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

Published 3 April 2017