Quantum Moduli Space

• Quantum moduli space is a parameter space that classifies quantum objects (e.g., quantum toric stacks, instantons) and generalizes classical moduli by incorporating deformation and quantization parameters.
• It features intricate geometric and combinatorial structures such as secondary fans and wall-crossing phenomena that enable transitions between different moduli strata.
• These spaces bridge noncommutative algebraic geometry, supersymmetric gauge theory, and quantum field theory, offering insights into deformations, stability, and quantum corrections.

A quantum moduli space is a parameter space whose points classify isomorphism classes of quantum (noncommutative, quantized, or stack-theoretic) objects—such as quantum toric stacks, noncommutative deformations of varieties, quantum gauge theory vacua or instantons, and moduli associated to quantum invariants—frequently generalizing or extending classical moduli spaces by relaxing integrality or rationality constraints, or by incorporating deformation and quantization parameters. Quantum moduli spaces arise naturally in the context of noncommutative algebraic geometry, higher representation theory, supersymmetric gauge theory, deformation quantization, and string/M-theory, and are characterized by intricate geometric, combinatorial, and categorical structures.

1. Quantum Toric Stacks and Their Moduli

Quantum toric stacks generalize classical toric varieties by replacing the rational lattice $N \cong \mathbb{Z}^d$ with a finitely generated subgroup $\Gamma \subset \mathbb{R}^d$ of full real rank, together with a fan $\Delta$ of strongly convex polyhedral cones generated by elements of $\Gamma$, and a calibration $h:\mathbb{Z}^n \twoheadrightarrow \Gamma$ with a set $I$ of "virtual generators" (generators whose images may lie in the span of the others) (Boivin, 2 Apr 2025). The data $(\Delta,h,I)$ defines a quantum fan, and the associated analytic stack is

$X_{\Delta, h, I} = [S(\Delta)/\mathbb{C}^{n-d}]$

where $S(\Delta)$ is a union of open sets modeled on $\mathbb{C}^n$ corresponding to maximal cones, and the $\mathbb{C}^{n-d}$ action is given by a Gale transform.

The quantum moduli space of toric stacks with fixed combinatorial type $D$ is the orbifold

$$\mathcal{M}_{d,n,D} = [\Omega(D)/\operatorname{Aut}_{\mathrm{Poset}}(D)]$$

where $\Omega(D) \subset \mathbb{R}^{d(n-d)}$ parametrizes calibrations $h$ realizing type $D$, and $\operatorname{Aut}_{\mathrm{Poset}}(D)$ is the finite group of automorphisms of the poset $D$. This space is typically of dimension $d(n-d)$ and connected, potentially with orbifold structure or stacky singularities depending on $D$ (Boivin, 2023).

The big moduli space is constructed by gluing local moduli of all combinatorial types via the secondary fan, yielding a global compact topological stack $K(d,n)$ containing all compactified moduli $\overline{\mathcal{M}_{d,n,D}}$ along with their wall-crossing loci and combinatorial degenerations (Boivin, 2 Apr 2025).

2. Geometry and Combinatorics: The Secondary Fan and Wall-Crossing

The quantum Geometric Invariant Theory (GIT) parameter space is organized via continuous weights $\chi \in \mathbb{R}^{n-d}$, corresponding to polytopes

$$P_b = \{x \in \mathbb{R}^d ~|~ \langle x, h(e_i) \rangle \geq -b_i\}$$

which are nonempty if $\chi$ lies in a certain cone and full-dimensional on its interior (open "admissible region" $U^{adm}$). The secondary fan $Sec(h)$ partitions $U^{adm}$ into chambers $\Gamma_{(\Delta, D)}$ labeled by combinatorial types, with their codimension-1 walls encoding transitions (via "elementary" birational modifications) between different moduli strata (Boivin, 2 Apr 2025).

Two canonical wall types arise:

• Divisorial walls: Crossing such a wall corresponds to a blow-up or blow-down along a torus-invariant divisor; the exceptional locus is codimension one.
• Flipping walls: Crossing such a wall realizes a "flip" transition—small birational modifications with exceptional locus of codimension at least two.

These birational correspondences define the transition maps for gluing the local moduli, reflecting classical phenomena in toric flips and their quantum analogues. The structure mirrors wall-crossing behavior in non-Kähler LVM/LVMB manifolds (Boivin, 2 Apr 2025).

3. Examples and Explicit Constructions

Quantum toric moduli spaces generalize classic moduli spaces even in low dimensions:

• For $(d,n) = (2, n)$ (e.g., Hirzebruch or Delzant fans), $\Omega(C_n)$ is contractible and $\mathcal{M}_{2,n,C_n} \simeq B D_n$ with $D_n$ the dihedral group.
• For $\mathbb{P}^d$, $\Omega(S_d) = \mathbb{R}^d_{<0}$ and $$\mathcal{M}_{d,d+1,S_d} = [\mathbb{R}^d_{<0} / S_{d+1}]$$, classifying quantum analogues of projective space (Boivin, 2023).

