Quantum Geometric Invariant Theory

• Quantum GIT is a framework integrating geometric invariant theory and genus-zero quantum cohomology to compute enumerative invariants of quotient spaces.
• It employs tools like the quantum Kirwan map, localization, and abelianization to translate equivariant data into concrete geometric results.
• The framework finds applications in noncommutative toric geometry and birational wall-crossing, bridging classical and quantum insights in modern geometry.

Quantum Geometric Invariant Theory (GIT) is an overview of geometric invariant theory and quantum (Gromov–Witten) geometry, providing a framework for relating enumerative invariants of quotient varieties and stacks to equivariant invariants of the original spaces. Quantum GIT extends the classical theory by incorporating genus-zero Gromov–Witten theory, quantum cohomology, and wall-crossing structures, with applications spanning from the study of quantum cohomology rings of GIT quotients to noncommutative toric geometry and wall-crossing phenomena in birational geometry (Gonzalez et al., 2008, Gonzalez et al., 2012, Katzarkov et al., 2020).

1. Foundations and Key Structures

Let $G$ be a connected complex reductive group acting linearly via a linearization $L$ on a smooth projective or convex quasi-projective variety $X$. Classical GIT constructs the quotient $X^{ss}(L)//G$ (where $X^{ss}(L)$ is the semistable locus) and relates the $G$-invariance in sheaf cohomology to geometric properties of quotients. Quantum GIT enriches this framework as follows:

• Equivariant Quantum Cohomology: For a given linearization, the equivariant quantum cohomology ring $QH_G(X)$ is defined over the equivariant Novikov field

$$\Lambda_G = \left\{ \sum_{d \in H_2^G(X;\mathbb{Z})} c_d q^d \;\middle|\; c_d \in H_G^*(pt),\ \forall C>0\ \#\{\langle\omega, d\rangle \leq C\} < \infty \right\}$$

with the genus-zero quantum product determined by equivariant Gromov–Witten invariants.

• Quantum Kirwan Map: The map $K_{X,G}: QH_G(X) \to QH(X//G)$ generalizes the classical Kirwan map, providing the transition from equivariant quantum cohomology to the quantum cohomology of the GIT quotient.
• Graph (Givental) Potentials and J-Functions: Fundamental solutions to quantum differential equations (qde) are captured via J-functions, encoding enumerative data in their expansion and fulfilling the Dubrovin–Givental connection.

This framework supports wall-crossing analysis, reduction to maximal tori, and twisted invariants, enabling the study of quantum invariants under variation of GIT data (Gonzalez et al., 2008, Gonzalez et al., 2012).

2. Quantum Localization and Wall-Crossing

Quantum GIT establishes a broad generalization of Witten's localization and wall-crossing principles:

• Quantum Witten Localization: For a smooth projective $X$ with a reductive $L$0-action, the quantum Witten localization theorem relates Gromov–Witten graph integrals on the quotient to virtual residues over fixed loci of certain stratified moduli spaces. Specifically,

$L$1

where $L$2 is the graph potential for $L$3, $L$4 the equivariant Witten trace, $L$5 the quantum Kirwan map, and each fixed-point contribution is computed via virtual localization on moduli of Mundet–semistable gauged maps with structure group centralizer $L$6 (Gonzalez et al., 2008).

• Wall-Crossing and Crepant Transformation: Upon varying the polarization, the quantum wall-crossing formula compares Gromov–Witten invariants of GIT quotients related by a change in stability. For crepant transformations, the difference in graph potentials manifests as a distribution in the quantum parameter that is zero almost everywhere—a quantum manifestation of the crepant transformation conjecture (Gonzalez et al., 2012).

3. Quantum Abelianization and Reduction Techniques

Analogous to the classical Martin formula, quantum abelianization expresses invariants for nonabelian quotients in terms of those for maximal tori:

• Quantum Martin Formula: For maximal torus $L$7 with Weyl group $L$8, the quantum formula states

$L$9

where $X$0 is the Euler class of the adjoint bundle (Gonzalez et al., 2008). The formula maintains equivariant Gromov–Witten theoretic data, with Novikov variables permuted by Weyl group action.

