Gromov-Witten Potential in Quantum Cohomology

• Gromov-Witten potential is a generating function that encodes virtual counts of holomorphic curves, forming the basis for quantum cohomology and mirror symmetry.
• Its formal power series structure, including descendant and ancestor potentials, allows reconstruction of higher-genus invariants via rigorous NF-convergence criteria.
• Givental’s quantization formalism and semisimple quantum cohomology guarantee analytic convergence, enabling applications in toric, flag varieties, and birational geometry.

The Gromov-Witten potential is the central generating function in Gromov-Witten theory, encoding the enumerative geometry of holomorphic curves on a smooth projective variety $X$. Each coefficient in its formal power series expansion is a Gromov-Witten invariant, which counts (virtually) stable maps from curves to $X$ with fixed numerical data and constrained conditions. The Gromov-Witten potentials—particularly the descendant and ancestor potentials—serve as the foundation of quantum cohomology, mirror symmetry, and the paper of moduli problems in algebraic geometry and mathematical physics. Their analytic and algebraic properties, most notably the convergent or formal nature of these series, have far-reaching implications: from the reconstruction of higher-genus invariants using only genus-zero data, to their role in the comparison of Gromov-Witten theories under birational transformations and mirror symmetry.

1. Definition and Structure of Gromov-Witten Potentials

Let $X$ be a nonsingular projective variety. The descendant genus-$g$ Gromov-Witten potential is the formal generating function: $$F_X^{(g)} = \sum_{n, d} \frac{Q^d}{n!} \langle \prod_{i=1}^n \tau_{k_i}(\phi_{\alpha_i}) \rangle_{g,n,d}$$ where $\langle \cdots \rangle_{g,n,d}$ are the genus-$g$ Gromov-Witten invariants with $n$ marked points, insertions $\tau_{k_i}(\phi_{\alpha_i})$ (descendants of cohomology classes $\phi_{\alpha_i} \in H^*(X)$), and Novikov variable $Q^d$ encodes the curve class $d \in H_2(X, \mathbb{Z})$.

The total (genus generating) descendant potential is

$$\mathcal{Z}_X = \exp\left( \sum_{g=0}^\infty \hbar^{g-1} F_X^{(g)} \right)$$

with formal variables $\hbar$ (genus), $t^a_k$ (descendant coordinates), and Novikov parameters $Q_1,\dots,Q_r$.

In parallel, the ancestor potentials are defined by similar, but slightly shifted, combinations to facilitate reconstruction results and connection with Frobenius manifold geometry.

The Taylor coefficients of these power series encode the full system of Gromov-Witten invariants of $X$, with an infinite-dimensional variable structure resulting from the countably many possible descendant insertions.

The role of Gromov-Witten potentials is foundational: they serve as generating functions for quantum products, control deformations of Frobenius manifolds, and appear as the partition functions of topological string theories.

2. Notions of Convergence in Gromov-Witten Potential Theory

In practice, Gromov-Witten potentials are initially constructed as formal power series—a product of the infinite-dimensional nature of the moduli problem and the dependence on descendant insertions and Novikov variables. However, analytic applications (such as comparing quantum invariants under mirror symmetry or birational transformations) require a rigorous notion of convergence.

The key analytic criterion introduced is NF-convergence (nuclear Fréchet convergence). Explicitly, a genus-$g$ descendant potential $F_X^{(g)}$ is NF-convergent if there exist constants $\epsilon>0$, a sequence $C_i > 0$, such that the power series converges absolutely and uniformly on a polydisc satisfying $|t^a_i| < \epsilon/(C_i i!)$ for all indices. For the ancestor potential, convergence is with respect to $|y_i^\alpha| < \epsilon C^i$ in each variable.

These conditions are tailored to formal series in infinitely many variables, capturing absolute convergence in the topology of a nuclear Fréchet space. Moreover, a stronger notion (“Definition 3.13 convergence”) requires convergence together with rationality and holomorphic dependence under a so-called dilaton shift, supporting the reconstruction theorems of Frobenius manifold theory.

These refined convergence criteria allow rigorous comparisons between the formal side of quantum cohomology and the analytic structures arising in mirror symmetry, ensuring that, for appropriate geometries, the sum over holomorphic curves indeed represents an analytic function on a neighborhood of the large-radius limit.

