Aganagic-Vafa Brane in Toric CY3s

• Aganagic–Vafa brane is a real Lagrangian submanifold in toric Calabi–Yau threefolds characterized by explicit toric construction and framing conditions.
• It underpins open Gromov–Witten invariants, mirror symmetry relations, and large N dualities, linking enumerative geometry with quantum invariants and knot theory.
• Its analysis leads to advances in integrality proofs, SYZ mirror corrections, and the construction of algebraic structures like Frobenius manifolds and cohomological Hall algebras.

The Aganagic–Vafa brane is a distinguished class of real Lagrangian (A-brane) submanifolds in toric Calabi–Yau threefolds, originally introduced in the context of open topological string theory to provide calculable examples of large N dualities, mirror symmetry, and integrality structures in open Gromov–Witten theory. Its geometric, enumerative, and mirror-theoretic properties have become central in the paper of open-closed string duality, quantum periods, and the construction of algebraic structures such as formal Frobenius and F-manifolds arising from open Gromov–Witten invariants. The Aganagic–Vafa brane, notable for its explicit toric construction and rich enumerative geometry, connects open-string BPS invariants to knot theory, quiver varieties, and quantum algebraic structures.

1. Toric Geometry and the Definition of the Aganagic–Vafa Brane

The Aganagic–Vafa (AV) brane $L_{AV} \subset X$ is constructed as a special Lagrangian submanifold in a smooth semi-projective toric Calabi–Yau threefold $X$. For $X$ defined by a toric fan $\Sigma \subset N_\mathbb{R}$ with complex dimension three, $L_{AV}$ is characterized by imposing two real linear relations and a phase condition on the moment-map image: $$|x_3|^2 - |x_1|^2 = c, \qquad |x_2|^2 - |x_1|^2 = 0,$$ where $(x_1,\ldots,x_{k+3})$ are coordinates on the dense torus $(\mathbb{C}^*)^{k+3}$ of $X$ and $c \in \mathbb{R}$ is a real parameter (Hu et al., 8 Dec 2025). Topologically, $L_{AV} \cong S^1 \times \mathbb{R}^2$, and it is invariant under a one-dimensional subtorus $T_{L,f} \subset (\mathbb{C}^*)^2$ (the Calabi–Yau torus), corresponding to an integer "framing" $f$.

In the toric diagram, the AV brane is realized as a noncompact leg attached to an external edge. The brane’s homotopy type and boundary data generalize to the orbifold case, where for a toric Calabi–Yau 3-dimensional Deligne–Mumford (DM) stack $X$ with generic stabilizer $K$, the AV brane becomes $L \cong (S^1 \times \mathbb{R}^2)/G_\tau$ and $\pi_1(L) \cong \mathbb{Z} \times G_\tau$. Framing is encoded through the choice of subtorus in the real moment-map picture and modifies the localization weights in equivariant Gromov–Witten theory (Fang et al., 2012, Fang et al., 2011).

2. Open Gromov–Witten Invariants, Framing, and Enumerative Structures

Open Gromov–Witten invariants $N^{X,L}_{g, \beta', \mu}(f)$ enumerate stable holomorphic maps from bordered Riemann surfaces to $(X, L_{AV})$, in relative homology class $\beta' \in H_2(X, L)$ and winding number $\mu \in \mathbb{Z}$ around the $S^1$ factor. For $g=0$, the disk potential generating function is

$$F_{0,1}(Q; f) = \sum_{w > 0} \sum_{\beta \in H_2(X)} N_{0, \beta + w b, w}^{X, L}(f) Q_0^w Q^\beta,$$

with $b$ denoting the basic disk class and $Q_0$, $Q$ as open/closed Kähler parameters. Framing $f \in \mathbb{Z}$ appears explicitly as a twist both in localization computations and in the open mirror map, shifting disk invariants by quadratic expressions in $f$ (Zhou, 2010, Fang et al., 2011, Luo et al., 2016, Fang et al., 2012). For orbifold branes, the invariants acquire additional group-theoretic data and values in Chen–Ruan cohomology (Fang et al., 2012).

LMOV (Labastida–Mariño–Ooguri–Vafa) invariants $n_{\mu,g,Q}(f)$, extracted from the open Gromov–Witten generating functions through plethystic and Möbius inversion formulas, are proven to be integers, reflecting the BPS counting nature of open strings ending on $L_{AV}$ (Luo et al., 2016, Zhu, 2019).

3. Mirror Symmetry, Mirror Curves, and Open Mirror Theorems

Under central developments in mirror symmetry, $L_{AV}$ is mirrored by a specific curve in the B-model geometry. For toric CY3s, the Hori–Vafa mirror construction yields

$$uv = H(x, y; q),\qquad H(x, y; q) = -x y^{-f} + y + 1 + \sum_i c_i(q) x^{a_i} y^{b_i},$$

where $(x, y) \in (\mathbb{C}^*)^2$, $u, v \in \mathbb{C}$, and $c_i(q)$ depend on the closed string (complex structure) parameters (Fang et al., 2011). For strip geometries (toric CY3s without compact surfaces), the mirror curve simplifies to

$$y\prod_i (1 - \alpha_i x) + \prod_j (1 - \beta_j x) = 0,$$

with $\alpha_i$, $\beta_j$ determined by products of Kähler parameters attached to specific "up" or "down" vertices in the toric diagram (Hu et al., 8 Dec 2025).

