Quantum Toric Stacks: Geometric & Moduli Insights

• Quantum toric stacks are stack-theoretic generalizations of toric varieties that use quantum lattices to incorporate irrational generators and noncommutative structures.
• They extend classical toric geometry by employing analytic stacks, categorical equivalences, and controlled wall-crossing phenomena to model birational transitions.
• Their framework supports rich moduli spaces, mirror symmetry applications, and quantum cohomology analysis, unveiling deep insights into complex geometric structures.

A quantum toric stack is a stack-theoretic generalization of a toric variety in which the underlying combinatorial and geometric data allow non-rational, finitely generated subgroups—called “quantum lattices”—instead of the integral lattices characteristic of classical toric geometry. This extension admits irrational rays, non-commutative structures, and a much richer deformation and moduli theory than classical toric stacks, while preserving the local and global combinatorial tractability that makes toric geometry powerful for explicit computations. The formalism accommodates both analytic and categorical enhancements, interfaces effectively with the minimal model program and mirror symmetry, and provides a systematic framework for wall-crossing, birational transitions, and the study of moduli spaces with combinatorial stratifications.

1. Foundational Structure: Quantum Fans, Calibrations, and Stacks

The classical construction of a toric variety employs a rational simplicial fan $\Delta$ in $\mathbb{R}^d$ and a lattice $N\cong\mathbb{Z}^d$. Quantum toric geometry replaces $N$ with a “quantum lattice,” i.e., a finitely generated subgroup $\Gamma\subset\mathbb{R}^d$ such that the real span $\mathbb{R}\cdot\Gamma = \mathbb{R}^d$ holds. The key combinatorial object is a quantum fan, denoted $(\Delta,h,I)$, where:

• $\Delta$ is a collection of strongly convex polyhedral cones in $\mathbb{R}^d$ generated by elements in $\Gamma$, closed under faces and intersections;
• $h: \mathbb{Z}^n \to \Gamma$ is a calibration, recording a choice of virtual and actual generators, such that the images of the standard basis vectors $e_1,\ldots,e_d$ form a basis for (possibly irrational) $\mathbb{Z}^d\subset\Gamma$ and the “virtual generators” $I\subset\{1,\ldots,n\}$ encode additional gerbe data;
• The 1-cones of $\Delta$ are in bijection with $\mathbb{R}_{\geq 0}\cdot h(e_i)$ for $i\notin I$.

To each cone $$\sigma=\mathrm{Cone}(h(e_{i_1}),\ldots, h(e_{i_k}))$$ corresponds an analytic stack chart

$$U_\sigma = [\, \mathbb{C}^k \times (\mathbb{C}^*)^{d-k} \, / \, \mathbb{Z}^{n-d} \,],$$

with the group action defined via the exact sequence $$0\to \mathbb{Z}^{n-d} \to \mathbb{Z}^n \xrightarrow{h} \Gamma \to 0$$ plus the exponential mapping to $(\mathbb{C}^*)^n$. These local pieces glue along common substacks to form the global quantum toric stack $X_{\Delta,h,I}$, formally a colimit over the category of $\sigma\in\Delta$.

The entire construction admits a presentation via a stack quotient: $$X_{\Delta,h,I} \cong [\,S(\Delta) / \mathbb{C}^{n-d}\,],$$ where $S(\Delta)$ is the union of toric charts and the $\mathbb{C}^{n-d}$-action is specified through Gale duality and the calibration data (Boivin, 2023, Katzarkov et al., 2020, Boivin, 14 Jan 2026).

2. Categorical Equivalence and Generalization of Toric Stacks

Quantum toric stacks generalize classical toric Deligne-Mumford stacks in two critical aspects:

1. Equivalence of categories: The assignment $(\Delta,h,I) \mapsto X_{\Delta,h,I}$ provides an equivalence between the category of simplicial quantum fans (with morphisms given by compatible pairs of linear and group homomorphisms respecting calibrations and virtual generators) and the category of analytic toric stacks with prescribed dense quantum torus $[(\mathbb{C}^*)^d/\mathbb{Z}^{n-d}]$ (Boivin, 2023, Katzarkov et al., 2020, Boivin, 14 Jan 2026).
2. Non-rational phenomena: Allowing $\Gamma$ to be non-integral introduces non-Hausdorff transversals, infinite isotropy subgroups (in the irrational case), and new moduli phenomena, such as the presence of gerbe degrees of freedom and noncommutative deformations arising from non-integrality in the local torus charts (Katzarkov et al., 2020).

