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A descendent tropical Landau-Ginzburg potential for \(\mathbb{P}^2\). (English) Zbl 1415.14021
The role of Mirror Symmetry in enumerative geometry was a triumph of string theory dating to 1991 [P. Candelas et al., Nucl. Phys., B 359, No. 1, 21–74 (1991; Zbl 1098.32506)], leading to the theory of Gromow-Witten [GW] invariants. This has been formalized over the years as mathematical conjectures, notably Kontsevich’s homological mirror symmetry and the torus-fibration structure of SYZ [A. Strominger et al., ibid. 479, No. 1–2, 243–259 (1996; Zbl 0896.14024)]. In particular, SYZ suggest the importance of a middle-dimensional real manifold described by tropical geometry. The Gross-Siebert programme followed this idea very successfully and a major step was by Mikhalkin who computed general GW invariants on \(\mathbb{P}^2\) to that of the associated tropical curves.
In general, by the works of A. Givental [Prog. Math. 160, 141–175 (1998; Zbl 0936.14031); in: Proceedings of the international congress of mathematicians, ICM 1994. Basel: Birkhäuser. 472–480 (1995; Zbl 0863.14021)], the mirror of a toric Fano manifold \(X\) is a Landau-Ginzburg model \(\hat{X}\) consisting of a manifold as well as a potential \(W\) which is map \(W : \hat{X} \to \mathbb{C}\). For example, the mirror to \(\mathbb{P}^2\) is given by the hypersurface \(x_0x_1x_2=1\) in \(\mathbb{C}[x_0,x_1,x_2]\) together with \(W = x_0 + x_1 + x_2\).
The purpose of the current paper is to nicely follow the direction of Gross and establish a family of Landau-Ginzburg potentials for \(\mathbb{P}^2\). Oscillatory integrals of this family compute an enhancement of Givental’s J-function, encoding many descendent Gromov-Witten invariants. This construction can be seen as yielding a canonical family of Landau-Ginzburg potentials on a refinement of a sector of the big phase space, and the resulting descendent J-function is the natural lift given by the constitutive equations of R. Dijkgraaf and E. Witten [“Mean field theory, topological field theory, and multi-matrix models”, Nucl. Phys. B 342, No. 3, 486–522 (1990; doi:10.1016/0550-3213(90)90324-7)].

14T05 Tropical geometry (MSC2010)
14J33 Mirror symmetry (algebro-geometric aspects)
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
35Q56 Ginzburg-Landau equations
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