Local Toric Calabi–Yau Threefolds

• Local toric Calabi–Yau threefolds are noncompact complex manifolds with a torus action and trivial canonical bundle, constructed as the total space of the canonical bundle over smooth toric surfaces.
• They enable explicit computations of enumerative invariants such as Gromov–Witten and Donaldson–Thomas counts by leveraging toric diagrams, mirror curves, and quiver representations.
• Advanced techniques like the topological vertex and modular topological recursion reveal deep links between partition functions, wall-crossing phenomena, and integrable systems.

A local toric Calabi–Yau threefold is a noncompact complex threefold equipped with a torus action, typically realized as the total space of the canonical bundle over a smooth toric surface or, more generally, as a gluing of affine pieces corresponding to a convex lattice polytope with one interior point. The noncompactness preserves the Calabi–Yau condition ($K_X \cong \mathcal O_X$ and $H^1(X,\mathcal O_X)=0$), with rich applications in enumerative geometry, string theory, and representation theory. The toric structure enables concrete computational techniques, ranging from explicit toric fans and mirror curves to powerful links between Gromov–Witten and Donaldson–Thomas invariants, wall-crossing, and integrability hierarchies.

1. Construction and Toric Data

A local toric Calabi–Yau threefold $X$ arises as the total space $X = \operatorname{Tot}_S(K_S)$ for a smooth toric surface $S$, or more generally via the combinatorics of a three-dimensional toric fan $\Sigma \subset N_{\mathbb R}$. Each primitive generator $v_i \in N$ of the 1-cones satisfies $\langle m, v_i \rangle = 1$ for some $m \in M = \operatorname{Hom}(N,\mathbb Z)$, ensuring the anticanonical bundle is trivial. This construction guarantees a residual $(\mathbb{C}^*)^k$-action, with $H^1(X,\mathcal O_X)=0$0 in the threefold case, and equips $H^1(X,\mathcal O_X)=0$1 with a concrete toric diagram.

Explicit examples include:

Model	Toric Surface $H^1(X,\mathcal O_X)=0$2	Total Space Description
Resolved conifold	$H^1(X,\mathcal O_X)=0$3	$H^1(X,\mathcal O_X)=0$4
Local $H^1(X,\mathcal O_X)=0$5	$H^1(X,\mathcal O_X)=0$6	$H^1(X,\mathcal O_X)=0$7
Local del Pezzo	$H^1(X,\mathcal O_X)=0$8	$H^1(X,\mathcal O_X)=0$9
Orbifold resolution	(e.g. $X$0)	crepant resolution with toric fan determined by irreducible representations

The toric data encode both the combinatorial geometry and the intersection theory, crucial for subsequent invariants and categories (Lau et al., 2010, Banerjee et al., 2019).

2. Mirror Symmetry and Mirror Curves

Mirror symmetry for local toric Calabi–Yau threefolds is formulated via explicit mirror geometries. The (B-model) mirrors are families of affine algebraic curves—mirror curves—embedded in $X$1, capturing the complex moduli dual to Kähler parameters of $X$2. For instance, the resolved conifold with standard framing yields the mirror curve

$X$3

where $X$4 encodes the complexified Kähler modulus (Banerjee et al., 2019).

In higher complexity cases (e.g., crepant resolutions of $X$5), one obtains genus-two mirror curves

$X$6

which can be put in hyperelliptic form and have a modular structure reflecting the nontrivial geometry of the fan (Ruan et al., 2019).

The topology and moduli of these mirror curves serve as the foundation for B-model calculations via topological recursion and encode the spectra of D-brane states and their central charges.

3. Enumerative Invariants: Gromov–Witten, Donaldson–Thomas, and Open Theory

Local toric Calabi–Yau threefolds admit powerful techniques for computing both closed and open enumerative invariants:

• Closed Gromov–Witten invariants count holomorphic maps from closed Riemann surfaces to $X$7, and are accessible via toric localization. For example, for the resolved conifold, genus-zero Gromov–Witten invariants with curve class $X$8 yield

$X$9

(Lau et al., 2010).

• Open Gromov–Witten invariants evaluate virtual counts of bordered holomorphic curves (disks) with boundary on Lagrangian torus fibers. A crucial result is the reduction of these open invariants, via Chan’s comparison and birational modifications (blow-up and flop), to closed Gromov–Witten invariants of a birational model:

$X = \operatorname{Tot}_S(K_S)$0

where $X = \operatorname{Tot}_S(K_S)$1 is the transform of a base class in $X = \operatorname{Tot}_S(K_S)$2, and $X = \operatorname{Tot}_S(K_S)$3 is a projective bundle over a blown-up surface (Lau et al., 2010).

• Donaldson–Thomas invariants (DT), both classical and generalized, count ideal sheaves or stable coherent sheaves on $X = \operatorname{Tot}_S(K_S)$4 (interpreted as D0/D2/D6-brane bound states). They are computed as (weighted) Euler characteristics of sheaf moduli, often via their quiver and superpotential descriptions (Hua, 2011).

