Tropical Disk Potentials

• Tropical disk potentials are combinatorial invariants that count holomorphic disks bounded by Lagrangians, using tropical graphs to encode mirror symmetry data.
• They are computed as sums over rigid tropical graphs with weights from local multiplicities, automorphism groups, and boundary homology classes.
• Their formulation bridges symplectic topology, tropical enumerative geometry, and mirror symmetry, enabling explicit calculations in toric, del Pezzo, and flag varieties.

A tropical disk potential is a combinatorial invariant that encodes the count of holomorphic disks with boundary on a Lagrangian submanifold, captured via tropical geometry. These potentials play a central role in the tropical approach to mirror symmetry, particularly for toric and almost-toric varieties, and are closely related to the Landau–Ginzburg superpotential in the mirror correspondence. The tropical disk potential is computed as a sum over rigid tropical graphs, with weights reflecting local multiplicities, automorphism symmetries, and the homology class of the boundary. This construction bridges symplectic topology, tropical geometry, and enumerative mirror symmetry, underpinning explicit calculations in del Pezzo surfaces, flag varieties, and Gromov–Witten theory.

1. Foundational Definitions and Setup

Let $X$ be a compact symplectic manifold equipped with a Lagrangian torus fibration $\Phi: X \to B$, where $B$ admits an integral affine structure outside a finite set of singularities, such as focus–focus points in almost-toric four-manifolds. A Lagrangian submanifold $L \subset X$ is assumed monotone, so $\omega(A) = \kappa \mu(A)$ for all $A\in\pi_2(X,L)$, with constant $\kappa > 0$ and Maslov index $\mu$.

The disk (Landau–Ginzburg) potential $W_L$ associates to $L$ a Laurent polynomial function on rank-1 local systems: $\Phi: X \to B$0 where the sum extends over rigid Maslov-2 disks $\Phi: X \to B$1 and the weights are determined by the holonomy of the local system along the boundary $\Phi: X \to B$2.

2. Tropical Degeneration and Correspondence Principle

A rational polyhedral decomposition $\Phi: X \to B$3 of the base $\Phi: X \to B$4 is chosen, resulting in a collection of cut spaces $\Phi: X \to B$5, each with normal-crossing relative divisors. The manifold $\Phi: X \to B$6 is degenerated through neck-stretching along these hypersurfaces, and in the limit ($\Phi: X \to B$7) holomorphic disks break up into configurations mapping into the $\Phi: X \to B$8, glued along matching conditions at the nodes.

Each such broken disk configuration corresponds bijectively to a tropical graph $\Phi: X \to B$9 in the dual polyhedral complex $B$0:

• Vertices $B$1 represent disk or sphere components mapped to $B$2.
• Edges $B$3 encode neck regions, weighted with integer direction vectors $B$4.
• A tropical embedding $B$5 is affine on each edge and obeys a balancing condition at sphere vertices: $B$6.

Rigidity is defined: Only graphs for which the space of vertex positions is zero-dimensional (finite automorphism group $B$7) contribute nontrivially. The space of broken disks of type $B$8 is finite-dimensional if and only if $B$9 is rigid.

3. Tropical Disk Potential: Definition and Formula

The tropical disk potential for a monotone Lagrangian $L \subset X$0 is computed as: $L \subset X$1 where:

• The sum runs over isomorphism classes of rigid tropical graphs $L \subset X$2.
• $L \subset X$3 is the local multiplicity of a vertex $L \subset X$4, determined by its local type (see below).
• $L \subset X$5 is the order of the automorphism group of $L \subset X$6.
• $L \subset X$7 is the boundary class in $L \subset X$8 associated to the tropical disk.
• $L \subset X$9 is the associated monomial in the coordinates $\omega(A) = \kappa \mu(A)$0 on $\omega(A) = \kappa \mu(A)$1, for example, $\omega(A) = \kappa \mu(A)$2.

Vertex multiplicities $\omega(A) = \kappa \mu(A)$3 in four-dimensional almost-toric manifolds are:

• For a bivalent sphere vertex with two parallel edges (cylinder): $\omega(A) = \kappa \mu(A)$4.
• For a trivalent sphere vertex of "pair–of–pants" type: $\omega(A) = \kappa \mu(A)$5.
• At a focus–focus singularity, with edge direction $\omega(A) = \kappa \mu(A)$6: $\omega(A) = \kappa \mu(A)$7.
• The unique disk vertex (root): $\omega(A) = \kappa \mu(A)$8. This complete description ensures that the count is well-defined after summing over all possible resolutions of higher-valency vertices (Venugopalan et al., 3 Apr 2026, Venugopalan et al., 2020).

