Bulk-deformed potentials for toric Fano surfaces, wall-crossing and period

Abstract

We provide an inductive algorithm to compute the bulk-deformed potentials for toric Fano surfaces via wall-crossing techniques and a tropicalholomorphic correspondence theorem for holomorphic discs. As an application of the correspondence theorem, we also prove a big quantum period theorem for toric Fano surfaces which relates the log descendant Gromov-Witten invariants with the oscillatory integrals of the bulk-deformed potentials.

Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces

“…This is of course consistent with the generalized Maslov index of a holomorphic disc defined in [33] (see Section 2.5.1, also). Note that if there is no constraint, then M I agrees with the Maslov index of the tropical disc above.…”

“…• 1 L and the resulting W b is called the bulk-deformed potential. In [33], the second and third authors considered a special type of a bulk-deformation of the Floer theory of toric fibres in a Fano surface, for which b is taken to be a linear combination t i q i of generic points with t 2 i = 0. One needs to impose l-many point-constraints to discs of Maslov index µ = 2l + 2 in order for them to contribute to W b .…”

“…One needs to impose l-many point-constraints to discs of Maslov index µ = 2l + 2 in order for them to contribute to W b . The dimension formula of the disc moduli (see Section 2.5.1) tells us that these constrained discs behave like Maslov 2 discs, and more generally, the generalized Maslov index µ of a holomorphic disc was introduced in [33] as twice the number of point-constraints subtracted from the ordinary Maslov index µ (i.e., µ = µ − 2l). Notice that (2.1) is precisely its tropical analogue.…”

“…The unique rational curve in Lemma 3.1 corresponding to relative γ is different from those A 1 -curves contributing to N γ even topologically. We refer readers to Example 5.3 [33].…”

Reconstruction of $T_{\mathbb{P}^2}$ via tropical Lagrangian multi-section

“…Hence Equation ( 3) can be understood as the wall-crossing diagram as shown in Figure 2. Furthermore, in view of [29], the walls of the mirror Lagrangian should concentrate on a small neighborhood of 2 k=0 R ≥0 n k as → 0. The closed string wall-crossing phenomenon has been studied in [32, 24,12].…”