The A-Model and Gromov-Witten Theory - Key Properties

Gromov-Witten theory exhibits remarkable structural properties that make it amenable to computation and reveal deep connections with other areas of mathematics. These properties encode both geometric intuition and algebraic formalism.

Quantum Cohomology

The quantum cohomology ring deforms the classical cup product by including quantum corrections from curves.

DefinitionQuantum Product

The quantum product $\alpha \star_q \beta$ on $H^*(X)$ is defined by: $(\alpha \star_q \beta, \gamma) = \sum_{\beta, k\geq 0} \frac{q^\beta}{k!} \langle\alpha, \beta, \gamma, t, \ldots, t\rangle_{0,\beta,k+3}$

where $(,)$ is the Poincaré pairing and $t$ represents additional insertions.

For $q=0$ , the quantum product reduces to the classical cup product. The quantum corrections arise from threepointin variant counting genus zero curves through three cycles. This product is associative and commutative, forming the quantum cohomology ring $QH^*(X)$ .

Remark

The quantum cohomology ring encodes the operator product expansion (OPE) in the A-model topological field theory. Associativity of the quantum product corresponds to the WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equations, a system of nonlinear PDEs satisfied by Gromov-Witten invariants.

WDVV Equations

The associativity of the quantum product imposes stringent constraints on Gromov-Witten invariants.

DefinitionWDVV Equations

The genus zero Gromov-Witten potential $F^{(0)}(t)$ satisfies the WDVV equations: $\frac{\partial^3 F^{(0)}}{\partial t^i \partial t^j \partial t^a} \eta^{ab} \frac{\partial^3 F^{(0)}}{\partial t_b \partial t^k \partial t^l} = \frac{\partial^3 F^{(0)}}{\partial t^i \partial t^k \partial t^a} \eta^{ab} \frac{\partial^3 F^{(0)}}{\partial t_b \partial t^j \partial t^l}$

where $\eta^{ab}$ is the inverse Poincaré metric.

These equations dramatically constrain the structure of Gromov-Witten invariants, allowing recursive computation of higher point functions from lower ones.

Divisor and Point Axioms

The Gromov-Witten invariants satisfy fundamental axioms that reduce computations to simpler cases.

The divisor axiom states that for a divisor $D \in H^2(X)$ : $\langle\alpha_1, \ldots, \alpha_k, D\rangle_{g,\beta} = \left(\int_\beta D\right) \langle\alpha_1, \ldots, \alpha_k\rangle_{g,\beta}$

The point axiom says insertions of the fundamental class can be removed: $\langle\alpha_1, \ldots, \alpha_k, [X]\rangle_{g,\beta} = \langle\alpha_1, \ldots, \alpha_k\rangle_{g,\beta}$ .

Virtual Fundamental Class Properties

The virtual fundamental class construction via derived algebraic geometry has specific properties.

Remark

For Calabi-Yau threefolds, the moduli space of curves typically has the correct expected dimension due to vanishing obstruction theory. The virtual class often equals the actual fundamental class when the moduli space is smooth of expected dimension.

The virtual class respects natural morphisms:

Gluing maps: When curves degenerate, the virtual class on the boundary matches predictions from gluing
Forgetful maps: Forgetting marked points induces push-forwards compatible with virtual classes

Gopakumar-Vafa Invariants

For Calabi-Yau threefolds, Gromov-Witten invariants can be reorganized into more fundamental integer invariants.

DefinitionGopakumar-Vafa Invariants

The Gopakumar-Vafa (GV) invariants $n_{\beta}^g$ are integers defined by reorganizing the generating function: $\sum_{g,\beta} N_{g,\beta} q^\beta \lambda^{2g-2} = \sum_{\beta,g,k} \frac{n_\beta^g}{k}(2\sin(k\lambda/2))^{2g-2} q^{k\beta}$

where $N_{g,\beta}$ are reduced Gromov-Witten invariants.

The GV invariants are conjectured to count BPS states in string theory and often exhibit simpler structure than raw Gromov-Witten invariants. They satisfy integrality and positivity properties consistent with physical interpretation as counting supersymmetric solitons.