabi-Yau threefold. The method is to express the local invariants of the vertex in terms of ordinary Gromov-Witten invariants of a certain blowup of P 3 and then to compute those invariants via the geometry of the Cremona transformation.

Gromov–Witten theory ofP1×P1×P1<

“…Our starting point is the observation that both these varieties come equipped with an explicit isomorphism with the blowup of X Π 3 at 2 points. The equivalence of the all genus virtual dimension zero GW theories of X and X for nonexceptional (see below) classes was proved by Bryan and the first author [1,Lemma 7]; in this work we complete the square. The non-exceptional property considered here is as follows.…”

“…The Cremona symmetry on the Gromov-Witten theory of P 2 was observed and studied independently in [3,8]. The Cremona symmetry on P 3 was studied in the context of the closed topological vertex [1]. We remark that such toric symmetries seem to be relatively rare phenomena -by the results of [10], the Cremona symmetry is the only nontrivial toric symmetry that gives rise to relations between the invariants of P n for n = 2, 3.…”

The local Gromov–Witten invariants of configurations of rational curves

“…Note that Y 1 is the closed topological vertex. By the results in Faber-Pandharipande [10] and Bryan-Karp [4], N …”

“…In [5], Bryan, Katz and Leung computed local invariants of certain rational curves with nodal singularities, and in particular, contractible ADE configurations of rational curves. The local invariants of the closed topological vertex, which is a configuration of three ‫ސ‬ 1 's meeting in a single triple point, were computed by Bryan and the first author [4].…”

Lectures on the Topological Vertex

“…It contains three P 1 touching at a single point. The local Gromov-Witten theory of this geometry was studied by Bryan and Karp (2005), who called it the closed topological vertex, and also by Karp et al (2005). The vertex rules give the following expression for the total partition function: …”