3. Givental’s Quantization Formalism and Ancestor Potentials

Under the assumption of semisimple genus-0 quantum cohomology and analytic convergence, Givental’s quantization formalism provides an explicit formula for the total ancestor potential. The key components are:

• The symplectic loop space $H((z^{-1}))$ endowed with the Givental symplectic form

$\Omega(f, g) = \text{Res}_{z=0} (f(-z), g(z)) dz,$

• The abstract ancestor potential

$$\mathcal{A}_X^{\text{abs}} = \exp\left( \sum_{g \geq 0} \hbar^{g-1} F_X^{(g,\text{abs})} \right),$$

where $F_X^{(g, \text{abs})}$ are constructed via symplectic transformations (the $R$- and $\Psi$-matrices) encoding genus-0 data, the discriminant, and genus-0 fundamental solutions: $$S = \Psi R \exp(U/z), \quad U = \text{diag}(u_0, \dots, u_N).$$ Teleman’s classification (of semisimple cohomological field theories) proves that the geometric ancestor potential coincides with Givental’s abstract construction.

Semisimplicity here ensures that higher-genus corrections can be assembled entirely by quantizing the genus-0, semisimple Frobenius manifold data, with explicit dependence on the $R$-matrix and vanishing monodromy.

The convergence of the genus-0 non-descendant potential, together with analytic semisimplicity, guarantees that Givental’s formula is analytic and that the resulting higher-genus descendant and ancestor potentials are likewise NF-convergent.

4. Main Convergence Results and Theoretical Framework

The principal theorem, established using the methods above, is:

If the non-descendant genus-0 Gromov-Witten potential $F_X^{(0)}$ converges and the quantum cohomology of $X$ is analytic and generically semisimple, then the total descendant and ancestor potentials are NF-convergent.

The structure of the proof proceeds via:

• Establishing convergence in a neighborhood of a semisimple point in the quantum cohomology manifold,
• Constructing the $R$-matrix and discriminant factor from the analytic genus-0 data,
• Showing the existence and analytic dependence of fundamental solutions to the Dubrovin connection on a semisimple locus,
• Deduction of ancestor and descendant convergence from Givental’s quantization and Teleman’s classification,
• Application of rationality and tameness for the ancestor potential, guaranteeing the applicability of the full analytic apparatus.

This convergence theorem paves the way for analytic continuation and detailed comparisons in birational geometry, as well as rigorous implementations of mirror symmetry predictions.

5. Applications to Toric and Flag Varieties

These convergence results are shown to apply to several deeply studied geometric classes:

• Compact toric varieties: Via mirror symmetry and explicit calculations, the genus-0 potential converges analytically. Since the quantum cohomology is semisimple for these targets (except for very special degenerations), the Givental-Teleman machinery applies and both total ancestor and descendant potentials are NF-convergent.
• Complete flag varieties: The geometry of flag varieties and work by Kostant provides analytic convergence and semisimplicity, so Gromov-Witten potentials in these cases satisfy the strong analytic properties needed.
• Non-compact toric varieties such as total spaces of negative line bundles: If $X$ is the total space of negative line bundles over a compact toric $Y$, then the Gromov-Witten theory of $X$ can often be related explicitly to that of $Y$, transferring convergence results and enabling analytic computations in settings beyond the compact Fano case.

The ability to conclude convergence of the Gromov-Witten potential in these settings ensures the analytic robustness of quantum cohomology and supports computations of enumerative invariants, analytic continuation, and explicitly analytic mirror maps across different birational models.

6. Significance for Mirror Symmetry and Birational Geometry

The analytic behavior of Gromov-Witten potentials is crucial in making the link between formal algebraic structures (used to define quantum cohomology, Gromov-Witten invariants, and their deformations) and the holomorphic/analytic world of mirror symmetry and moduli spaces.

• Mirror symmetry: Many applications of mirror symmetry require matching analytic functions defined near large radius limit points (A-model) and near maximally unipotent monodromy points (B-model). The convergence criteria ensure that virtual counts of curves in $X$ give rise to actual analytic functions in the relevant moduli.
• Birational geometry and birational invariance: Analyticity of Gromov-Witten potentials underlies comparisons of Gromov-Witten and quantum cohomology for birational or $K$-equivalent varieties.
• Computational implications: Analytic convergence provides the foundations for reconstructing higher-genus invariants, applying wall-crossing arguments, and applying analytic continuation—essential in both arithmetic and geometric enumerative applications.

These principles provide analytic justification for much of the extrinsic structure arbitrated by mirror symmetry, Frobenius manifold theory, and the analytic geometry of quantum invariants in modern algebraic geometry.

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In conclusion, the Gromov-Witten potential encapsulates the full system of enumerative curve counts on a variety $X$ as an infinite-dimensional formal (or analytic) generating function, whose analytic and algebraic structure is controlled by genus-0 quantum cohomology and semisimplicity. The convergence analysis of these potentials, pioneered using Givental’s formalism and codified using Teleman’s classification, ensures these potentials can be compared, continued, and manipulated systematically across a broad range of geometric situations, including toric and flag varieties, opening the field to both rigorous computational and conceptual advances in algebraic geometry and mathematical physics (Coates et al., 2012).