[Paper Review] The Gromov-Witten axioms for symplectic manifolds via polyfold theory

Wolfgang Schmaltz|arXiv (Cornell University)|Dec 31, 2019

Geometric and Algebraic Topology31 references3 citations

TL;DR

This paper establishes the Gromov–Witten axioms for polyfold-theoretic invariants in symplectic geometry, proving that the invariants constructed via Hofer–Wysocki–Zehnder's polyfold theory satisfy the algebraic relations originally proposed by Kontsevich and Manin. The proof relies on constructing perturbations of evaluation maps and analyzing intersection numbers through branched integrals on universal curve polyfolds, confirming consistency with the axiomatic framework for all genus and closed symplectic manifolds.

ABSTRACT

Polyfold theory, as developed by Hofer, Wysocki, and Zehnder, is a relatively new approach to resolving transversality issues that arise in the study of $J$-holomorphic curves in symplectic geometry. This approach has recently led to a well-defined Gromov-Witten invariant for $J$-holomorphic curves of arbitrary genus, and for all closed symplectic manifolds. The Gromov-Witten axioms, as originally described by Kontsevich and Manin, give algebraic relationships between the Gromov-Witten invariants. In this paper, we prove the Gromov-Witten axioms for the polyfold Gromov-Witten invariants.

Motivation & Objective

• To establish the Gromov–Witten axioms for the polyfold Gromov–Witten invariants constructed via Hofer–Wysocki–Zehnder's polyfold theory.
• To resolve transversality issues in the moduli space of $J$-holomorphic curves using abstract perturbations and branched integrals.
• To verify that the polyfold invariants satisfy the algebraic relations (axioms) of Kontsevich and Manin, including the gluing and associativity properties.
• To analyze the behavior of evaluation maps and forgetful maps on polyfold moduli spaces, particularly under gluing and identification of marked points.
• To confirm that the invariants are well-defined and consistent under natural geometric operations such as forgetting a marked point or identifying two marked points.

Proposed method

• Constructs the universal curve polyfold as a fiber bundle over the logarithmic Deligne–Mumford orbifold to model stable curves with marked points.
• Uses abstract perturbations of the Cauchy–Riemann section to achieve transversality in the polyfold setting, enabling the definition of branched integrals.
• Applies the concept of branched integrals to compute polyfold Gromov–Witten invariants as intersection numbers of evaluation maps with suborbifolds.
• Analyzes the $k$th-marked point forgetting map on polyfold moduli spaces, identifying obstructions due to non-submersion and ghost components.
• Uses pullbacks of perturbations via forgetful and marked-point-identifying maps to compare invariants across different moduli spaces.
• Establishes equality of intersection numbers via two-to-one correspondence under marked point identification, leading to the key identity involving the factor of 2 in the gluing axiom.

Experimental results

Research questions

• RQ1 Do the polyfold Gromov–Witten invariants constructed via Hofer–Wysocki–Zehnder theory satisfy the Kontsevich–Manin axioms?
• RQ2 How does the forgetful map on the $k$th marked point behave in the polyfold setting, particularly regarding differentiability and continuity?
• RQ3 Can the gluing axiom (involving identification of two marked points) be verified using polyfold perturbation theory despite the failure of transversality due to non-submersive projection?
• RQ4 What is the precise relationship between the invariants on the split genus $g-1$ moduli space and the original genus $g$ space under marked point identification?
• RQ5 How do perturbations of suborbifolds and evaluation maps interact under pullback via forgetful and gluing maps in the polyfold framework?

Key findings

• The polyfold Gromov–Witten invariants satisfy the full set of Kontsevich–Manin axioms, including the gluing and associativity relations.
• The forgetful map on the $k$th marked point fails to be continuous on subsets with constant destabilizing ghost components, revealing a key technical obstruction.
• Despite non-transversality, the intersection numbers of evaluation maps with perturbed suborbifolds are well-defined and finite, allowing for branched integral computation.
• The marked point identification map induces a two-to-one correspondence between intersection points in the genus $g-1$ and genus $g$ moduli spaces, leading to a factor of 2 in the invariant equality.
• The equality $\operatorname{GW}_{A,g-1,k+2}(\alpha_1,\dots,\alpha_k,\operatorname{PD}(e_\nu),\operatorname{PD}(e^\nu);\beta) = 2 \cdot \operatorname{GW}_{A,g,k}(\alpha_1,\dots,\alpha_k;\psi_*\beta)$ confirms the gluing axiom in the polyfold setting.
• The construction confirms that the polyfold invariants are consistent with the classical axiomatic framework, validating their role in enumerative geometry.