The A-Model and Gromov-Witten Theory - Applications

Gromov-Witten theory has far-reaching applications beyond pure enumerative geometry, connecting to topology, physics, and representation theory. These applications demonstrate the theory's fundamental role in modern mathematics.

Quantum Cohomology and Floer Theory

Quantum cohomology provides a deformation of classical cohomology rings that encodes symplectic topology.

TheoremQuantum Cohomology as Frobenius Manifold

The quantum cohomology $QH^*(X)$ equipped with the Gromov-Witten potential forms a Frobenius manifold structure. This includes:

A flat metric (the Poincaré pairing)
A compatible multiplication (quantum product)
A unit vector field
An Euler vector field

satisfying the WDVV equations and associativity constraints.

This structure appears in various contexts: singularity theory, integrable systems, and topological field theories. The Frobenius manifold axioms encode the algebraic-geometric content of 2D topological field theory.

Mirror Symmetry Verification

The most celebrated application is verifying mirror symmetry predictions.

ExampleQuintic Mirror Symmetry Proof

Givental (1996) and Lian-Liu-Yau (1997) rigorously proved that Gromov-Witten invariants of the quintic threefold match period calculations on the mirror. The proof uses:

Localization to compute quantum differential equation
Birkhoff factorization to relate it to Picard-Fuchs equation
Analytic continuation matching the mirror map

This confirmed the 1991 predictions of Candelas-de la Ossa-Green-Parkes.

Donaldson-Thomas Theory

Gromov-Witten invariants connect to Donaldson-Thomas invariants counting ideal sheaves.

TheoremGW/DT Correspondence

For a Calabi-Yau threefold $X$ , the generating series of Gromov-Witten and Donaldson-Thomas invariants are related by: $Z_{\text{GW}}(X;q,\lambda) = M(-q)^{\chi(X)/2} \cdot Z_{\text{DT}}(X;q,\lambda)$

where $M(q) = \prod_{n\geq 1}(1-q^n)^{-n}$ is the MacMahon function.

This profound relationship connects curve counting with sheaf counting, two a priori different enumerative theories on Calabi-Yau threefolds.

Topological String Theory

In physics, Gromov-Witten invariants compute topological string amplitudes.

Remark

The topological A-model partition function on a Calabi-Yau $X$ is: $Z_A(X; t, g_s) = \exp\left(\sum_{g=0}^\infty g_s^{2g-2} F_g(t)\right)$

where $F_g$ are the genus $g$ Gromov-Witten potentials and $g_s$ is the string coupling. The genus zero part $F_0$ is the quantum cohomology prepotential.

Higher genus contributions $F_g$ encode quantum corrections and satisfy the BCOV holomorphic anomaly equations, providing recursive computational tools.

Integrable Hierarchies

Gromov-Witten invariants generate solutions to integrable PDEs.

TheoremDubrovin-Zhang Hierarchy

The genus zero Gromov-Witten potential of a Fano variety generates a solution to an integrable hierarchy (Toda, KdV, or generalized Witten-Kontsevich hierarchies depending on the geometry).

The tau-function of the hierarchy equals the total descendant potential: $\tau(t) = \exp\left(\sum_{g,k} \frac{\hbar^{g-1}}{k!} \langle\tau, \ldots, \tau\rangle_{g,k}\right)$

This connection imports tools from integrable systems into enumerative geometry and vice versa.

Algebraic Curve Counting

Gromov-Witten theory provides modern foundations for classical curve-counting problems.

Example27 Lines on a Cubic Surface

The classical result that every smooth cubic surface in $\mathbb{P}^3$ contains exactly 27 lines can be recovered as the Gromov-Witten invariant: $\langle H, H, H, H\rangle_{0,1} = 27$

where $H$ is the hyperplane class and we count degree 1 curves (lines) through four points.

These applications show Gromov-Witten theory unifies classical enumerative geometry with modern quantum and topological methods.