• Abelianization for qde-Solutions: For localized J-functions, the theory yields

$X$1

up to Novikov variable shifts determined by root data.

This framework enables explicit calculation of invariants for GIT quotients by nonabelian groups via abelian localization techniques, substantially generalizing classical theory.

4. Quantum Lefschetz Principles and Holomorphic Symplectic Reductions

The quantum Lefschetz theorem quantifies the impact of passing to zero loci of equivariant maps:

• Quantum Lefschetz Principle: If $X$2 is the zero locus of a $X$3-equivariant map $X$4, with $X$5 a representation, and $X$6 is smooth and proper, then

$X$7

where $X$8 is the Euler-twisted graph potential, and at the quantum cohomology level,

$X$9

demonstrating that the quantum theory of $X^{ss}(L)//G$0 is obtained as an Euler twist of the ambient $X^{ss}(L)//G$1 theory (Gonzalez et al., 2008).

In holomorphic symplectic reduction—such as Nakajima quiver varieties—quantum Lefschetz facilitates reduction of enumerative data to more computable structures.

5. Quantum GIT for Noncommutative and Toric Geometries

Quantum GIT has been extended to noncommutative and quantum toric contexts:

• Quantum Toric Varieties and QGIT: Quantum toric geometry generalizes the classical theory by replacing tori with quantum tori. The construction of quantum toric stacks $X^{ss}(L)//G$2 via gluing of local quotients $X^{ss}(L)//G$3 shows categorical equivalence between quantum fans and quantum toric stacks. The QGIT construction realizes $X^{ss}(L)//G$4 as a global quotient $X^{ss}(L)//G$5, directly paralleling the Cox construction for classical toric varieties, but with non-commutative structure manifested via stacky and groupoid methods (Katzarkov et al., 2020).
• Moduli and Homological Invariants: Moduli spaces for quantum toric varieties are constructed as real orbifolds, with periodic cyclic homology yielding flat local systems over these moduli families, aligning with the classical paradigm while encompassing broader non-algebraic features.

6. Examples and Explicit Constructions

Two central examples illustrating the computational power of quantum GIT are:

Example	Description	Quantum GIT Technique
Moduli of $X^{ss}(L)//G$6 Points on $X^{ss}(L)//G$7	$X^{ss}(L)//G$8; GIT quotient analyzed using quantum Kirwan, abelianization, and I-functions	T-localization, mirror symmetry
Smoothed Moduli of Framed Sheaves on $X^{ss}(L)//G$9	Nakajima quiver variety as symplectic reduction; application of quantum Lefschetz and I-function formulae	Quantum Witten localization + Lefschetz

In both cases, explicit formulas for I- and J-functions emerge, enabling direct computation of fundamental solutions to the quantum differential equations associated to the quotient stacks or varieties (Gonzalez et al., 2008).

7. Applications, Wall-Crossing, and Future Directions

Quantum GIT provides a powerful toolkit for:

• Birational Wall-Crossing: Quantum Kalkman formulas precisely describe the structure of wall-crossing for Gromov–Witten invariants as polarizations change, capturing the enumerative content of birational geometry and crepant transformations. Under crepant wall-crossings, the difference in invariants becomes supported at roots of unity in the quantum parameter, aligning with predictions of the crepant transformation conjecture (Gonzalez et al., 2012).
• Intersection with Symplectic Geometry and Representation Theory: Applications include calculations of quantum cohomology rings of moduli spaces (points on projective lines, quiver varieties), and computations for cases relevant to mathematical physics, enumerative geometry, and symplectic reductions.
• Extension to Quantum and Noncommutative Settings: Generalization to quantum toric stacks and their moduli indicates the adaptability of quantum GIT beyond the classical algebraic paradigm toward noncommutative geometry (Katzarkov et al., 2020).

A plausible implication is that quantum GIT will continue to underpin the study of quantum invariants across birational and noncommutative geometries, further connecting enumerative invariants, wall-crossing behavior, and generalized moduli structures.