4. Explicit Calculations: Toric and Almost-Toric Manifolds

For toric surfaces and their almost-toric blow-ups (del Pezzo surfaces), the tropical disk potential recovers classical mirror symmetry formulas:

• $\omega(A) = \kappa \mu(A)$9: $A\in\pi_2(X,L)$0
• $A\in\pi_2(X,L)$1: $A\in\pi_2(X,L)$2
• Del Pezzo blow-ups—for $A\in\pi_2(X,L)$3 blow-ups, explicit formulas are:

$A\in\pi_2(X,L)$4	$A\in\pi_2(X,L)$5
1	$A\in\pi_2(X,L)$6
2	$A\in\pi_2(X,L)$7
5	$A\in\pi_2(X,L)$8

For the monotone cubic surface (Clifford torus), there are exactly 21 rigid tropical graphs, contributing via: $A\in\pi_2(X,L)$9 as shown by Sheridan and Pascaleff–Tonkonog (Venugopalan et al., 2020).

For flag varieties $\kappa > 0$0, the Gelfand–Cetlin fiber yields a potential with one Maslov-2 disk contribution per facet of the polytope, matching the sum over combinatorial data (Venugopalan et al., 2020).

5. Connections to Mirror Symmetry and Gromov–Witten Theory

The tropical disk potential $\kappa > 0$1 is identified with the Landau–Ginzburg superpotential in the mirror correspondence—explicitly, period integrals of $\kappa > 0$2 over suitable cycles compute modifications of Givental’s $\kappa > 0$3-function, thus encoding Gromov–Witten invariants, including descendent invariants (Overholser, 2015). In the case of $\kappa > 0$4, the descendent tropical Landau–Ginzburg potential takes the form: $\kappa > 0$5 where the terms and multiplicities are determined by tropical data, and period integrals expand as series of tropical correlators, matching the classical Gromov–Witten side (Overholser, 2015). The overall approach recovers maximal mutability and aligns with wall-crossing formulas and the structure of scattering diagrams inherent in the Gross–Siebert program (Venugopalan et al., 3 Apr 2026).

6. Methodological and Conceptual Context

The tropical disk potential paradigm was developed to generalize tropical enumerative geometry—previously restricted to closed curves—to the case of open invariants (holomorphic disks with Lagrangian boundary), providing a symplecto–tropical approach that bypasses the need for complex a priori Floer-theoretic transversality considerations (Venugopalan et al., 3 Apr 2026, Venugopalan et al., 2020). Mikhalkin’s genus-zero correspondence for spheres in $\kappa > 0$6 is now mirrored for disks, while full agreement with potentials derived by wall-crossing (Pascaleff–Tonkonog), and within the Gross–Siebert mirror program, confirms the tropical method’s completeness in the monotone and almost-toric settings. The identification of combinatorially rigid graphs as the essential contributors ensures the invariance of the count under wall-crossing and decomposition choices.

Further applications include explicit computation for flag varieties, almost-toric del Pezzo manifolds, and the extraction of structural Laurent polynomials underlying mutations and Fano mirror symmetry. There is a direct relationship to the tropical Fukaya algebra, where the $\kappa > 0$7 structure map is precisely the tropical disk count: $\kappa > 0$8 with the coefficients $\kappa > 0$9 fully specified by the local and global combinatorics of the tropicalization (Venugopalan et al., 2020).

7. Advanced Directions and Significance

Tropical disk potentials enable algorithmic and combinatorial methods for computing open Gromov–Witten invariants and reflecting mirror symmetry phenomena in monotone symplectic manifolds with toric or almost-toric degenerations. Subsequent developments generalize the technique to other open invariants (descendents, higher genus), contribute to the structure of the mirror map and scattering diagrams, and connect to the theory of cluster algebras and wall-crossing structures. The precise correspondence with classical invariants as established in $\mu$0 and beyond (Overholser, 2015), and the direct combinatorial reconstruction for Lagrangians in explicit models (Venugopalan et al., 3 Apr 2026, Venugopalan et al., 2020), position tropical disk potentials as foundational invariants at the intersection of symplectic geometry, tropical enumerative theory, and algebraic mirror symmetry.