Classical toric stacks are precisely those quantum toric stacks for which $\Gamma$ is a lattice, $h(e_i)$ are integral, and $I=\emptyset$. In this case, the construction recovers the usual Borisov–Chen–Smith stack $[Z(\Delta)/T^d]$ (Boivin, 14 Jan 2026).

3. Moduli Spaces and Combinatorial Stratification

The moduli of quantum toric stacks is organized by fixing a combinatorial type—a poset $D$ representing the face lattice of a fan—and then considering all calibrations $h$ that realize $D$ as the set of strongly convex cones. Up to isomorphism (i.e., modulo the natural action of $GL_d(\mathbb{R})$ and $GL_n(\mathbb{Z})$), the moduli space is presented as a global quotient: $M(d,n,D) \cong [\,\Omega(D)\,/\,G\,],$ where $\Omega(D)\subset \mathbb{R}^{d(n-d)}$ is a semialgebraic set corresponding to choices of quantum lattice generators ensuring all cones are simplicial with the desired intersection pattern; $G$ is the automorphism group of the combinatorics and calibration kernel. For complete simplicial types, $\Omega(D)$ is connected and $M(d,n,D)$ is an (analytic) orbifold (Boivin, 2023, Boivin, 2 Apr 2025).

A universal quantum toric stack exists over $M(d,n,D)$, constructed as a bundle of stack quotients over $\Omega(D)$. This universal family extends (via topological closure) to a compactification $\overline{M}$ stratified by degenerations of the combinatorics—points where some cones shrink or become non-convex—yielding a real semialgebraic orbifold stratified by degenerate quantum fans (Boivin, 2023, Boivin, 2 Apr 2025).

The more refined object, the “quantum secondary fan” $\Sigma^{\rm sec}_{\rm quantum}$, organizes the gluing of all such moduli spaces across all combinatorial types, encoding wall-crossings, flips, and birational transitions between chambers (see §5 below for wall-crossing formulas) (Boivin, 2 Apr 2025).

4. Birational Geometry, Weighted Blow-ups, and Cobordisms

Quantum toric stacks possess a natural theory of birational morphisms, strictly extending the classical toric context. A birational morphism of quantum fans is specified by invertible linear maps $L:\mathbb{R}^d \to \mathbb{R}^d$, $H:\mathbb{R}^n \to \mathbb{R}^{n'}$ such that $h'\circ H = L\circ h$ and the virtual generator structure is preserved or compatibly extended. These induce isomorphisms on open substacks and generate birational maps between the stacks $X_{\Delta,h,I}$ and $X_{\Delta',h',I'}$ (Boivin, 14 Jan 2026).

Weighted blow-ups are constructed via star subdivisions along arbitrary real (possibly irrational) weight vectors $\alpha\in\mathbb{R}^{I^c}_{>0}$. For integral weights, the process reproduces the classical weighted blow-up along toric strata; for rational or irrational $\alpha$, the operation factors through root stacks or minimal stack covers, endowing the exceptional divisors with quantum toric stack structures (often quantum projective bundles of lower polytopal complexity).

Cobordism-induced morphisms are defined using one higher-dimensional quantum fans projecting to the given stacks. A quantum cobordism between $(\Delta_0,h_0,I_0)$ and $(\Delta_1,h_1,I_1)$ is a complete fan in $\mathbb{R}^{d+1}$ with projections inducing the lower-dimensional fans away from a “critical” cone. The induced birationality generalizes classical flips and connects stacks within the same birational class by zig-zags of weighted blow-ups/downs and wall-crossings.

The class of all birational morphisms forms a groupoid; any two quantum fans with the same support can be related by compositions of these moves, providing a categorical and combinatorial model for the full toric birational geometry, including irrational wall crossings (Boivin, 14 Jan 2026).

