Tropical Refined Invariants for Toric Surfaces

• The paper introduces tropical refined invariants that assign q-refined multiplicities to tropical vertices for systematic curve counting on toric surfaces.
• It incorporates descendant information and logarithmic Gromov-Witten theory, establishing a correspondence with classical enumerative invariants.
• Combinatorial models like floor diagrams and recursive lattice-path algorithms enable precise computations and reveal polynomiality and deformation invariance.

Tropical refined invariants for toric surfaces provide a framework for enumerative curve counting in algebraic geometry, encoding rich structure that interpolates between classical Gromov-Witten invariants and real enumerative invariants via tropical geometry. Building upon foundational work by Block–Göttsche, further developments have incorporated descendants, higher-genus, and real refinements, yielding relations to logarithmic Gromov-Witten theory, double ramification cycles, and integrable hierarchies. These invariants exhibit deep combinatorial, algebro-geometric, and topological properties with profound implications for understanding curve moduli on toric surfaces.

1. Refined Tropical Invariants: Definitions and Multiplicities

Given a convex lattice polygon $\Delta \subset \mathbb{R}^2$ defining a toric surface $X_\Delta$, the enumeration of curves of degree $\Delta$ is enriched by the introduction of refined multiplicities. For tropical curves—realizations of algebraic curves in the piecewise-linear (tropical) world—each trivalent vertex $V$ with outgoing primitive edge directions $u, v, w$ (balanced, i.e., $u+v+w=0$) is assigned the refined Block–Göttsche vertex multiplicity: $$[\mu(V)]_q = \frac{q^{\mu(V)/2}-q^{-\mu(V)/2}}{q^{1/2}-q^{-1/2}}, \quad \mu(V) = |\det(u, v)|,$$ where $q \in \mathbb{C}^*$ is a formal parameter. The refined multiplicity of a tropical curve is the product of the multiplicities of its vertices: $m_{\text{BG}}(\Gamma;q) = \prod_{V} [\mu(V)]_q.$ For higher-valency vertices arising in refined descendant settings, the multiplicity is extended via cyclic symmetrization or recursive relations, typically involving sums over cyclic orderings and related to symmetric Laurent polynomials in $q$ (Kennedy-Hunt et al., 2023, Blechman et al., 2016).

2. Descendant and Lambda-Refined Correspondence

The construction of refined invariants further incorporates descendant information, i.e., insertion of $\psi$-classes and Hodge ($\lambda_g$) classes in Gromov-Witten theory. For a degree $\Delta^\circ$ and nonnegative integers $k_1,\ldots,k_n$ prescribing descendant valencies, define the logarithmic Gromov-Witten descendant invariant: $N^k_{g,\Delta} = \int_{[\Mbar_{g,\Delta}]^{\mathrm{vir}}} (-1)^g \lambda_g \prod_{i=1}^n \psi_i^{k_i} \ev_i^*(\mathrm{pt}),$ where $\Mbar_{g,\Delta}$ is the moduli of genus-$g$ logarithmic stable maps to $X_\Delta$ with toric boundary conditions.

A key result is the $q$-refined tropical correspondence theorem, equating the generating series of such logarithmic Gromov-Witten invariants with a $q$-refined count of genus-zero tropical maps with prescribed higher-valency vertices, using refined tropical multiplicities: $$\sum_{g\ge0} N_{g,\Delta}^k \, u^{2g-2+|\Delta^\circ|-\sum k_i} = \sum_{h \in T_{\Delta,p}^k} \prod_{V\in V(\Gamma)} m_V(q), \qquad q = e^{iu},$$ where $T_{\Delta,p}^k$ is the set of rigid genus-zero tropical maps passing through a generic point configuration (Kennedy-Hunt et al., 2023).

3. Proof Scheme: Logarithmic Degeneration and Double Ramification Cycles

The proof utilizes a sequence of algebraic and tropical reductions:

1. Toric Degeneration and Logarithmic Degeneration Formula: The toric surface is degenerated into a union of toric components governed by a polyhedral decomposition matching the tropical combinatorics, applying the Abramovich–Chen–Gross–Siebert log degeneration formula for the decomposition of Gromov–Witten invariants.
2. Logarithmic Gluing: The logarithmic gluing theorem (Ranganathan–Ramos–Wise) enables the reduction to vertex contributions—simpler log-GW invariants on toric pieces—multiplied by additional gluing weights.
3. Reduction to Double Ramification Integrals: At each vertex, the intersection numbers reduce to integrals involving $\lambda_g$ classes and double-ramification cycles:

$N_{g_V,V} = \int_{[\Mbar_{g_V,\Delta_V}]^\vir} (-1)^{g_V}\lambda_{g_V} \ev^*(\cdots)\psi^d \cdot \DR_{g_V}(\Delta_V^x) \cdot \DR_{g_V}(\Delta_V^y)$

Here, the double ramification cycles $\DR_{g}(-)$ encode relative conditions at the two vectors of the local contact matrix.