Quantum toric stacks admit continuous deformations through "irrational" choices of calibration, and their local and global moduli carry bundle structures over classical moduli. In the special case where $\Gamma$ is a lattice, the quantum moduli reduce to classical counterparts—usually as a discrete locus inside a much larger quantum parameter space.

4. Noncommutative and Quantum Deformation Moduli

A distinct but related notion of quantum moduli spaces arises in the study of noncommutative deformations of schemes, particularly for local Calabi-Yau threefolds $W_k$ (Ballico et al., 2023). Given a holomorphic Poisson structure $\pi$, deformation quantization yields a sheaf of algebras $\mathcal{A}_\hbar$ deforming $\mathcal{O}_X$, and one considers moduli of $\mathcal{A}_\hbar$-modules (quantum vector bundles). The quantum moduli space $M_j^\hbar$ of modules of fixed splitting type $j$ is pro-represented by a formal scheme, whose geometry (dimension, fiber rank, singular stratification) depends sensitively on $\pi$, typically realized as a constructible sheaf over the classical moduli $M_j(W_k)$. For "basic" Poisson structures, quantum and classical moduli coincide, whereas "extremal" Poisson structures generate a hierarchy of singularities and rank jumps in moduli fibers (Ballico et al., 2023).

5. Quantum Moduli Spaces in Gauge and Field Theories

Quantum moduli spaces also describe the spectrum of vacua, instantons, or domain walls in quantum field and string theory contexts:

• In $\mathcal{N}=2$ Chern–Simons quiver gauge theories, the quantum moduli space encompasses configurations of monopole operators and matter fields subject to quantum-corrected F- and D-term relations; its structure encodes classical vacua, quantized charges, and geometric branches isomorphic to symmetric products of Calabi-Yau cones and their resolutions (Benini et al., 2011).
• In dimensional reduction over quantum homogeneous spaces such as $M \times S^2_q$, vacuum and instanton moduli spaces emerge as $q$-deformed hyper-Kähler quotients, whose geometry is controlled by $q$-dependent moment maps. These moduli spaces exhibit enhanced compactness and smoothness, with additional stability constraints compared to the undeformed case (Landi et al., 2010).
• In the context of BPS solitons and domain walls, quantum corrections induce potentials on classically flat moduli spaces, breaking classical degeneracies and generating quantum forces that selectively stabilize certain configurations depending on coupling parameters (Alonso-Izquierdo et al., 2013).
• For conformal field theories and their limits, Soibelman describes a noncommutative (quantum) Riemannian moduli space, formulated as a category of functors from metric bordisms to Hilbert spaces, with topology defined via strong convergence and quantum Gromov-Hausdorff distance, capturing both commutative and noncommutative geometries (Soibelman, 1 Jun 2025).

6. Quantum Moduli Associated to Invariants and Geometric Quantization

Certain quantum moduli spaces parametrize the data underlying quantum topological invariants, representation varieties, or Wigner quantization:

• The moduli space $\mathcal{M}(G)$ associated to a planar trivalent graph $G$ encodes equivariant decorations corresponding to representations of the fundamental group of $S^3 \setminus G$ into $SU(N)$, constrained by conjugacy to fixed type elements. Its Euler characteristic coincides with the evaluation at $q=1$ of the quantum $sl(N)$ MOY polynomial of $G$ (Lobb et al., 2012).
• For Wigner quasiprobability distributions in $N$-level quantum systems, the moduli space of Stratonovich–Weyl kernels is the intersection of a coadjoint $SU(N)$ orbit with a unit $(N-2)$-sphere, yielding a spherical "polytope" parameterizing inequivalent quantum distributions (Abgaryan et al., 2018).

7. Compactification, Wall-Crossing, and Global Properties

Quantum moduli spaces often admit natural compactifications by inclusion of degenerate or "boundary" objects defined via combinatorial or geometric limits. For quantum toric stacks, compactification is achieved by extending the parameter space to the closure of calibration data in appropriate Grassmannians and attaching boundary strata corresponding to degenerations of the normal fan (Boivin, 2023, Boivin, 2 Apr 2025). Wall-crossing phenomena—transitioning between chambers of the secondary fan—correspond to birational morphisms such as flips and blowups, with quantum analogues exhibiting small or divisorial modifications, and combinatorial transition maps defined globally on the moduli stack. The resulting big moduli space is connected and contains all compactified local moduli spaces as locally closed substacks, glued precisely along their secondary-fan walls (Boivin, 2 Apr 2025).

These structures enable the systematic study of quantum deformations, nonrational and noncommutative phenomena, stability conditions, and universal behavior pertinent for mirror symmetry, cluster theory, and integrability, and render quantum moduli spaces fundamental objects bridging classical algebraic geometry, noncommutative geometry, and quantum field theory.