5. Wall-crossing, Secondary Fan, and Stratified Universal Moduli

The quantum secondary fan $\Sigma^{\rm sec}_{\rm quantum}$ in $\mathbb{R}^{n-d}$ controls the wall-crossing behavior and gluing of moduli spaces for quantum toric stacks with fixed numbers of rays and dimension. Each maximal cone (chamber) corresponds to a distinct combinatorial type of quantum fan; adjacent chambers share a non-simplicial refinement along a wall.

Transitions are governed by explicit wall-crossing formulas in the homogeneous coordinate description: $$z_{i} \mapsto \begin{cases} z_{i} & i \notin \{i_1,\ldots,i_r\} \ z_{i} \prod_{j \in \{i_1,\ldots,i_r\}} z_{j}^{-\langle h(e_i), h(e_j)^\vee\rangle} & i\in\{i_1,\ldots,i_r\} \end{cases}$$ where $h(e_j)^\vee$ denotes the dual basis in $\mathbb{R}^d$ (Boivin, 2 Apr 2025). This formula generalizes the GIT wall-crossing for classical toric stacks and enables the construction of a single augmented moduli stack $A(d,n)$ over which all quantum toric stacks of a given $(d,n)$ live, stratified by combinatorial type and glued along these explicit birational maps as the parameter $\chi$ moves in the secondary fan.

Exemplary computations, such as the deformation class of quantum Hirzebruch surfaces, demonstrate cycles of wall-crossings and birational maps that parameterize families of quantum stacks over the secondary fan, unifying classical and quantum moduli across polytopal subdivisions (Boivin, 2 Apr 2025).

6. Mirror Symmetry, Quantum Cohomology, and Birational Invariants

Quantum toric stacks provide a testbed for mirror symmetry and wall-crossing phenomena in quantum cohomology and derived categories. The equivariant quantum cohomology ring of a toric Deligne–Mumford stack—presented as a quantum Stanley–Reisner ring—admits an explicit Landau–Ginzburg mirror description in terms of oscillatory integrals and twisted de Rham complexes (Coates et al., 2016). The analytic structure constants and flat connections (quantum $D$-module) converge in the big equivariant setting; the mirror theorem asserts isomorphism between the quantum $D$-module and the Saito structure associated to the mirror superpotential.

Under birational transformations (including weighted blow-ups with arbitrary real weights), there exist formal decompositions of the quantum $D$-module corresponding to semiorthogonal decompositions of $K$-theory aided by analytic sectorial lifts and $\widehat{\Gamma}$-integral structures. These decompositions match the categorical semiorthogonal summands predicted by Orlov’s conjectures for classical birational geometry and extend naturally to the quantum case (Iritani, 2019).

Wall-crossing formulas for the I- and J-functions reflect the combinatorial modifications in the fan or quantum fan, while on the categorical level, wall-crossings induce semi-orthogonal decompositions in the derived categories via spherical functors associated to the birational modification (Boivin, 14 Jan 2026).

7. Notable Examples and Applications in Birational and Moduli Theory

Explicit geometric cases illustrate the full machinery:

• Quantum projective space $\mathbb{P}^d_q$: For the complete fan with $d+1$ rays, the moduli space is $[\mathbb{R}^d_{<0}/S_{d+1}]$ and its compactification is the simplex quotient, with degenerate boundary strata representing lower-dimensional polytopes (Boivin, 2023).
• Quantum Hirzebruch surfaces: Realizations of quantum fans with irrational parameters lead to a family of stacks glued along the secondary fan, exhibiting wall-crossings as explicit birational flips or blow-ups (Boivin, 2 Apr 2025).
• LVMB manifolds and non-commutative tori: When the generators of $\Gamma$ are irrational, the resulting stacks encode LVMB manifolds with non-Hausdorff transversals and infinite isotropy, displaying nontrivial foliation and quantum algebra phenomena (Katzarkov et al., 2020).

These constructions play a central role in the study of irrational toric birational phenomena, wall-crossing in mirror symmetry, and in extending the minimal model program to contexts beyond rational coefficients (Boivin, 14 Jan 2026).