1. Evaluation via DR/KdV Hierarchy: Generating series of these double-ramification integrals are matched with explicit expressions in the non-commutative KdV hierarchy, with Buryak–Rossi showing that they yield the quantum multiplicity assignments required for the tropical side (Kennedy-Hunt et al., 2023).

4. Combinatorial Models and Algorithmic Computation

The combinatorial approach to refined invariants involves:

• Floor Diagrams and Templates: Tropical curves are encoded by floor diagrams with edge weights (quantum numbers) and templates specifying the building blocks. The refined multiplicity of a floor diagram in genus $g$ is

$\mu(D;q) = \prod_{e \in E(D)} [\omega(e)]_q^2,$

summed over all isomorphism classes of marked diagrams (Brugallé et al., 2020, Block et al., 2014). For descendants, the rules are altered to encode $\psi$-classes through constrained pairings or valency patterns, with matching refined multiplicities.

• Recursive and Lattice-Path Algorithms: Caporaso–Harris-type recursions allow computation by decomposing the enumeration problem into subproblems matched by combinatorial rules for gluing and edge contributions. Lattice-path algorithms generalize this to the refinement, using admissible subdivisions of the Newton polygon and recursive computation of tile weights (Blechman et al., 2016).
• Algorithmic Invariance: These methods yield explicit polynomials in $q$ (Laurent polynomials), whose invariance under deformations of the constraints is established both combinatorially and via degeneration arguments.

5. Structural Properties and Polynomiality

Tropical refined invariants display several critical algebraic and enumerative properties:

• Polynomiality: For fixed genus and codegree, the coefficients of the refined invariants are polynomials in the Newton polygon parameters (e.g., $a,b,n$ for certain toric families) and in the descendant parameter $s$. This generalizes classical node-polynomial results and underlies the universality phenomena analogous to Göttsche's conjecture (Brugallé et al., 2020, Block et al., 2014).
• Specializations and Interpolations: Setting the refinement parameter to $q=1$ (unrefined) retrieves the classical complex curve counts, while $q=-1$ recovers real (Welschinger) enumerative invariants. For descendants, this interpolation continues to hold, matching the combinatorics and geometry (Blechman et al., 2016).
• Deformation Invariance: The invariants, both algebraic and tropical, are independent of the generic choices of point constraints. This is ensured by geometric arguments using degeneration formulas and combinatorial wall-crossing invariance in the moduli spaces.
• Relation to Motivic and $\chi_y$-Refinement: The polynomial structure suggests, though does not establish, a motivic or cohomological meaning—possibly via a yet-to-be-constructed $\chi_y$-refinement or algebro-geometric realization, extending the connection with relative Hilbert schemes (Brugallé et al., 2020).

6. Relation to Real and Positive Genus Refined Invariants

Refined invariants extend beyond complex enumerative questions to the real case and higher genus:

• Quantum Index Refinement: Mikhalkin's theory of quantum index attaches additional grading to real rational curves in toric surfaces, with corresponding tropical refinements matching Block–Göttsche multiplicities under tropicalization (Blomme, 2019, Blomme, 2020, Itenberg et al., 2024).
• Positive Genus and Collinear Cycle Corrections: For genus one and higher, additional correction factors corresponding to collinear cycles and orientation kits (for elliptic and higher genus floor diagrams) arise, requiring new combinatorial multiplicities that account for cycles and their orientation gluing (Schroeter et al., 2016, Shustin et al., 2024). These corrections reflect the topological and dual graph complexities of higher-genus tropical curves and are required for the invariants to maintain deformation invariance.
• Limitations: For genus $g \geq 2$ and multiple contact constraints, any purely local vertex-and-cycle rule fails to produce an invariant under all deformations—explicitly shown by degenerations in the moduli space that cannot be reconciled by any such assignment (Shustin et al., 2024).

7. Open Conjectures and Future Directions

Current understanding leaves several fundamental questions open:

• Universal Polynomiality and Geometric Realization: Seeking a universal polynomial governing refined invariants for arbitrary smooth toric surfaces and line bundles remains an open conjecture, paralleling Göttsche's node polynomials. An algebro-geometric model for the refined descendant invariants—including a motivic or $\chi_y$ interpretation—remains elusive (Brugallé et al., 2020).
• Refined Surgery and Recursion Relations: Analogues of real enumerative surgery formulas for the refined case are only partially established, hinting at possible extensions via tropical methods but lacking a full geometric proof or higher genus generalization.
• Stability, Asymptotics, and Combinatorial Positivity: The positivity and log-concavity of refined coefficients, combinatorial stabilization for fixed genus, and finer structure as both genus and codegree grow are observed but not fully explained or generalized.
• Extension to Broader Settings: Efforts are ongoing to extend constructions to arbitrary genus and constraint types—though fundamental obstructions arise for higher genus and multiple descendant (contact) insertions due to nonlocal wall-crossing behaviors (Shustin et al., 2024).

These topics continue to motivate research into the intricate connections between tropical geometry, algebraic curve counting, real enumerative invariants, and the interplay with moduli theory and integrable systems.