Local Toric Calabi–Yau Threefolds

• Local toric Calabi–Yau threefolds are non-compact Calabi–Yau spaces constructed as the total space of a canonical bundle over a toric surface or via a trivalent toric diagram.
• They enable precise computation of enumerative invariants using combinatorial methods, fermionic Fock space techniques, and categorical approaches.
• Their rich structure underpins integrable systems and wall-crossing phenomena, providing critical insights in modern topological string theory and gauge theory.

A local toric Calabi–Yau threefold is a non-compact Calabi–Yau 3-fold that admits a faithful algebraic action of the complex torus $(\mathbb{C}^*)^3$ and is defined, up to isomorphism, by a finite trivalent planar graph known as its "toric diagram." Equivalently, every such threefold may be realized as the total space $K_S = \mathrm{Tot}(K_S)$ of the canonical bundle over a smooth projective toric surface $S$. These spaces serve as central objects in modern enumerative geometry, topological string theory, and mathematical gauge theory, providing explicit testbeds for the interplay of algebraic, symplectic, and physical ideas. Local toric Calabi–Yau threefolds admit tractable algebraic, combinatorial, and categorical descriptions for all associated enumerative invariants and partition functions, enabling rigorous verification of string dualities, wall-crossing phenomena, and integrability predictions.

1. Construction and Combinatorial Representation

A local toric Calabi–Yau threefold is defined either as the total space $\mathrm{Tot}(K_S)$ of the canonical bundle over a smooth toric surface $S$ or, equivalently, as the toric Deligne–Mumford stack associated to a Gorenstein three-dimensional fan given by the cone over a two-dimensional triangulated lattice polygon $\Delta \subset \mathbb{Z}^2$. The structure is combinatorially encoded via a “toric diagram”: a trivalent planar graph whose vertices correspond to torus-fixed points and compact and non-compact curves, while edges represent invariant $\mathbb{P}^1$’s or legs.

In the toric formalism, the variety $X$ may be written as

$X = (\mathbb{C}^r \setminus Z(\Sigma))/G$

where $\Sigma$ is the fan, the $r$ rays $v_i$ specify homogeneous coordinates $x_i$, and $G$ is a torus acting via relations among the $v_i$. The canonical bundle is trivialized by a nowhere-vanishing torus-invariant holomorphic three-form constructed from the data of the fan and its relations.

2. The Topological Vertex and Gluing Formulas

The topological vertex $W_{\mu^1,\mu^2,\mu^3}(q) \in \mathbb{Q}(q)$, as introduced by Aganagic, Klemm, Mariño, and Vafa, provides the generating function for all open Gromov–Witten invariants of the basic geometry $\mathbb{C}^3$. Boundary conditions are labeled by partitions, and general toric Calabi–Yau threefolds are constructed by gluing vertices along internal edges, each labeled by a partition summed over:

$$Z_{\rm open}(Q_i;\lambda^1,\dots,\lambda^M) = \sum_{\mu^1,\dots,\mu^M} \left[\prod_{i=1}^M (-1)^{\gamma_i |\mu^i|} q^{(\gamma_i+1)\kappa_{\mu^i}/2} Q_i^{|\mu^i|}\right] \prod_{i=1}^M W_{\mu^{i-1},\,\lambda^i,\,\mu^i}$$

Here, $Q_i$ encode Kähler parameters, $\gamma_i$ framings, and $\kappa_\mu = \sum_j \mu_j (\mu_j - 2j + 1)$.

The gluing structure allows for the computation of all-genus generating functions for Gromov–Witten invariants via explicit sums over Young diagrams.

3. Fermionic and Categorical Descriptions

A profound simplification of vertex computations is achieved via the fermionic Fock space formalism, in which the topological vertex can be written as a matrix element involving free-fermion operators. For example,

$$W_{\lambda,\mu,\nu}(q) = \langle \lambda | \Psi_\mu(q)\, q^{-K/2} | \nu^t \rangle$$

where $K$ is the cut-and-join operator and $|\nu^t\rangle$ is the state for the transposed partition. This operator formalism underlies trace and determinant expressions for partition functions.

Categorically, for $Y = \mathrm{Tot}_{X_\Sigma}(K_{X_\Sigma})$, one constructs a quiver with potential (Chern–Simons superpotential) $W$ derived from a full strong exceptional collection on $X_\Sigma$. The moduli stack of sheaves or representations is exhibited as a global critical locus $\operatorname{Crit}(W)$, and the Donaldson–Thomas or motivic invariants are, respectively, weighted Euler characteristics or virtual motives of these critical loci (Hua, 2011).

