Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves

TL;DR

This paper develops algebraic structures on polyvector fields of algebraic tori to compute tropical curve multiplicities and explores their connections to mirror symmetry, string topology, and log Gromov-Witten theory.

Contribution

It introduces a tropical quantum field theory and an $L_{}$-structure on polyvector fields, linking tropical geometry with mirror symmetry and string topology.

Findings

Defined algebraic structures for tropical curve multiplicities

Connected tropical quantum field theory to mirror symmetry

Related structures to string topology and gravity algebra

Abstract

We introduce algebraic structures on the polyvector fields of an algebraic torus that serve to compute multiplicities in tropical and log Gromov-Witten theory while also connecting to the mirror symmetry dual deformation theory of complex structures. Most notably these structures include a tropical quantum field theory and an $L_{\infty}$-structure. The latter is an instance of Getzler's gravity algebra, and the $l_2$-bracket is a restriction of the Schouten-Nijenhuis bracket. We explain the relationship to string topology in the appendix (thanks to Janko Latschev).