For a quasi-homogeneous polynomial \(W\) with weights \((w_1, \ldots , w_N)\) and degree \(d\) (satisfying certain conditions), let \(X_W\) be the hypersurface in the weighted projective space \(\mathbb{P}(w_1, \ldots , w_N)\) defined by the vanishing of \(W\). Then, the Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence asserts the equivalence of two cohomological field theories (CohFTs): the Fan-Jarvis-Ruan-Witten (FJRW) theory of \(W\), and the Gromov-Witten (GW) theory of \(X_W\). The FJRW theory of \(W\) is defined via intersection numbers of the moduli spaces \(\mathcal{R}^d_{\vec k}, \epsilon \vec{l}\), parametrising \(\vec{l}\)-twisted \(d\)-spin structures on \((d,\epsilon)\)-stable curves. These intersection numbers are defined by integrating \(\psi\)-classes over the Witten class of \(\mathcal{R}^d_{\vec k}\).
It is expected that the LG/CY correspondence could be proved using the theory of gauged linear sigma model (GLSM) of Witten (for a mathematically rigorous definition of the GLSM see [H. Fan et al., Geom. Topol. 22, No. 1, 235–303 (2018; Zbl 1388.14041)]. In the case of hypersurfaces, the GLSM is a one-dimensional family of CohFTs parametrised by the rational numbers different from zero. The CohFTs lying over \(\mathbb{Q}_{>0}\) (the so-called geometric phase) are related to certain versions of quasi-maps, while those over \(\mathbb{Q}_{<0}\) (the Landau-Ginzburg phase) correspond to the FJRW theory of \(W\).
The main results of the the paper under review concerns the CohFTs lying over \(\mathbb{Q}_{<0}\). For a Fermat polynomial \(W\), the authors define the genus zero descendant potential \(\mathcal{F}^\epsilon_W\), for any \(\epsilon \in \mathbb{Q}_{>0}\) (the authors normilize the GLSMs in such a way to work with positive \(\epsilon\)). This is a generating function for the intersection numbers of the moduli spaces \(\mathcal{R}^d_{\vec k}\) above. For \(\epsilon >1\) (denoted \(\epsilon = \infty\)) one recovers the narrow FJRW descendant potential \(\mathcal{F}^\infty\). Following Givental, the authors define a certain infinite-dimensional vector space \(\mathcal{H}\), and the graph of the differential of \(\mathcal{F}^\infty\) is the formal germ of a Lagrangian cone \(\mathcal{L} \subset \mathcal{H}\), whose geometry reflects the properties of \(\mathcal{F}^\infty\). The derivatives of \(\mathcal{F}^\epsilon_W\) with respect to \(t\) yields the so-called large \(\mathcal{J}^\epsilon (t,u,-z)\)-functions. The main theorem of the paper (Theorem 1.11) says that, for any \(\epsilon >0\), \(\mathcal{J}^\epsilon (t,u,-z)\) is an \(\mathcal{H} (u)\)-valued point of \(\mathcal{L}\). This means that \(\mathcal{J}^\epsilon (t,u,-z)\) is a formal series of a certain form.
In Section 3 the authors derive several important consequences of the previous main theorem. The first one is a formula that relates \(\mathcal{J}^{\epsilon_1}\) with \(\mathcal{J}^{\epsilon_2}\), for different \(\epsilon_1, \epsilon_2 >0\). Furthermore, for \(\epsilon \to 0\) they obtain a new geometric interpretation of the LG mirror theorem.