A Landau-Ginzburg (LG) model is a pair \((W,G)\), often interpreted as a singularity, where \(W\) is a non-degenerate homogeneous polynomial on \(\mathbb{C}^N\) and \(G\) is a finite group of its automorphisms. In the early 1990s Vafa, Witten, etc., predicted existence of invariants of LG models that correspond to the Gromov-Witten invariants of a Calabi-Yau (CY) manifold. In 2013 Fan, Jarvis and Ruan defined the condidate invariants (now called the FJRW invariants) mathematically as intersection numbers on the moduli space of “\(W\) curves”. Let \(W:=x_1^5+x_2^5+\dots +x_5^5\) and \(\jmath\) be the coordinatewise multiplication by the fifth primitive root of unity. For the Fermat quintic \(M:=\{W=0\}\) the LG/CY correspondence with \((W,\langle\jmath\rangle)\) follows from the recent work of A. Chiodo and Y. Ruan [Adv. Math. 227, No. 6, 2157–2188 (2011; Zbl 1245.14038)].
The story unfolds in two steps. First, let \(W_\psi:=W-\psi x_1\cdots x_5\) and define a family of mirror quintics \(\mathcal{W}_\psi:=\{W_\psi=0\}\subset\mathbb{P}^4/(\mathbb{Z}/5\mathbb{Z})^3\). By the Givental’s mirror theorem, after a change of variables, components of the Givental’s \(J\)-function of \(M\) give a basis of solutions to the Picard-Fuchs equations for \(\mathcal{W}_\psi\) around \(\psi=\infty\). Chiodo and Ruan showed that the \(J\)-function of the FJRW invariants of \((W,\langle\jmath\rangle)\) is similarly related to the Picard-Fuchs equations for \(\mathcal{W}_\psi\) around \(\psi=0\). The correspondence then follows via analytic continuation in \(\psi\) and a linear transformation.
The authors prove a similar correspondence for the mirror quintic \(\mathcal{W}:=\mathcal{W}_0\). This is a first such example for a space which is not a complete intersection in a weighted projective space. The \(\psi=\infty\) part has already been proved by Y.-P. Lee and M. Shoemaker [Geom. Topol. 18, No. 3, 1437–1483 (2014; Zbl 1305.14025)]. Namely, after a change of variables, components of \(J^{\mathcal{W}}\) give solutions to the Picard-Fuchs equations of a holomorphic \((3,0)\) form on \(M_\psi:=\{W_\psi=0\}\subset\mathbb{P}^4\), the mirror family of \(\mathcal{W}\). The first derivatives of \(J^{\mathcal{W}}\) give solutions to the Picard-Fuchs equations of non-holomorphic families of 3-forms. The main result of the paper is that the FJRW \(J\)-function of \((W,(\mathbb{Z}/5\mathbb{Z})^4)\), and its first derivatives, after a change of variables give solutions to the Picard-Fuchs equations around \(\psi=0\). Analytic continuation combined with a linear symplectic transformation \(\mathbb{U}\) complete the correspondence, the first derivatives are needed to determine \(\mathbb{U}\) uniquely.
In Givental’s geometric formalism the correspondence can be rephrased as identifying analytic continuation of the “small slices” of Lagrangian cones determining the respective \(J\)-functions. It is conjectured that quantization of \(\mathbb{U}\) identifies the analytic continuation of the higher genus Gromov-Witten and FJRW theories. Analogous conjecture is still open even for the Fermat quintic.