4. Partition Functions, Fermion Fluxes, and Integrable Systems

Partition functions with nontrivial topology (including loops in the toric diagram) require the inclusion of fermion-number fluxes $N \in \mathbb{Z}$ through the associated cycles, encoded by inserting shift and charge operators in the free-fermion formalism:

$$Z_\lambda^{(N)}(Q_i,\gamma_i;\Xi) = \sum_{\mu \in \mathcal{P}} \langle \mu | R^N \Xi^C \left( \Psi_\lambda\, q^{-(\gamma_1+2)K}(-1)^{\gamma_1 L}Q_1^L \right) \cdots ( \cdots ) R^{-N} | \mu \rangle$$

Summing over all $N$ and external representations yields the total partition function

$$Z^{\rm total}(\mathbf t;Q_i,\gamma_i; \Xi) = \sum_{N \in \mathbb{Z}} \sum_{\lambda \in \mathcal{P}} Z_\lambda^{(N)}(Q_i,\gamma_i;\Xi) s_\lambda(\mathbf t)$$

where $s_\lambda$ are Schur functions.

It has been rigorously established (Wang et al., 13 Nov 2025) that $Z^{\rm total}$ is a multi-component KP tau-function: the underlying operator satisfies the Kyoto-school bilinear relation

$$[G \otimes G,\; \sum_r \psi_r^*\otimes\psi_{-r}] = 0,$$

which is both necessary and sufficient for KP integrability. The partition function’s Schur expansion has determinantal structure characteristic of tau-functions, encoding integrable hierarchies and confirming predictions from physical enumerative string theory.

5. Moduli Spaces, D-critical Loci, and Motivic Invariants

Hilbert schemes of points and moduli spaces of sheaves on local toric Calabi–Yau threefolds possess an algebro-geometric structure as $d$-critical loci, as defined by Joyce and collaborators. For example, the Hilbert scheme $\mathrm{Hilb}^n(X)$ of 0-dimensional subschemes is locally modeled as the critical locus of a function $W$ (e.g., $W(X,Y,Z,v) = \operatorname{Tr}([X,Y]Z)$ on suitable framed representation varieties). The $d$-critical structure, together with canonical orientation data (often derived from universal families on the product with $X$), enables the construction of motivic Donaldson–Thomas invariants refined by the motivic vanishing cycle (Katz et al., 2020). These invariants agree with previous definitions via stratification and integration, and plethystic generating functions such as

$$Z_{\omega_{\mathbb{P}^2}}(t) = \mathrm{Exp} \left( \frac{\mathbb{L}^{-1/2} (\mathbb{L}^2+\mathbb{L}+1) t}{(1-\mathbb{L}^{1/2} t)(1-\mathbb{L}^{-1/2} t)} \right)$$

can be computed for explicit geometries.

6. Gromov–Witten Theory, Birational Invariance, and Open Invariants

Closed and open genus-zero Gromov–Witten invariants of local toric Calabi–Yau threefolds are accessible via recursive localization, wall-crossing, and birational invariance. A fundamental wall-crossing formula relates invariants of the projective bundle $X = \mathbb{P}(K_S \oplus \mathcal{O}_S)$ and its flopped counterpart $W = \mathbb{P}(K_{S_n} \oplus \mathcal{O}_{S_n})$:

$$\left\langle \underbrace{[\mathrm{pt}],\dots,[\mathrm{pt}]}_n \right\rangle^{X}_{0,n,\beta} = \left\langle\,\right\rangle^{W}_{0,0,\beta'}$$

where $\beta'$ is related to $\beta$ under an explicit correspondence (Lau et al., 2010). This identification, combined with Chan's result comparing open and closed invariants, allows computation of all basic Maslov-index-two disk invariants from closed Gromov–Witten data, leading to explicit open superpotential expansions (e.g., $W_{\mathrm{conifold}}(z) = z + Q z^{-1}$ for the resolved conifold).

7. Quivers, BPS Graphs, and Wall-Crossing Phenomena

For every local toric Calabi–Yau threefold, one can associate a quiver with relations and a superpotential $W$ derived from the toric data and exceptional collections on the base surface or stack. These quiver moduli spaces reproduce the deformation theory and enumerative invariants of the threefold (Hua, 2011, Banerjee et al., 2019). For simple cases (conifold, $\mathbb{C}^3/\mathbb{Z}_2$), the quivers have two nodes, with arrows and superpotentials matching the topology of the exponential BPS graphs arising from non-abelianization procedures. The resulting frameworks capture all unframed and framed BPS spectra, account for wall-crossing at the level of Kontsevich–Soibelman transformations, and encode both 3d and 5d numerical Donaldson–Thomas and Gopakumar–Vafa invariants.

A plausible implication is that the interplay between quiver representations, motivic and cohomological DT invariants, and integrable hierarchy structures provides a universal, computationally explicit package for the paper of enumerative and physical invariants of local toric Calabi–Yau threefolds.

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This multidimensional structure unites explicit algebraic, combinatorial, and physical approaches to the enumerative geometry of non-compact Calabi–Yau threefolds, confirming integrality, furnishing wall-crossing rules, and yielding explicit formulas for all genus Gromov–Witten, DT, and open invariants, while simultaneously revealing the underlying integrable and categorical frameworks. The precise relationship between the total partition functions, moduli-theoretic $d$-critical loci, and the inductive/combinatorial formulas in the context of toric Calabi–Yau threefolds positions these spaces as central objects in modern enumerative geometry and mathematical physics.