[Paper Review] Categorical Saito theory, II: Landau-Ginzburg orbifolds

Junwu Tu|arXiv (Cornell University)|Sep 30, 2019

Algebraic structures and combinatorial models25 references5 citations

TL;DR

This paper constructs a $G$-equivariant version of Saito's theory of primitive forms using $G$-equivariant matrix factorizations and Variation of Semi-infinite Hodge Structures (VSHS), proving the existence of a canonical categorical primitive form for invertible polynomials with finite diagonal special linear symmetries. It verifies LG/LG mirror symmetry for the quintic family by showing the genus-zero prepotential of the $B$-model orbifold $(W, (ield{Z}/5\mathbb{Z})^4)$ matches the FJRW prepotential of its mirror $A$-model $(W, \mathbb{Z}/5\mathbb{Z})$ via a comparison of flat sections and power series expansions.

ABSTRACT

Let $W\in \mathbb{C}[x_1,\cdots,x_N]$ be an invertible polynomial with an isolated singularity at origin, and let $G\subset {\sf SL}_N\cap (\mathbb{C}^*)^N$ be a finite diagonal and special linear symmetry group of $W$. In this paper, we use the category ${\sf MF}_G(W)$ of $G$-equivariant matrix factorizations and its associated VSHS to construct a $G$-equivariant version of Saito's theory of primitive forms. We prove there exists a canonical categorical primitive form of ${\sf MF}_G(W)$ characterized by $G_W^{\sf max}$-equivariance. Conjecturally, this $G$-equivariant Saito theory is equivalent to the genus zero part of the FJRW theory under LG/LG mirror symmetry. In the marginal deformation direction, we verify this for the FJRW theory of $\big(\frac{1}{5}(x_1^5+\cdots+x_5^5),\mathbb{Z}/5\mathbb{Z}\big)$ with its mirror dual B-model Landau-Ginzburg orbifold $\big(\frac{1}{5}(x_1^5+\cdots+x_5^5), (\mathbb{Z}/5\mathbb{Z})^4\big)$. In the case of the Quintic family $\mathcal{W}=\frac{1}{5}(x_1^5+\cdots+x_5^5)-ψx_1x_2x_3x_4x_5$, we also prove a comparison result of B-model VSHS's conjectured by Ganatra-Perutz-Sheridan.

Motivation & Objective

• To develop a $G$-equivariant extension of Saito's theory of primitive forms for Landau-Ginzburg orbifolds with invertible singularities.
• To establish a categorical framework using $G$-equivariant matrix factorizations and their associated VSHS.
• To prove the existence of a canonical $G_{W}^{{\rm max}}$-equivariant splitting of the non-commutative Hodge filtration.
• To verify mirror symmetry between the $B$-model $G$-equivariant Saito theory and the genus-zero FJRW theory for the quintic family.
• To compare the categorical VSHS of the $B$-model with Griffiths' geometric VSHS, supporting a conjecture by Ganatra-Perutz-Sheridan.

Proposed method

• Construct a polarized VSHS from the category ${\sf MF}_{G}(W)$ of $G$-equivariant matrix factorizations using Hodge-to-de-Rham degeneration.
• Apply a bijection between splittings of the non-commutative Hodge filtration and categorical primitive forms via results from [1].
• Prove the existence of a canonical $G_{W}^{{\rm max}}$-equivariant splitting using non-commutative Hochschild and cyclic homology computations.
• Use the comparison isomorphism $HC^{-}_{\rm odd}({\sf MF}_{G}(W)) \cong H^{*}(\Omega_{\mathbb{C}^5}^{*}[[u]], dW + u d_{DR})^{G}$ to model the VSHS in de Rham cohomology.
• Compute flat extensions of basis forms along the marginal deformation direction $\psi$ using the isomorphism $e^{(x_1\cdots x_5)\psi / u}$, relating the Gauss-Manin connection to the trivial connection.
• Derive the primitive form $\zeta$ as the positive part of the flat extension of $s_0$, leading to $\zeta = \frac{1}{\omega_0} dx_1\cdots dx_5$ with $\omega_0$ a power series in $\psi$.

Experimental results

Research questions

• RQ1 Does a canonical $G$-equivariant primitive form exist for the category ${\sf MF}_{G}(W)$ of $G$-equivariant matrix factorizations of an invertible polynomial $W$?
• RQ2 Is the $G$-equivariant Saito theory constructed here mirror dual to the genus-zero FJRW theory under LG/LG duality?
• RQ3 Can the categorical VSHS of the $B$-model $({\sf MF}_{G}(W))$ be compared to the geometric VSHS of Griffiths for the mirror quintic family?
• RQ4 Does the prepotential function of the $B$-model $G$-equivariant Saito theory match the FJRW prepotential in the marginal deformation direction?
• RQ5 Is the canonical splitting of the non-commutative Hodge filtration for $G = (\mathbb{Z}/5\mathbb{Z})^4$ on $W = \frac{1}{5}(x_1^5 + \cdots + x_5^5)$ computable via power series in $\psi$?

Key findings

• A canonical $G_{W}^{{\rm max}}$-equivariant splitting of the non-commutative Hodge filtration exists for ${\sf MF}_{G}(W)$, yielding a unique categorical primitive form.
• The primitive form $\zeta$ is explicitly computed as $\zeta = \frac{1}{\omega_0} dx_1 dx_2 dx_3 dx_4 dx_5$, where $\omega_0 = \sum_{k \geq 0} \frac{[(5k-4)(5k-9)\cdots(1)]^5}{(5k)!} \psi^{5k}$.
• The flat extension of the basis form $s_0$ satisfies $s_0^{\sf flat} = \frac{1}{\omega_0} dx_1\cdots dx_5 + u^{-1} \frac{\omega_1}{\omega_0} s_1^{\sf flat} - u^{-2} \frac{\omega_2}{\omega_0} s_2^{\sf flat} + u^{-3} \frac{\omega_3}{\omega_0} s_3^{\sf flat}$, with $\omega_i$ defined as power series in $\psi$.
• The $B$-model VSHS of $({\sf MF}_{(\mathbb{Z}/5\mathbb{Z})^4}(W))$ is isomorphic to the geometric VSHS of the mirror quintic family, confirming a conjecture of Ganatra-Perutz-Sheridan in this case.
• The genus-zero prepotential of the $B$-model $({\sf MF}_{(\mathbb{Z}/5\mathbb{Z})^4}(W))$ restricted to the marginal direction $\tau = \frac{\omega_1}{\omega_0}$ matches the FJRW prepotential of the mirror $A$-model $(W, \mathbb{Z}/5\mathbb{Z})$, verifying LG/LG mirror symmetry.