Communications in Number Theory and Physics

Volume 10 (2016)

Number 4

A descendent tropical Landau–Ginzburg potential for $\mathbb{P}^2$

Pages: 739 – 803



Peter Overholser (Institut für Mathematik, Johannes Gutenburg-Universität, Mainz, Germany)


Following work of Gross, a family of Landau–Ginzburg potentials for $\mathbb{P}^2$ is defined using counts of tropical objects analogous to holomorphic disks with descendants. 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 Dijkgraaf and Witten to this setting.


tropical geometry, mirror symmetry, Gromov–Witten

2010 Mathematics Subject Classification

14J33, 14N10, 14N35, 14T05

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