From symplectic cohomology to Lagrangian enumerative geometry

Abstract

We build a bridge between Floer theory on open symplectic manifolds and the enumerative geometry of holomorphic disks inside their Fano compactifications, by detecting elements in symplectic cohomology which are mirror to Landau-Ginzburg potentials. We also treat the higher Maslov index versions of the potentials.We discover a relation between higher disk potentials and symplectic cohomology rings of smooth anticanonical divisor complements (themselves conjecturally related to closed-string Gromov-Witten invariants), and explore several other applications to the geometry of Liouville domains. arXiv:1711.03292v3 [math.SG] 27 Jun 2019As a general prediction, rings of regular functions are mirror to degree zero symplectic cohomology. (This was proven by Ganatra and Pomerleano [35] for the complement to the anticanonical divisor itself, see Section 2.) So one expects:These rings can be complicated (much bigger than C[r]), but they carry a distinguished element, the restriction of W :It is natural to ask what is the symplectic counterpart, or the mirror, of this element. This is answered by Theorem 1.1, which can be summarised in the language of mirror symmetry as follows:The Borman-Sheridan class BS ∈ SH 0 (M ) is mirror to W |M .

The wall-crossing formula and Lagrangian mutations

“…The key tools in the proof are the relative Floer theory of the pair (X, D) which relates Floer theory in X to Floer theory in X \ D, described in section 2, and the locality properties of the Floer cohomology of compact Lagrangians, described in section 3. An alternative approach to this result that uses symplectic cohomology of X \ D and a stretching argument is presented in the paper [34] by the second author.…”

Liouville domains from Okounkov bodies

“…In Homological Mirror Symmetry (HMS), the pair pX, Dq is expected to have a partner Landau-Ginzburg model pX _ , W q consisting of a complex variety with a regular function W P OpX _ q called potential. The work of Tonkonog [31] suggests that Lagrangians L Ă X that are monotone with respect to ω D and become exact in a Liouville subdomain of X should correspond to subschemes U Ă X _ , with W |U being a generating function of rigid pseudo-holomorphic curves. These curves are half-cylinders that have boundary on L, and are obtained by neckstretching from global pseudo-holomorphic disks.…”

Fukaya 𝐴_{∞}-structures associated to Lefschetz fibrations. IV

“…The behaviour of pseudo-holomorphic curves in such a situation is a well-explored topic, see e.g. [9,7,38]; our approach borrows from [33, Section 7] as well. The outcome of our discussion will be another noncommutative divisor, which encodes pseudo-holomorphic maps that intersect the hypersurface.…”

“…One can think of the relation between this and the previous object as analogous to that between two distinguished elements in the symplectic cohomology of the hypersurface complement, namely the unit and the Borman-Sheridan class (for the latter, see e.g. [38]). From that point of view, it is not surprising that the combination of the two natural transformations recovers the disc-counting superpotential [8,3].…”