The tropical critical point and mirror symmetry

TL;DR

This paper proves the existence and uniqueness of a positive critical point for complete Laurent polynomials over generalized Puiseux series, introduces a tropical critical point, and applies these results to cluster varieties and toric geometry.

Contribution

It establishes a canonical positive critical point for complete Laurent polynomials over Puiseux series and links it to tropical geometry and cluster mutations.

Findings

Unique positive critical point exists for complete Laurent polynomials.

Introduces a canonical tropical critical point via valuation.

Applications to non-displaceable Lagrangian tori in toric symplectic manifolds.

Abstract

Call a Laurent polynomial $W$ `complete' if its Newton polytope is full-dimensional with zero in its interior. We show that if $W$ is any complete Laurent polynomial with coefficients in the positive part of the field $K$ of generalised Puiseux series, then $W$ has a unique positive critical point $p_{crit}$. Here a generalised Puiseux series is called `positive' if the coefficient of its leading term is in $\mathbb R_{>0}$. Using the valuation on $K$ we obtain a canonically associated `tropical critical point' $d_{crit}$ in $\mathbb R^{r}$ for which we give a finite recursive construction. We show that this result is compatible with a general form of mutation, so that it can be applied in a cluster varieties setting. We also give applications to toric geometry including, via the theory of [FOOO], to the construction of canonical non-displaceable Lagrangian tori for toric symplectic manifolds.