Skein-Valued Holomorphic Curve Counting

• Skein-valued holomorphic curve counting is a framework that assigns skein module elements to holomorphic curves, capturing topological and geometric boundary data.
• It employs moduli space construction, equivariant localization, and combinatorial recursion to transform classical curve invariants into quantum-topological objects.
• This method bridges enumerative geometry with knot invariants, mirror symmetry, and wall-crossing phenomena using rigorous analytic and polyfold techniques.

Skein-valued counting of holomorphic curves is a framework which assigns to each holomorphic curve—not a mere number, but an element in a skein module that encodes topological/geometric data of boundaries, intersections, framings, and local moves analogous to those in quantum knot theory. This approach transforms classical enumerative invariants of holomorphic maps (closed or open, real or complex, possibly with marked points or boundary) into quantum-topological objects interconnected with knot invariants, cluster algebras, wall-crossing phenomena, and the geometry of symplectic and Calabi–Yau manifolds. Originating in the context of open Gromov–Witten theory and inspired by mirror symmetry, Chern–Simons theory, and quantum topology, the skein-valued construction rigorously blends moduli space analysis, localization, Fredholm and polyfold theory, and algebraic structures intrinsic to skein algebras.

1. Moduli Spaces of Holomorphic Curves and Real Structures

Skein-valued holomorphic curve counting requires the intricate construction of moduli spaces that parameterize holomorphic (or pseudo-holomorphic) maps from Riemann surfaces (possibly with boundary or marked points) into a symplectic or Calabi–Yau manifold $X$, often subject to extra symmetries. Real curve settings—such as maps invariant under anti-symplectic involutions—necessitate moduli spaces for both disk-type and non-disk-type configurations, as in the genus-zero real invariants of projective spaces (Tehrani et al., 2012).

The central technical aspects include:

• Construction of Kuranishi structures and virtual fundamental classes on these moduli spaces to resolve non-transversality.
• Choice of orienting data: real square roots of the canonical bundle and spin structures on fixed Lagrangian loci, ensuring the orientation compatibility necessary for gluing together disk and real structures.
• For moduli spaces with boundary (arising from nodal degenerations), a key tool is the "gluing" of disk-type and non-disk-type moduli to obtain total moduli with cancellation of unwanted boundaries. This enables rigorous invariant definitions that retain the topological data necessary for skein-theoretic enhancement.

The moduli space construction is highly sensitive to orientation and gluing data, and it is this sensitivity that makes skein-valued refinement both meaningful and technically subtle.

2. Equivariant Localization and Combinatorial Recursion

Equivariant localization, particularly via torus actions, is central for explicit computations and for connecting enumerative geometry with combinatorial and representation-theoretic frameworks (Tehrani et al., 2012, Ekholm et al., 2020). For example, in the enumerative theory over $\mathbb{P}^{2m-1}$ or $\mathbb{P}^{4m-1}$, moduli spaces admit torus actions whose fixed loci correspond to decorated graphs—encoding the geometry of holomorphic maps.

The localization formula expresses skein-valued invariants as weighted sums over combinatorial structures (such as graphs or partitions), with weights furnished by:

• Euler classes of deformation-obstruction bundles,
• Insertions corresponding to cohomology evaluations at fixed points,
• Orientation data arising from the intricate structure of the moduli.

In the open-strand context, similar localization and recursion phenomena appear: boundaries at infinity correspond to operators (in the skein algebra or quantum torus) acting on the space of boundary data, yielding operator recurrence or "quantum curve" equations for partition functions valued in the skein (Ekholm et al., 2020, Ekholm et al., 13 Jul 2024).

3. Quantum Topology, Skein Modules, and Operator Relations

The skein module is a formal algebraic construction generated by isotopy classes of links (or more generally, curves) in a 3-manifold, modulo local HOMFLYPT skein relations and possible framing corrections. In skein-valued holomorphic curve counting, each curve is assigned a skein value:

• The boundary $\partial u$ of an open holomorphic curve $u$ maps to an element $[\partial u]$ in the skein of the boundary Lagrangian (Ekholm et al., 2019).
• The weight attached to a holomorphic curve includes quantum parameters (such as $z$, $a$), linking numbers, and possible automorphism factors, as in partition functions

$$\langle\langle L \rangle\rangle_{X, d, \lambda} = \sum_{u \in \mathcal{M}_{d, \lambda}} w(u) z^{-\chi(u)} a^{u \uplus L} [\partial u] \in \text{Sk}(L).$$

• Codimension-one boundary degenerations (hyperbolic or elliptic nodes) in a one-parameter family of almost complex structures produce local wall-crossings corresponding precisely to skein relations:
  • Hyperbolic node: (overcrossing) – (undercrossing) = $z\times$(smoothing)
  • Elliptic node: addition of a trivial unknot weighted by $a$.

