Scattering diagrams, theta functions, and refined tropical curve counts

Abstract

Working over various monoid-graded Lie algebras and in arbitrary dimension, we express scattering diagrams and theta functions in terms of counts of tropical curves/disks, weighted by multiplicities given in terms of iterated Lie brackets. Over the tropical vertex group, already important in the Gross-Siebert mirror symmetry program, our tropical curve counts give descendant log Gromov-Witten invariants. Upcoming work will use this to prove the Gross-Hacking-Keel Frobenius structure conjecture for cluster varieties. The non-degeneracy of the trace-pairing for this conjecture is also proven here. Working over the quantum torus algebra yields theta functions for quantum cluster varieties, and our tropical description sets up for geometric interpretations of these. As an immediate application, we prove the quantum Frobenius conjecture of [FG09]. We also prove a refined version of the [CPS] result on consistency of theta functions.

Refined count of real oriented rational curves

“…Concretely, it means that the multiplicity is not a product of vertex multiplicities anymore. However, for a specific choice of enumerative problem involving some choice of 2-form ω, we recover such a simple recipe providing real tropical invariants [11]. This suggests the existence of a corresponding real refined count which is precisely the content of this paper: we enlarge the definition of the quantum index in higher dimension, and we provide a corresponding real refined count.…”

“…In the case S = 0, to get Laurent polynomial in one variable, one can evaluate a 2-form on the exponents, for instance ω. However, it is worth noticing that in this case we have a coarser invariant since the 2-form that we evaluate on the exponents needs not to be ω, as it is yet the case in [11].…”

“…This allows us to relate R ∆,ω to some already known tropical invariants considered in [4]. We briefly recall their definition and refer to [4] or [11] for a more complete description.…”

“…The resulting Laurent polynomial does in fact not depend on the choice of (P, µ) as long as it is generic. See [4] or [11] for a proof.…”

“…Filippini and J. Stoppa [8]. See also the work of T. Mandel [11] and [12] for more details on the use of refined tropical invariants, and higher dimensional scattering diagrams.…”