Strong Positivity for the Skein Algebras of the 4-Punctured Sphere and of the 1-Punctured Torus

Stable maps to Looijenga pairs

“…Now, by the q ! 1 limit of (1-7), the genus-zero LMOV and GW invariants of .D/ to be three-dimensional affine space with a single toric Lagrangian at framing one (see Construction 6.4) -and as expected, in this case the quiver construction in [102, Section 5.1] returns exactly the quiver with one vertex and two loops we had found in (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16). In general, this connection leads to some nontrivial implications for the Gopakumar-Vafa/Donaldson-Thomas theory of CY4 local surfaces from log Gromov-Witten theory, which we describe precisely in Sections 1.6.1 and 1.6.3.…”

“…We state our results in a slightly discursive form below, including pointers to their precise versions in the main body of the text. Moreover, we provide a closed-form solution to the calculation of both sets of invariants in (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17).…”

“…The statement of the theorem for tame cases follows by subsequently comparing with a closed-form solution of the local theory via Givental-style mirror theorems: the proof follows from a general statement valid for local invariants of toric Fano varieties in any dimension twisted by a sum of concave line bundles (Lemma 3.1), and the notion of tameness is shown to coincide here with the vanishing of quantum corrections to the mirror map. For non-quasi-tame cases, we use a blowup formula which allows to restrict to the case of highest Picard number; the proof of the equality (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) in this case, in Theorem 3.3, requires a highly nontrivial mirror map calculation.…”

“…Moreover, we provide a closed-form solution to the calculation of the invariants in (1-18)- (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19).…”

“…The discussion of Section 1.4.2 also opens the door to pushing the log/open correspondence beyond the maximal contact setting: it is tempting to see how the maximal tangency condition could be removed from Conjecture 1.2, with the splitting of contact orders amongst multiple points on the same divisor being translated to windings of multiple boundary disks ending on the same Lagrangian. The multicovering factor of (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) would then be naturally given by a product of individual contact orders/disk windingsan expectation that the reader can verify to be fulfilled in the basic example presented there of .Y; D/ D .P 1 A 1 ; A 1 [ A 1 /. More generally, the link to the topological vertex and open GW invariants of arbitrary topology raises a fascinating question how much the topological vertex knows of the log theory of the surface -and how it can be effectively used in the construction of (quantum) SYZ mirrors.…”