[Paper Review] Gromov-Witten theory of orbicurves, the space of tri-polynomials and Symplectic Field Theory of Seifert fibrations

Paolo Rossi|ArXiv.org|Aug 19, 2008

Algebraic Geometry and Number Theory12 references8 citations

TL;DR

This paper establishes a mirror isomorphism between Frobenius manifolds defined on spaces of tri-polynomials $F(x,y,z) = -xyz + P_1(x) + P_2(y) + P_3(z)$ and the orbifold Gromov-Witten theory of $π^1_{\alpha_1,\ldots,\alpha_a}$, using extended Symplectic Field Theory to compute genus-0 potentials and proving that polynomial quantum cohomology arises precisely for $\mathbb{P}^1_{2,2,r}$, $\mathbb{P}^1_{2,3,3}$, $\mathbb{P}^1_{2,3,4}$, and $\mathbb{P}^1_{2,3,5}$, with explicit SFT-Hamiltonians for Seifert fibrations over these orbifolds.

ABSTRACT

We compute, with Symplectic Field Theory techniques, the Gromov-Witten theory of the complex projective line with orbifold points. A natural subclass of these orbifolds, the ones with polynomial quantum cohomology, gives rise to a family of (polynomial) Frobenius manifolds and integrable systems of Hamiltonian PDEs, which extend the (dispersionless) bigraded Toda hierarchy. We then define a Frobenius structure on the spaces of polynomials in three complex variables of the form F(x,y,z)= -xyz+P_1(x)+P_2(y)+P_3(z) which contains as special cases the ones constructed on the space of Laurent polynomials. We prove a mirror theorem stating that these Frobenius structures are isomorphic to the ones found before for polynomial P1-orbifolds. Finally we link rational Symplectic Field Theory of Seifert fibrations over S^2 and three singular fibers with orbifold Gromov-Witten invariants of the base, extending a known result valid in the smooth case.

Motivation & Objective

• To compute the genus-0 orbifold Gromov-Witten potential of $\mathbb{P}^1_{\alpha_1,\ldots,\alpha_a}$ using an extension of Symplectic Field Theory to orbifold cobordisms.
• To identify the subclass of orbicurves with polynomial quantum cohomology and construct associated polynomial Frobenius manifolds.
• To define a Frobenius structure on the space of tri-polynomials $F(x,y,z) = -xyz + P_1(x) + P_2(y) + P_3(z)$ of given degrees $p,q,r$ with $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} > 1$.
• To prove a mirror theorem establishing an isomorphism between these tri-polynomial Frobenius manifolds and the quantum cohomology of $\mathbb{P}^1_{p,q,r}$-orbifolds.
• To extend the correspondence between rational SFT of Seifert fibrations and orbifold Gromov-Witten invariants of the base, generalizing a known result from the smooth case.

Proposed method

• Extends Symplectic Field Theory to include $\mathbb{Z}_k$-singularities in 2-dimensional target cobordisms, modifying the index formula and grading for orbifold maps.
• Computes the genus-0 Gromov-Witten potential of $\mathbb{P}^1_{\alpha_1,\ldots,\alpha_a}$ as a power series in orbifold cohomology variables, using Hurwitz numbers.
• Defines a Frobenius manifold structure on the space of tri-polynomials $F(x,y,z) = -xyz + P_1(x) + P_2(y) + P_3(z)$ with fixed degrees $p,q,r$, using the residue pairing and Euler vector field.
• Derives an explicit integral formula relating the rational SFT potential of a Seifert fibration $V$ over $\mathbb{P}^1_{\alpha_1,\ldots,\alpha_a}$ to the orbifold Gromov-Witten potential of the base via a Fourier-type integral over $x \in [0, 2\pi \alpha_1\cdots\alpha_a]$.
• Uses Dubrovin and Zhang’s reconstruction theorem to prove isomorphism between $M_{2,2,r}$ and the Frobenius manifold $M(D_{r+2},r)$ by matching the unity vector, Euler vector, and intersection metric.
• Applies the orbifold Poincaré pairing and degree-shifting numbers $\iota_n = \frac{l}{\alpha_i}$ to define the Poisson structure and grading in the SFT framework.

Experimental results

Research questions

• RQ1 For which orbicurves $\mathbb{P}^1_{\alpha_1,\ldots,\alpha_a}$ does the Gromov-Witten potential truncate to a polynomial in the even orbifold cohomology variables?
• RQ2 Can a Frobenius manifold be constructed on the space of tri-polynomials $F(x,y,z) = -xyz + P_1(x) + P_2(y) + P_3(z)$ with fixed degrees $p,q,r$, and under what conditions is it polynomial?
• RQ3 Is there a mirror isomorphism between the Frobenius manifold of tri-polynomials $M_{p,q,r}$ and the orbifold quantum cohomology $QH^*_{\text{orb}}(\mathbb{P}^1_{p,q,r})$?
• RQ4 How can the rational SFT-Hamiltonians of Seifert fibrations over $\mathbb{P}^1_{\alpha_1,\ldots,\alpha_a}$ be expressed in terms of the orbifold Gromov-Witten invariants of the base?
• RQ5 What is the precise relation between the SFT potential of a prequantization bundle $V$ over an orbicurve and the Gromov-Witten potential of the base, generalizing the smooth case?

Key findings

• The Gromov-Witten potential of $\mathbb{P}^1_{\alpha_1,\ldots,\alpha_a}$ truncates to a polynomial if and only if the orbicurve is $\mathbb{P}^1_{2,2,r}$ for $r \geq 4$, $\mathbb{P}^1_{2,3,3}$, $\mathbb{P}^1_{2,3,4}$, or $\mathbb{P}^1_{2,3,5}$, yielding polynomial Frobenius manifolds.
• The Frobenius manifold $M_{p,q,r}$ on the space of tri-polynomials $F(x,y,z) = -xyz + P_1(x) + P_2(y) + P_3(z)$ with $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} > 1$ is isomorphic to $QH^*_{\text{orb}}(\mathbb{P}^1_{p,q,r})$, establishing a mirror theorem.
• The space $M_{2,2,r}$ is isomorphic to the Frobenius manifold $M(D_{r+2},r)$ associated with the extended affine Weyl group $\tilde{D}_l$, confirmed by matching the unity vector, Euler vector, and intersection metric.
• The rational SFT potential of a Seifert fibration $V$ over $\mathbb{P}^1_{\alpha_1,\ldots,\alpha_a}$ is given by an integral over $x \in [0, 2\pi \alpha_1\cdots\alpha_a]$ of the orbifold Gromov-Witten potential with shifted parameters and a phase factor $e^{-i c_1(V) x}$.
• The SFT-Hamiltonians $\mathbf{h}^j_V$ are related to the Gromov-Witten descendants $\mathbf{f}^j_P$ via a Fourier-type integral involving the degree-shifting numbers $\iota_n = \frac{l}{\alpha_i}$ for twisted sectors.
• The construction provides explicit SFT-Hamiltonians for contact 3-manifolds such as Lens spaces (A-type), Prism manifolds (D-type), and exotic quotients of the Poincaré sphere (E-type), extending the smooth case.