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

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).

Intro rewritten; improved definition of Tropical QFT; corrected algebraic characterization of TrQFT (Theorem 3.6); added appendix on relation to string topology; various other improvements; 37 pages; comments welcome!

References in corpus (9)

• Enumerative tropical algebraic geometry in R2
• Smoothing toroidal crossing spaces
• Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties
• Tropical Lagrangian Hypersurfaces are Unobstructed
• Mirror symmetry for P^2 and tropical geometry
• Theta bases and log Gromov-Witten invariants of cluster varieties
• A tropical view on Landau-Ginzburg models
• Tropical Lagrangians in toric del-Pezzo surfaces
• On refined count of rational tropical curves

Cited by in corpus (9)

• Stable maps to Looijenga pairs
• Tropically constructed Lagrangians in mirror quintic threefolds
• On the log-local principle for the toric boundary
• Sheaves of maximal intersection and multiplicities of stable log maps
• On refined count of rational tropical curves
• Refined count of real oriented rational curves
• BV operators of the Gerstenhaber algebras of holomorphic polyvector fields on toric varieties
• Tropical Mirror
• Tropicalized quantum field theory and global tropical sampling