These invariants exhibit intricate wall-crossing phenomena and are interrelated via generating series, factorization identities, and motivic constructions.

4. Quiver Descriptions, Superpotentials, and Moduli

The geometry and representation theory of local toric Calabi–Yau threefolds admit a presentation in terms of quivers with superpotentials:

• Full strong exceptional collections of line bundles on $X = \operatorname{Tot}_S(K_S)$5 yield a quiver $X = \operatorname{Tot}_S(K_S)$6 with vertices and arrows determined by the collection.
• The path algebra, augmented by relations, is realized as $X = \operatorname{Tot}_S(K_S)$7, and its derived equivalence gives an explicit link between the geometry and algebraic representations.
• For the total space $X = \operatorname{Tot}_S(K_S)$8, the quiver is expanded to account for higher Ext-groups, and a cyclic superpotential $X = \operatorname{Tot}_S(K_S)$9 is constructed, often as a trace of commutators or cyclic words in the arrows (e.g., for $S$0, the superpotential is given explicitly in (Hua, 2011)).

The moduli stack of coherent sheaves is identified with the critical locus of $S$1 modulo gauge equivalence, providing a concrete approach to moduli spaces, their virtual motives, and DT theory. Dimension reduction formulas further relate the virtual motives to simpler (framed) quiver moduli (Hua, 2011).

5. Integrable Hierarchies, Topological Vertex, and Partition Functions

Enumerative invariants and partition functions of local toric Calabi–Yau threefolds are encapsulated in generating functions with deep integrable structure:

• The topological vertex formalism expresses all-genus Gromov–Witten invariants in terms of combinatorics of partitions and Schur functions, interpreted geometrically as interactions of D-brane boundary conditions along the external legs of the toric diagram (Wang et al., 13 Nov 2025).
• Representation-theoretically, the total partition function is realized as the trace of an operator in the fermionic Fock space:

$S$2

and it can be expanded as a tau-function of the multi-component KP hierarchy, substantiating predictions from the physics literature regarding the relationship between topological strings and integrable systems.

• The determinant formulae and Plücker relations associated with the Schur expansion ensure satisfaction of the Hirota bilinear equations, and the construction allows explicit extraction of all genus Gromov–Witten invariants, including nontrivial fermion number fluxes through cycles of the toric diagram (Wang et al., 13 Nov 2025).

This correspondence gives a profound unification between string theory partition functions and classical soliton equations, opening avenues for algorithmic computation of enumerative invariants.

6. Modular Structure and Topological Recursion

For those local toric Calabi–Yau threefolds whose mirror curves have higher genus (notably genus two), enumerative potentials acquire an additional modular structure:

• The Eynard–Orantin topological recursion enables recursive computation of multi-point correlators $S$3 using the Bergman kernel on the mirror curve and produces free energies $S$4 corresponding to higher-genus Gromov–Witten potentials.
• These invariants are shown to be quasi-Siegel modular forms (closed sector) or quasi-Siegel Jacobi forms (open sector) of genus two, with explicit differential ring structure and finite-generation properties. For example, under the mirror map, $S$5 is a Siegel modular function of weight $S$6, and $S$7 is (up to anomaly) a holomorphic Siegel modular form of weight zero (Ruan et al., 2019).
• The modularity provides algebraic and analytic constraints on the generating functions, relates them to classical automorphic forms, and enables the computation of periods, theta constants, and holomorphic anomaly equations.

This structure underlies the universal features of local mirror symmetry and the enumerative geometry of local toric Calabi–Yau threefolds.

7. Wall-Crossing, BPS Spectra, and BPS Quivers

The enumerative invariants and moduli spaces also exhibit wall-crossing phenomena, governed by the behavior of BPS spectra and their quivers:

• In the mirror, the charge lattice $S$8 of the mirror curve $S$9 supports central charges for D0 and D2 brane cycles. Jumping loci in the parameter space correspond to wall-crossing of framed or unframed BPS indices $\Sigma \subset N_{\mathbb R}$0 (Banerjee et al., 2019).
• Wall-crossing is realized via explicit K-wall symplectomorphisms acting on abelian or nonabelian flat connections, with generating functions for DT invariants matching predictions from the topological vertex and noncommutative Donaldson–Thomas theory.
• At degenerate points (the “Roman locus”) all BPS central charges align, leading to combinatorial objects termed exponential BPS graphs. These directly encode the structure of the associated BPS quiver and its superpotential, with nodes and arrows reconstructed from the ribbon graph, and cyclic words contributing to the superpotential governing bound states.

These results establish a precise dictionary between soliton data, wall-crossing, modular generating functions, and the representation theory of quivers and their potentials for local toric Calabi–Yau threefolds (Banerjee et al., 2019, Hua, 2011, Wang et al., 13 Nov 2025).