Quantum Multiplication for Hypertoric Varieties

• Quantum multiplication for hypertoric varieties is defined as a deformation of classical intersection theory through genus-zero Gromov–Witten invariants and explicit circuit relations.
• The construction uses a hyperkähler quotient and combinatorial hyperplane arrangements to generate a cohomology ring whose relations encode essential divisor classes.
• Mirror symmetry is established via GKZ-type hypergeometric differential systems, linking quantum connections with period integrals and revealing enumerative geometrical insights.

Quantum multiplication for hypertoric varieties refers to the operation in their (equivariant) quantum cohomology ring, encoding the deformation of classical intersection theory by contributions of genus-zero Gromov–Witten invariants associated to rational curves. For smooth hypertoric varieties, this structure exhibits a remarkable dependence on the combinatorial data of the associated hyperplane arrangement, is captured by explicit algebraic relations among divisor classes, and possesses a mirror-theoretic interpretation via hypergeometric-type differential systems. The interplay between quantum multiplication, the structure of circuits in the arrangement, and mirror symmetry forms the foundation for computational and theoretical advancements in this area.

1. Construction of Hypertoric Varieties and Classical Relations

A smooth hypertoric variety $M$ arises as a hyperkähler quotient $M = \mu_K^{-1}(\lambda) // T^k$, with $\mu_K$ the moment map for the torus $T^k$ acting on $T^*\mathbb{C}^n$. Choosing appropriate stability data ensures $M$ is smooth, equipped with residual symmetries from a torus $T^d$ and a scaling $\mathbb{C}^*$-action by $\hbar$. The equivariant cohomology ring is generated by divisor classes $u_1,\ldots,u_n$ associated to the coordinate hyperplanes.

The key combinatorial feature is the hyperplane arrangement: for each "circuit" $S$ (minimal subset of indices such that the intersection of hyperplanes $H_i$ for $i\in S$ is empty), the classical relations read: $$\prod_{i\in S^+} u_i \cdot \prod_{j\in S^-} (\hbar - u_j) = 0$$ with $S = S^+ \sqcup S^-$ determined by a sign convention. This presentation realizes the cohomology as a Stanley–Reisner-type ring with specific circuit relations controlling the product structure.

2. Quantum Multiplication: Algebraic Formulation

Quantum cohomology deforms the classical cup product by quantum corrections indexed by the primitive effective curve classes, themselves determined by circuits in the hyperplane arrangement. The genus-zero Gromov–Witten invariants provide these corrections. The quantum product by a divisor $u$ is given by: $$u * (-) = u \cup (-) + \hbar \sum_S \frac{q^S}{1-q^S} (u, \beta_S) \cdot L_S(-)\tag{1}$$ where:

• $q^S = (-1)^{|S|} q^{\beta_S}$ encodes the quantum deformation parameter for the circuit $S$ and associated curve class $\beta_S$,
• $(u, \beta_S)$ is the pairing of the class $u$ with the curve $\beta_S$,
• $L_S$ is a Steinberg correspondence arising from the specialized two-point Gromov–Witten invariant along a subvariety $P^S$ (a $\mathbb{P}^{|S|-1}$-bundle).

The quantum product is not simply the classical cup product: the sum runs over all circuits, and each term encodes enumerative data of rational curves related to the arrangement configuration.

The quantum cohomology relations deform the classical circuit relations to: $$\prod_{i\in S^+} u_i * \prod_{j\in S^-} (\hbar - u_j) = q^{\beta_S} \prod_{i\in S^+} (\hbar - u_i) * \prod_{j\in S^-} u_j\tag{2}$$ with $*$ indicating the quantum product. Setting $q^{\beta_S}=0$ recovers the classical situation.

3. Quantum Differential Equation and Mirror Formula

Quantum multiplication endows the cohomology with a quantum connection, formulated as: $$\nabla_i = q_i \frac{\partial}{\partial q_i} + u_i *$$ for chosen coordinates $q_i$ on the torus dual to $H^2(M)$ (with $q_i = \exp(2\pi i\langle e_i, \tilde{v}\rangle)$). The flat sections of this connection solve the quantum differential equation (QDE), which is a GKZ-type hypergeometric system determined by the arrangement.

The mirror theorem constructs a mirror family of complex manifolds: $$M_q = (T^d)^\vee \setminus \bigcup_{i\in A} \{ t \mid q_i t^{a_i} = -1 \}$$ with $(T^d)^\vee$ the dual torus and $q_i t^{a_i} = -1$ defining multiplicative hyperplanes. A $\mathbb{C}$-local system $\mathcal{L}$ with monodromy $\hbar$ winds around these hyperplanes.

Period integrals

$$J_\gamma(q) = \int_\gamma \Omega, \quad \text{with}\quad \Omega = \prod_{i\in A} (1 + q_i t^{a_i})^\hbar \prod_{j=1}^d t_j^{-c_j} \frac{dt_j}{t_j}$$

span the solution space of the QDE, with the critical locus of the "mirror superpotential" extracted from $\Omega$ encoding the structure constants for quantum multiplication.

The theorem states the Gauss–Manin connection on $H^d(M_q, \mathcal{L})$ coincides with the quantum connection on $H^*_{\mathbb{C}}(M, \mathbb{C})$, realizing quantum multiplication as the categorical mirror to variation of complex structures on $M_q$.

4. Steinberg Correspondences and Circuit Contributions

For every circuit $S$, the map $L_S$ emerges from specializing the two-point Gromov–Witten invariants along $P^S$. The subvariety $P^S$ is a projective bundle $\mathbb{P}^{|S|-1}$ and its fibers reflect the circuit’s geometry. These Steinberg correspondences implement the quantum correction terms in the operator formula for quantum multiplication (1). The explicit dependence of quantum multiplication on the arrangements of hyperplanes is manifested in the sum over $S$ and the associated structure of $P^S$.

5. Summary and Implications

The quantum multiplication operation for hypertoric varieties is characterized by:

• A ring presentation with generators and circuit-modified relations,
• Operator formulas encoding divisor quantum multiplication via enumerative corrections,
• Deep combinatorial dependence on the hyperplane arrangement,
• Mirror symmetry interpretation via a hypergeometric system and period integrals,
• Global structure constants identified with critical loci of mirror superpotentials.

This construction ensures that both classical intersection theory and quantum correction data are encapsulated within the quantum cohomology ring. The mirror theorem bridges quantum geometry and variation of complex structures, while the GKZ formalism offers computational tools for extracting quantum multiplication structure constants and exploring degenerations or wall-crossing phenomena. These results provide a uniform, fully explicit framework for understanding quantum multiplication and enumerative geometry in hypertoric and related symplectic algebraic varieties.