Crucially, these operator relations promote the skein-valued partition function (analogous to a wave-function or quantum state) to satisfy polynomial relations in the skein algebra, often interpreted as a $D$-module structure (Ekholm et al., 13 Jul 2024).

4. Recursion, Deformation Invariance, and Quantum Curve Structures

Recursion relations and quantum difference equations (mirror curve equations) arise because skein-valued curve counts, as quantum partition functions, must be annihilated by operator polynomials encoding the allowed wall-crossings and degenerations (Ekholm et al., 2020, Ekholm et al., 13 Jul 2024, Ekholm et al., 19 Dec 2024):

• For toric branes in $\mathbb{C}^3$ or the resolved conifold, the basic recursion is

$(\circ - P_{1,0} + a_L \gamma P_{0,1}) \Psi = 0,$

where $P_{1,0}$ and $P_{0,1}$ act as difference operators on the skein basis indexed by partitions.

• The recursion relations are often a skein-theoretic quantization of geometric or mirror symmetry relations, connecting the enumerative A-model (holomorphic curve counts) with the B-model (mirror algebraic geometry).
• Deformation invariance is established at the level of skein modules: any wall-crossing changes in the naive counts are killed by the skein relations, rendering the skein-valued invariant robust to changes in auxiliary geometric structures (Ekholm et al., 2019).
• Uniqueness of the invariant is proved algebraically—partition functions are determined recursively by initial data and the operator equations, aligning with the physical intuition that holomorphic curve counts in these settings solve universal recursion/quantum curve problems (Ekholm et al., 19 Dec 2024).

5. Applications to Knot Invariants, Mirror Symmetry, and Wall-Crossing

Skein-valued counting of holomorphic curves provides a rigorous mathematical framework for physical predictions connecting open Gromov–Witten invariants and knot invariants such as the HOMFLYPT polynomial (Ekholm et al., 2019, Ekholm et al., 2021, Ekholm et al., 19 Dec 2024):

• In the resolved conifold, the colored HOMFLYPT invariant of a link becomes the generating function of skein-valued holomorphic curve counts with boundary on the conormal Lagrangian,
• Multiple covers and disconnected curves exponentiate, corresponding to the combinatorics of Wilson line insertions in Chern–Simons theory and the expansion of knot polynomials in symmetric and colored representations.

This formalism rigorously matches the Ooguri–Vafa conjecture and mirrors topological vertex constructions of open string partition functions (Ekholm et al., 19 Dec 2024). The operator equations satisfied by the skein-valued partition functions dequantize (as $q \rightarrow 1$) to define the mirror moduli spaces (augmentation varieties), confirming deep aspects of homological mirror symmetry.

Wall-crossing phenomena and cluster transformations are also captured:

• Skein-valued lifts of the Kontsevich–Soibelman wall-crossing formula are realized via flow-graph analysis and holomorphic curve counts in geometric transitions, notably in the setting of cotangent bundles and Legendrian/Lagrangian surgeries (Ekholm et al., 21 Oct 2025, Scharitzer et al., 2023, Scharitzer, 23 Oct 2024).
• The adjacency with cluster algebra structures comes from the identification of quantum skein relations with quantum mutations, which are realized geometrically via disk surgeries and Lagrangian cobordisms.

6. Analytic and Polyfold Foundations

The rigorous analytic implementation of skein-valued counting necessitates sophisticated perturbation and transversality schemes:

• Construction of coherent perturbations of the Cauchy–Riemann equations (in the polyfold sense) that vanish on "ghost" (zero symplectic area) components, ensuring only nondegenerate ("bare") curves are counted (Ekholm et al., 2 Jun 2024).
• Polyfold methods allow inductive compatibility and "ghost bubble censorship", guaranteeing compactness and orientation of the moduli space of bare curves both in closed and boundary-marked settings.

This analytic backbone justifies the assignment of skein values to open holomorphic curves and ensures the resulting count is indeed an invariant of the underlying geometric data—a requirement for its physical and topological applications.

7. Generalizations and Future Directions

The skein-valued curve counting philosophy is actively expanding:

• To higher-genus, multiple-boundary, and quiver-like cases, with formal expansions in the HOMFLYPT skein algebra reflecting BPS state counts (Nakamura, 19 Jan 2024).
• Connections with quantum traces, nonabelianization maps, and wall-crossing in wild character varieties (Ekholm et al., 21 Oct 2025).
• Skein-valued lifts of quantum dilogarithm identities connected to $A_n$ quiver sequences, with implications for cluster algebras, categorification, and wall-crossing theory (Scharitzer, 23 Oct 2024).
• Deep interplay with Legendrian and Lagrangian invariants, augmentation varieties, and cluster algebra structures via disk surgery and symplectic field theory (Scharitzer et al., 2023).

This body of work supports a unified picture: skein-valued counting of holomorphic curves not only refines classical enumerative invariants but also serves as a foundational bridge between symplectic topology, low-dimensional topology, quantum algebra, and mathematical physics.