[Paper Review] Mirror symmetry and quantum cohomology of projective bundles

Artur Elezi|ArXiv.org|Oct 25, 2006

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper proves a conjecture on mirror symmetry for projective bundles over semiample complete intersections in toric varieties, showing that the quantum D-module of the projective bundle arises from a twisted hypergeometric series. The key result is that the classical cohomology relation $\prod_{i=0}^{n}(z - c_1(L_i)) = 0$ deforms to $z\prod_{i=1}^{n}(z - c_1(L_i)) = q_1$ in the small quantum cohomology ring, and relations from $X$ lift to the bundle via differential operators.

ABSTRACT

In an earlier paper we conjectured a relation between the quantum $\mathcal D$-modules of a smooth variety $X$ and the projectivisation of a direct sum of line bundles over it. In this paper we prove the conjecture when $X$ is a complete intersection in a toric variety. We also use the conjecture to show that the relations of the small quantum cohomology ring of $X$ that come from differential operators lift to the projective bundle. The basic cohomology relation of the projective bundle deforms to a relation in the small quantum cohomology.

Motivation & Objective

• To prove a conjecture on mirror symmetry for projective bundles $\mathbb{P}(V) \to X$, where $X$ is a semiample complete intersection in a toric variety.
• To establish that the $J$-function of $\mathbb{P}(V)$ matches a twisted hypergeometric series $I(\mathbb{P}(V))$, extending the Fano case to non-Fano settings.
• To demonstrate that quantum cohomology relations from the base $X$ lift to the projective bundle $\mathbb{P}(V)$ via differential operators.
• To identify the quantum deformation of the classical cohomology relation $\prod_{i=0}^{n}(z - c_1(L_i)) = 0$ in the small quantum cohomology ring of $\mathbb{P}(V)$.

Proposed method

• Construct a twisted hypergeometric series $I(\mathbb{P}(V))$ using a twisting factor $\mathcal{T}_{\nu,\beta}$ that encodes quantum corrections from fiber classes.
• Define the $J$-function $J(\mathbb{P}(V))$ as a generating function for one-point Gromov-Witten invariants and gravitational descendants.
• Use the conjectural equality $J(\mathbb{P}(V)) = I(\mathbb{P}(V))$ to derive quantum cohomology relations via differential operators.
• Lift relations from $QH^*_s X$ to $QH^*_s \mathbb{P}(V)$ by constructing a lifted differential operator $\tilde{\mathcal{P}}$ that acts on $J(\mathbb{P}(V))$.
• Derive the quantum deformation of the classical relation by showing that the operator $\Delta = \prod_{i=0}^{n}(\hbar\partial/\partial t_{k+1} - \sum_j a_{ij}\hbar\partial/\partial t_j) - q_1$ annihilates $J(\mathbb{P}(V))$, implying $\prod_{i=0}^{n}(z - c_1(L_i)) = q_1$ in $QH^*_s \mathbb{P}(V)$.
• Verify the conjecture in the toric case by proving injectivity of the map $i_*: H_2(X) \to H_2(\mathbb{P}(V))$, which allows transfer of $J$-functions and $I$-functions between $X$ and $\mathbb{P}(V)$.

Experimental results

Research questions

• RQ1 Does the conjectured mirror symmetry relation $J(\mathbb{P}(V)) = I(\mathbb{P}(V))$ hold for projective bundles over semiample complete intersections in toric varieties?
• RQ2 How do quantum cohomology relations from the base manifold $X$ deform in the quantum cohomology ring of the projective bundle $\mathbb{P}(V)$?
• RQ3 What is the quantum deformation of the classical cohomology relation $\prod_{i=0}^{n}(z - c_1(L_i)) = 0$ in $H^*\mathbb{P}(V)$?
• RQ4 Can differential operators that annihilate $J(X)$ be lifted to annihilate $J(\mathbb{P}(V))$, and what is the resulting relation in $QH^*_s \mathbb{P}(V)$?
• RQ5 How does the twisting factor $\mathcal{T}_{\nu,\beta}$ encode quantum corrections from fiber-wise rational curves in $\mathbb{P}(V)$?

Key findings

• The conjecture $J(\mathbb{P}(V)) = I(\mathbb{P}(V))$ is proven when $X$ is a semiample complete intersection in a toric variety, establishing mirror symmetry for such projective bundles.
• The classical cohomology relation $\prod_{i=0}^{n}(z - c_1(L_i)) = 0$ deforms to $z\prod_{i=1}^{n}(z - c_1(L_i)) = q_1$ in the small quantum cohomology ring of $\mathbb{P}(V)$, generalizing the $\mathbb{P}^n$ case.
• Relations in $QH^*_s X$ arising from differential operators lift to $QH^*_s \mathbb{P}(V)$, with the quantum product of $p_i$ deformed by a factor involving $\prod_{i=1}^{n}(z - c_1(L_i))^{L_i(\alpha)}$.
• The $J$-function of $\mathbb{P}(V)$ satisfies $\Delta J(\mathbb{P}(V)) = 0$, where $\Delta = \prod_{i=0}^{n}(\hbar\partial/\partial t_{k+1} - \sum_j a_{ij}\hbar\partial/\partial t_j) - q_1$, implying the quantum relation $\prod_{i=0}^{n}(z - c_1(L_i)) = q_1$.
• When the map $i_*: H_2(X) \to H_2(\mathbb{P}(V))$ is injective, the $J$-functions and $I$-functions of $X$ and $\mathbb{P}(V)$ are compatible under pullback, enabling the proof of the conjecture in the toric case.
• The twisting factor $\mathcal{T}_{\nu,\beta}$ is invariant under shifts by $\alpha \in H_2(X)$ with $c_1(\tilde{L}_j) \cdot \alpha = 0$, ensuring consistency in the construction of $I_{\nu,\beta}$.