[Paper Review] Equivariant Open Gromov-Witten Theory of $\mathbb{R}\mathbb{P}^{2m} \hookrightarrow \mathbb{C}\mathbb{P}^{2m}$

Amitai Netser Zernik|arXiv (Cornell University)|Sep 26, 2017

Geometric and Algebraic Topology13 references3 citations

TL;DR

This paper introduces equivariant open Gromov-Witten invariants for the Lagrangian embedding $\mathbb{R}\mathbb{P}^{2m} \hookrightarrow \mathbb{C}\mathbb{P}^{2m}$ using a resolution of moduli spaces of stable disc-maps via spherical blowups and extended equivariant forms. It establishes a Stokes' theorem for these forms, enabling the definition of invariants as integrals over resolution spaces, which encode the quantum deformation of the equivariant cohomology of $\mathbb{R}\mathbb{P}^{2m}$ and specialize to Welschinger invariants in the non-equivariant limit for $m=1$. The construction underpins a fixed-point formula for these invariants in subsequent work.

ABSTRACT

We define equivariant open Gromov-Witten invariants for $\mathbb{R}\mathbb{P}^{2m} \hookrightarrow \mathbb{C}\mathbb{P}^{2m}$ as sums of integrals of equivariant forms over resolution spaces, which are blowups of products of moduli spaces of stable disc-maps modeled on trees. These invariants encode the quantum deformation of the equivariant cohomology of $\mathbb{R}\mathbb{P}^{2m}$ by holomorphic discs in $\mathbb{C}\mathbb{P}^{2m}$ and, for $m=1$, specialize to give Welschinger's signed count of real rational planar curves in the non-equivariant limit.

Motivation & Objective

• To define equivariant open Gromov-Witten invariants for the pair $(\mathbb{C}\mathbb{P}^{2m}, \mathbb{R}\mathbb{P}^{2m})$ using a resolution of the moduli space of stable disc-maps.
• To construct a complex of extended equivariant differential forms satisfying Stokes’ theorem, enabling integration to define invariants.
• To provide a framework that encodes the quantum deformation of the equivariant cohomology of $\mathbb{R}\mathbb{P}^{2m}$ via holomorphic discs in $\mathbb{C}\mathbb{P}^{2m}$.
• To lay the foundation for a fixed-point formula computing these invariants, as developed in subsequent work [18].
• To extend Welschinger’s signed count of real rational curves to the equivariant setting for $m=1$.

Proposed method

• Constructs resolution spaces $\widetilde{M}_b^r$ via recursive spherical blowups of products of moduli spaces of stable disc-maps, indexed by labeled trees.
• Defines a complex $(\Omega_b, D)$ of extended equivariant forms that behave as if on a closed manifold with torus action, satisfying $\int_b D\omega = 0$.
• Uses a forgetful map $\operatorname{For}_b^r: \widetilde{M}_b^r \to \widehat{M}_b^r$ to relate resolution spaces to moduli spaces of stable maps with fewer marked points.
• Applies a spherical blowup construction along diagonal submanifolds in $L \times L$, using transversality to define $\widetilde{M}_b^r$ as a cartesian square involving evaluation maps.
• Introduces a relative orientation system on the moduli spaces of stable disc-maps to ensure consistent integration.
• Employs the homological perturbation lemma to reduce choices in the construction of weak bounding cochains, linking the integration and A∞-algebra perspectives.

Experimental results

Research questions

• RQ1 How can equivariant open Gromov-Witten invariants be consistently defined for the Lagrangian $\mathbb{R}\mathbb{P}^{2m} \hookrightarrow \mathbb{C}\mathbb{P}^{2m}$ despite boundary obstructions in the moduli space?
• RQ2 What is the role of resolution spaces and spherical blowups in constructing a well-defined integration theory for open invariants?
• RQ3 How can Stokes’ theorem be formulated and applied in the context of orbifolds with corners and torus actions to ensure invariance?
• RQ4 In what way do these invariants encode the quantum deformation of the equivariant cohomology of $\mathbb{R}\mathbb{P}^{2m}$?
• RQ5 How do the invariants specialize to Welschinger’s signed count of real rational curves when $m=1$ and the equivariant parameters are set to zero?

Key findings

• The equivariant open Gromov-Witten invariants are defined as integrals $I(k,\vec{l},\beta) = \int_b \omega_{(k,\vec{l},\beta)}$ over resolution spaces $\widetilde{M}_b^r$, where $\omega_{(k,\vec{l},\beta)}$ is an extended equivariant form.
• The complex of extended forms $\Omega_b$ supports a Stokes’ theorem: $\int_b D\omega = 0$, ensuring the invariants are well-defined and independent of choices.
• For $m=1$, the invariants specialize to Welschinger’s signed count of real rational planar curves passing through $k$ points in $\mathbb{R}\mathbb{P}^2$ and $l$ conjugation-invariant pairs in $\mathbb{C}\mathbb{P}^2$ in the non-equivariant limit.
• The resolution spaces $\widetilde{M}_b^r$ are constructed as iterated spherical blowups of products of moduli spaces of stable disc-maps, indexed by labeled trees, resolving boundary components of the moduli space.
• The construction establishes a bridge between the A∞-algebraic perspective of Solomon and Tukachinsky and the integration-theoretic approach, with the potential $\Phi(w)$ encoding the quantum deformation of the equivariant cohomology of $\mathbb{R}\mathbb{P}^{2m}$.
• The framework developed here provides the foundation for a fixed-point formula in [18], which simplifies the computation of the invariants by reducing them to contributions from fixed-point loci.