The wall-crossing formula and Lagrangian mutations

Abstract

We prove a general form of the wall-crossing formula which relates the disk potentials of monotone Lagrangian submanifolds with their Floertheoretic behavior away from a Donaldson divisor. We define geometric operations called mutations of Lagrangian tori in del Pezzo surfaces and in toric Fano varieties of higher dimension, and study the corresponding wall-crossing formulas that compute the disk potential of a mutated torus from that of the original one.In the case of del Pezzo surfaces, this justifies the connection between Vianna's tori and the theory of mutations of Landau-Ginzburg seeds. In higher dimension, this provides new Lagrangian tori in toric Fanos corresponding to different chambers of the mirror variety, including ones which are conjecturally separated by infinitely many walls from the chamber containing the standard toric fibre.

From symplectic cohomology to Lagrangian enumerative geometry

“…-An alternative proof of a general form of the wall-crossing formula for Lagrangian mutations due to Pascaleff and the author [47]. -A connection between seemingly distant properties of a Liouville domain M , under certain additional assumptions: the existence of a Fano compactification, the existence of an exact Lagrangian torus inside, the finite-dimensionality of SH 0 (M ), and split-generation of Fuk(M ) by simply-connected Lagrangians.…”

“…It is explained in [18] how to choose Σ so that L becomes exact in the complement, in the general symplectic case. However, the degree of this Σ may in general be large; see also the discussion in [47]. Using the tools developed in the proof of Theorem 1.1, it easy to argue that the Viterbo map respects the Borman-Sheridan classes.…”

SYZ Mirror Symmetry for Hypertoric Varieties

“…In the following we construct this resolution. The construction can be justified by Lagrangian Floer theory of immersed Lagrangians which is explained in [HL18,HKL18] (see also [Sei97] and [PT17] for more Floer theoretical aspects on gluing the chambers). We will study more about Lagrangian Floer theory in future work.…”

“…Auroux [Aur07,Aur09] provided a symplectic approach to SYZ and the Gross-Siebert program. Moreover, Floer theory of wall-crossing was developed in Pascaleff-Tonkonog [PT17] based on the work of Seidel [Sei]. Furthermore, based on the works of Fukaya-Oh-Ohta-Ono [FOOO09,FOOO10], Seidel [Sei11] and Akaho-Joyce [AJ10], deformation and moduli theory of Lagrangian immersions are being developed by Cho-Hong-Lau [CHL17, CHL,HL18] which enhance and generalize the SYZ program.…”

“…We glue together a resolution using local charts coming from the wall and chamber structure of the SYZ base. The gluing can be explained using Floer-theoretical techniques as in [Sei,PT17,HL18], but we will leave this in future work. The variety admits another resolution by a multiplicative hypertoric variety (Section 4.9).…”