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).

arXiv commentIntro 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 (28)

• String Topology
• Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields
• Enumerative tropical algebraic geometry in R2
• Two-dimensional topological gravity and equivariant cohomology
• The tropical vertex
• First Steps in Tropical Intersection Theory
• Refined curve counting with tropical geometry
• Symplectic cohomology and Viterbo's theorem
• Skeletons of stable maps I: Rational curves in toric varieties
• Smoothing toroidal crossing spaces
• On the symplectic cohomology of log Calabi-Yau surfaces
• Tropical descendant Gromov-Witten invariants
• Lagrangian pairs of pants
• Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties
• Intersection Theory on Tropicalizations of Toroidal Embeddings
• Tropical Lagrangian Hypersurfaces are Unobstructed
• An $L_{\infty }$ algebra structure on polyvector fields
• Mirror symmetry for P^2 and tropical geometry
• String topology with gravitational descendants, and periods of Landau-Ginzburg potentials
• A Log PSS morphism with applications to Lagrangian embeddings
• Theta bases and log Gromov-Witten invariants of cluster varieties
• Refined descendant invariants of toric surfaces
• Intersections on tropical moduli spaces
• A tropical view on Landau-Ginzburg models
• Scattering diagrams from asymptotic analysis on Maurer-Cartan equations
• Lagrangian submanifolds from tropical hypersurfaces
• Tropical Lagrangians in toric del-Pezzo surfaces
• On refined count of rational tropical curves

Cited by in corpus (11)

• Local Gromov-Witten Invariants are Log Invariants
• Stable maps to Looijenga pairs
• Tropically constructed Lagrangians in mirror quintic threefolds
• On the log-local principle for the toric boundary
• Scattering diagrams, theta functions, and refined tropical curve counts
• 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