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).

On the log–local principle for the toric boundary

“…We thank Helge Ruddat for getting this project started by asking us to compute some local Gromov-Witten invariants of ℙ 4 and for helpful discussions all along. We thank Travis Mandel for detailed explanations on [23,24]. We are grateful to Miles Reid and Andrea Petracci for some discussions on the classification of toric varieties with numerically effective divisors.…”

“…There are a number of ways of defining the multiplicity Mult(Γ) of Γ. The version we use was formulated (for 𝑋 smooth) in [23,Theorem 1.2]. We state it for our setting.…”

“…For the other cases, our proof proceeds by calculating both sides to obtain explicit closed formulae for these invariants for all (𝑋, 𝐷) (Theorems 3.2 and 3.3). To compute the log invariants we use the tropical correspondence result [24] and an algorithm of [23] for the tropical multiplicity. The log-local principle of Conjecture 1 then predicts an explicit formula in all degrees for the local invariants, which we verify using local mirror symmetry techniques and a reconstruction result from small to big quantum cohomology.…”

The degeneration formula for stable log maps

“…We give detailed arguments for the comparison results of virtual classes by proving the commutativity of the relevant maps of triangles, see (9.10), (9.12). Novel is the elaboration of the tropical point of view for the degeneration formula inspired by [26,27,31]. The tropical point of view in log Gromov-Witten theory was first established in [14].…”

Log Smooth Deformation Theory via Gerstenhaber Algebras

“…There has been recent interest in Gerstenhaber algebras of polyvector fields also in tropical geometry. In [23] integral polyvector fields on algebraic tori are studied in order to compute Gromov-Witten invariants via tropical curves. Moreover, it turns out that wall-crossing transformations, i.e.…”