The open mirror map relates A- and B-model parameters. The disk potential, viewed as the generating function for open holomorphic curves, matches termwise with the Abel-Jacobi integral over the mirror curve. Explicit recursive and hypergeometric formulas for the B-model disk superpotential, and proofs of the disk mirror theorem for all genera, are established for the resolved conifold and general strip geometries (Zhou, 2010, Fang et al., 2011, Hu et al., 8 Dec 2025). In the orbifold case, the open mirror theorem extends this correspondence to Chen–Ruan-valued disk invariants (Fang et al., 2012).

Quantum corrections and "skein-valued" quantizations of mirror curves encode higher-genus information and integrate non-commutative variables subject to $q$-deformation ($\hat y \hat x = q \hat x \hat y$). At the operator level, the unique solution to the quantized mirror curve gives a plethystic exponential formula governed by $q$-dilogarithms, in precise agreement with the topological vertex and open Gromov–Witten sums (Hu et al., 8 Dec 2025).

4. Integrality, Large N Duality, and Algebraic Structures

Integrality properties of open GW (or LMOV) invariants, proven via Möbius inversion and combinatorial reduction, reflect deep BPS/Chern–Simons dualities. For the resolved conifold with a single AV brane, the open string partition function matches the SU(N) Chern–Simons partition function on $S^3$ with a single framed unknot, establishing the large N duality at all genera and boundary multiplicities (Luo et al., 2016, Zhu, 2019).

For the $\mathbb{C}^3$ geometry with one AV brane, the Ooguri–Vafa invariants coincide with Betti numbers of Nakajima quiver varieties for certain framings, and the open string partition function factors as a plethystic product of $q$-series, formally related to Rogers–Ramanujan identities for special framings (Zhu, 2017).

In general, the reduced open partition function aligns with the Hilbert–Poincaré series of the cohomological Hall algebra (CoHA) of the $|\tau|$-loop quiver for $\tau \leq -1$, providing an open-string GW/DT correspondence (Luo et al., 2016). These correspondences transitively relate open BPS spectra to moduli of representations and cohomological invariants of quivers, extending the GW/DT correspondence beyond the closed sector.

The open Gromov–Witten theory of $(X, L_{AV})$ admits rich algebraic structures. The disk and closed potentials enable the construction of a semi-simple formal Frobenius manifold (with a metric and potential) and a flat formal F-manifold (product and connection, generically without a metric or unit), equipped with WDVV-type equations that encapsulate all open/closed relationships among invariants (Yu et al., 2023).

5. SYZ Mirror Construction and Quantum Corrections

The SYZ approach reconstructs the B-model mirror of the AV brane by first considering a dual special Lagrangian torus fibration and then constructing the semi-flat mirror. For an AV brane defined by charge vectors $l^{(1)}, l^{(2)}$, the naive SYZ mirror subvariety enforces monomial relations in complex torus variables. However, this naive construction omits quantum (disk) corrections arising from Maslov index 2 open Gromov–Witten disks. Correcting the SYZ mirror by adding the generating series of disk counts (quantum corrections), one obtains the expected enumerative B-model superpotential, in agreement with the predictions of Aganagic–Vafa and physicists' mirror symmetry (Chung, 2016). This result demonstrates that B-model disk amplitudes cannot be recovered from semi-flat geometry alone but require explicit enumeration of open worldsheet instantons.

6. Logarithmic Structures, Open/Closed Correspondence, and Higher-Genus Phenomena

For local del Pezzo geometries $K_P$, the $q$-refined quantum period of the mirror curve defines a $q$-refined open mirror map. The coefficients in the expansion of the $q$-theta function encode all-genus logarithmic two-point invariants (log-Gromov–Witten invariants) of the pair $(P, E)$, and the degree one (winding-1) open BPS invariants correspond precisely to closed Gopakumar–Vafa invariants of a blow-up of $K_P$ (Gräfnitz et al., 26 Feb 2025). At higher genus, discrepancies between all-genus open GW invariants and logarithmic invariants are attributed to relative elliptic curve invariants, and explicit gluing/flop formulas via the topological vertex relate open curves on $K_P$ with outer AV brane to closed GW invariants of the blow-up geometry.

This open/closed correspondence generalizes the Ooguri–Vafa and Lau–Leung–Wu results, structurally equating open disk counts with degenerate closed curve counts under open mirror symmetry. In genus-zero, the correspondence is exact; at higher genus, correction terms are described via degeneration of the mirror curve and relative geometry (Gräfnitz et al., 26 Feb 2025).

7. Applications, Generalizations, and Algebraic Structures

The framework of the Aganagic–Vafa brane has deep implications across multiple mathematical domains:

• Topological recursion and quantum curves: Skein-valued and $q$-deformed quantizations of mirror curves arising from AV branes provide a constructive route to topological recursion and quantum holonomic D-modules (Hu et al., 8 Dec 2025).
• Orbifolds and stacks: AV branes and their invariants extend naturally to the orbifold setting, giving orbifold disk potentials valued in the Chen–Ruan cohomology and matching with Abel–Jacobi maps for the B-model mirror curves (Fang et al., 2012).
• Frobenius and F-manifolds: Open Gromov–Witten theory on $(X, L_{AV})$ defines, via the open WDVV equations, novel algebraic structures—semi-simple Frobenius manifolds and flat F-manifolds—generalizing quantum cohomology to the open-string sector (Yu et al., 2023).
• Knot-quiver correspondence: Explicit AV brane invariants on the resolved conifold and $\mathbb{C}^3$ initiate a systematic program relating open string BPS invariants, knot theory, and the cohomology of quiver varieties (Zhu, 2017).

A plausible implication is that the AV brane construction functions as a canonical embedding of "boundary conditions with integrable and quantizable enumerative geometry"—serving as a touchstone for future developments in both open Gromov–Witten theory and higher structures in mathematical physics.