Open WDVV Equations in Gromov–Witten Theory

• The open WDVV equations are a system of nonlinear PDEs that extend the classical WDVV framework by incorporating boundary effects in Gromov–Witten theory.
• They establish a rich algebraic and geometric structure by coupling Frobenius manifold theory with open string invariants and integrable hierarchies.
• These equations drive recursion relations, mirror symmetry, and higher-genus deformations, offering explicit solutions in various geometric and topological settings.

The open WDVV equations are a fundamental system of nonlinear partial differential equations that govern the algebraic structure of genus-zero open Gromov–Witten invariants, integrating bulk and boundary effects in both mathematics and mathematical physics. They generalize the original (closed) Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations―the associativity relations for Frobenius manifolds and quantum cohomology rings―to stratified settings where boundary conditions and open string states are present. The open WDVV system produces a rich algebraic and geometric framework encompassing flat F-manifolds, recursion for open invariants, integrable hierarchies, and connections to mirror symmetry, singularity theory, and higher-genus deformation theory.

1. Formal Structure of the Open WDVV Equations

Let $F=F(t^1,\dots,t^N)$ be a solution to the closed WDVV equations, associated with the structure constants

$$c_{\alpha\beta\gamma} = \frac{\partial^3 F}{\partial t^\alpha \partial t^\beta \partial t^\gamma},$$

where $\eta^{\alpha\beta} = c_{1\alpha\beta}$ is a constant, nondegenerate metric (unit vector field $e_1=\partial_{t^1}$). The closed WDVV equations

$$\sum_{\lambda,\mu=1}^N \frac{\partial^3 F}{\partial t^\alpha\partial t^\beta\partial t^\lambda}\, \eta^{\lambda\mu}\, \frac{\partial^3 F}{\partial t^\gamma\partial t^\delta\partial t^\mu} = \sum_{\lambda,\mu=1}^N \frac{\partial^3 F}{\partial t^\gamma\partial t^\beta\partial t^\lambda}\, \eta^{\lambda\mu}\, \frac{\partial^3 F}{\partial t^\alpha\partial t^\delta\partial t^\mu}$$

encode associativity of the commutative, unital algebra.

For the open sector, introduce an extra variable $s$, and define an "open" potential $F^o = F^o(t^1,\dots,t^N, s)$. In flat coordinates, the open WDVV equations (for $1\le\alpha, \beta, \gamma \le N$) are: $$\sum_{\mu, \nu=1}^{N} \frac{\partial^3 F}{\partial t^\alpha \partial t^\beta \partial t^\mu}\, \eta^{\mu\nu}\, \frac{\partial^2 F^o}{\partial t^\nu \partial t^\gamma} + \frac{\partial^2 F^o}{\partial t^\alpha \partial t^\beta} \frac{\partial^2 F^o}{\partial s \partial t^\gamma} = \sum_{\mu, \nu=1}^{N} \frac{\partial^3 F}{\partial t^\gamma \partial t^\beta \partial t^\mu}\, \eta^{\mu\nu} \frac{\partial^2 F^o}{\partial t^\nu \partial t^\alpha} + \frac{\partial^2 F^o}{\partial t^\gamma \partial t^\beta} \frac{\partial^2 F^o}{\partial s \partial t^\alpha}.$$ For the "all open" sector,

$$\sum_{\mu, \nu=1}^{N} \frac{\partial^3 F}{\partial t^\alpha \partial t^\beta \partial t^\mu}\, \eta^{\mu\nu}\, \frac{\partial^2 F^o}{\partial t^\nu \partial s} + \frac{\partial^2 F^o}{\partial t^\alpha \partial t^\beta}\frac{\partial^2 F^o}{\partial s^2} = \frac{\partial^2 F^o}{\partial t^\alpha \partial s}\frac{\partial^2 F^o}{\partial t^\beta \partial s}.$$

These equations are typically supplemented by unit and homogeneity conditions, ensuring the correct normalization and scaling with respect to the Euler vector field determined by $F$ (Basalaev et al., 2019, Sheng, 26 Apr 2024, Basalaev et al., 2019, Alexandrov et al., 2022).

2. Geometric and Algebraic Interpretation

Open WDVV equations capture the compatibility of genus-zero bulk and boundary operations in open Gromov–Witten theory. Mathematically, they describe how the structure on the tangent sheaf of a closed (Frobenius) manifold can be flatly extended by adding a boundary/open coordinate, typically producing a (formal) flat F-manifold of one higher rank (Basalaev et al., 2019, Yu et al., 2023, Ma, 3 Dec 2025). In this setting, the structure constants

$$(C_{IJ}^K) = \begin{cases} c_{\alpha\beta}^\gamma & I,J,K \leq N \ ..., \text{(certain cross terms for open sector)} \end{cases}$$

encode a commutative, associative multiplication, and the open WDVV system is precisely the associativity condition in the extended $N+1$ variable algebra.

This extension underlies the modern approach to disk invariants and open topological field theories: the open WDVV equations are the constraints arising from flatness and associativity in the presence of a nontrivial boundary sector, and determine the higher genus recursion, topological recursion relations, and Virasoro constraints (Basalaev et al., 2019, Alexandrov et al., 2022).

3. Explicit Solutions and Classification

Closed-form polynomial and algebraic solutions to the open WDVV equations are classified for all finite irreducible Coxeter groups and some extended affine Weyl groups. For the $A_N$ Frobenius manifold, the open potential has the explicit closed formula: $$F^o_{A_N}(t,s) = \sum_{n\ge2} \sum_{1\le a_1,\ldots,a_n\le N} \sum_{k\ge0} \left\langle \tau_{a_1}\cdots\tau_{a_n}\, \sigma^k\right\rangle^{A_N} \frac{t_{a_1}\cdots t_{a_n}}{n!} \frac{s^k}{k!}$$ with correlator rule

$$\left\langle \tau_{a_1}\cdots\tau_{a_n}\, \sigma^k\right\rangle^{A_N} = (n+k-2)! \quad \text{if } \sum_{i=1}^n(N+2-a_i)+k = N+2.$$

For type $D_N$, open extension formulas involve Laurent polynomials with coefficients arising from transition functions of the miniversal deformation (Basalaev et al., 2019, Basalaev, 2022).

Basalaev–Buryak establish that for $W\neq A_N,B_N,I_2(k)$, no nontrivial homogeneous polynomial solution exists for the open WDVV equations with normalization and homogeneity constraints (Basalaev et al., 2019, Sheng, 26 Apr 2024). For rational/Whitham reductions and infinite-dimensional Frobenius manifolds, the open WDVV admits explicit solutions associated with infinite-rank flat F-manifolds (Ma, 3 Dec 2025).

4. Open Gromov–Witten Theory and Recursion

In open Gromov–Witten theory, the open WDVV system produces recursion relations for open Gromov–Witten invariants (OGW). The Frobenius superpotential $\Omega(s,t)$, a generating function for open correlators with $k$ boundary and $\ell$ interior insertions, encodes the full OGW data. The open WDVV relations for $(\Phi,\Omega)$ (with $\Phi$ the closed sector potential) take the schematic form: $$\sum_{\mu,\nu} \partial_a\partial_\mu\Omega\,g^{\mu\nu}\,\rho^*\!\partial_b\partial_c\partial_\nu\Phi - (\partial_a\partial_s\Omega)(\partial_b\partial_c\Omega) = \sum_{\mu,\nu} \rho^*\!\partial_a\partial_b\partial_\mu\Phi\,g^{\mu\nu} \partial_c\partial_\nu\Omega - (\partial_a\partial_b\Omega)(\partial_s\partial_c\Omega)$$ The resulting hierarchy enables recursive computation of all genus-zero open invariants from finite initial data, generalizing the Kontsevich–Manin theorem for the closed sector. Invariant vanishing and structure results are established for products of projective spaces and symplectic manifolds with high symmetry (Blumberg et al., 16 Feb 2024).

5. Integrable Hierarchies and Principal Hierarchy Construction

The open WDVV equations are intimately connected to integrable hierarchies of hydrodynamic type. For every solution to the open WDVV system, there exists a commuting family of dispersionless flows extending those defined on the underlying closed Frobenius manifold. In type $A_N$, the principal hierarchy arising from the open extension coincides with the dispersionless modified KP (dmKP) hierarchy in its Fay form: $$z\,e^{-D(z)\,\partial_{t_0}f} - w\,e^{-D(w)\,\partial_{t_0}f} = (z-w)\,e^{D(z)D(w)f}$$ where $D(z)$ is a generating operator. For $D_N$, the corresponding system is a dispersionless modified BKP hierarchy (1-component), with the open sector providing additional compatible flows parametrized by the extended flat F-manifold (Basalaev, 2022, Ma, 3 Dec 2025).

For infinite-dimensional Frobenius manifolds (universal Whitham hierarchy), Ma–Wu–Zuo construct explicit open potentials and the associated principal hierarchies, showing compatibility with all finite-dimensional reductions and recovering classical polynomial open solutions as specializations (Ma, 3 Dec 2025).

6. Frobenius Structures, Mirror Symmetry, and Higher-Genus Deformations

The open WDVV equations support sophisticated algebraic structures: semi-simple formal Frobenius manifolds (with open/closed units and semi-simple idempotent splitting), flat (but not necessarily unital) formal F-manifolds with nilpotent boundary directions, and extensions to genus expansions via open total descendant potentials (Yu et al., 2023, Alexandrov et al., 2022).

In the context of toric Calabi–Yau 3-folds with Aganagic–Vafa branes, open WDVV equations describe the flatness and associativity of the full space of closed and open moduli. This underpins the open/closed mirror map and the Hodge-theoretic reconstruction of higher-genus open amplitudes on the B-model side (Yu et al., 2023).

All-genera extensions, deformation theory, and open Virasoro constraints are also accessible from the open WDVV system, especially for semi-simple cases. Using an open analog of Givental's quantization and calibration machinery, the open total descendant potential is constructed in full generality and shown to satisfy open topological recursion relations and string/dilaton equations (Alexandrov et al., 2022, Basalaev et al., 2019).

7. Applications and Further Directions

Applications of the open WDVV system span open Gromov–Witten theory, mirror symmetry, singularity theory (universal unfoldings, $r$-spin disks), and the analysis of integrable hierarchies with boundary. Open WDVV equations also underlie explicit computation and vanishing results for OGW invariants in high-dimensional cases (e.g., products of complex projective spaces), and have been shown to arise for arbitrary Hurwitz Frobenius manifolds via Landau–Ginzburg superpotentials and their primitives (Almeida, 12 Mar 2025).

Table: Classes of Open WDVV Solutions

Class	Structure	Example Reference
Polynomial (Coxeter)	Flat F-manifold extension	(Basalaev et al., 2019)
Rational (Whitham)	Infinite-dimensional, universal	(Ma, 3 Dec 2025)
Toric CY with brane	Semi-simple Frobenius, Hodge-theoretic	(Yu et al., 2023)
General Hurwitz	LG primitive via Dubrovin's method	(Almeida, 12 Mar 2025)

Theoretical development continues in the classification of boundary extensions for orbifold and higher-genus Frobenius manifolds, quantization of open integrable hierarchies, and refinement of open/closed mirror correspondences. The consistent theme is the unifying role that open WDVV equations play in constructing and understanding algebraic structures blending closed and open sectors in geometry and physics.