15 problems

For an admissible LG pair $(W,G)$, let the Fan–Jarvis–Ruan–Witten, Polishchuk–Vaintrob, and Chang–Li–Li–Kiem–Li constructions define three cohomological field theories. Folklore eq…

Let $(W,\langle J\rangle)$ be an admissible LG pair, and let $I_{\rm FJRW}^{\rm sm}(t,-z)$ be its small FJRW $I$-function. Let $\tau(t)$ be defined by the coefficient of $z^0$ in t…

Let $(W,\langle J\rangle)$ be an admissible LG pair of general type with index $\nu>0$. For each integer $\ell\in\{1-\nu,\ldots,0\}$, let ${\mathbb C}(\ell)^{\rm st}$ denote the co…

Let $W$ be a nondegenerate quasihomogeneous singularity of general type. Let $I_{\rm FJRW}^{\rm sm}(t,z)$ be the small FJRW $I$-function, let $\tau(t)$ be determined by the coeffic…

Let $(W,\langle J\rangle)$ be an admissible LG pair, where $W$ is a nondegenerate quasihomogeneous polynomial of general type with weight system $(d;w_1,\ldots,w_N)$. Let…

Generating-function conjecture. According to the established cases $i=1,2$, one conjectures that

Let $\mathcal{X}$ be a Calabi–Yau variety and let $(W,G_{max})$ be a pair defining an FJRW theory. Suppose that the Gromov–Witten theory of $\mathcal{X}$ and the FJRW theory of…

Let $\mathcal{Z}^{GW}$ and $\mathcal{Z}^{FJRW}$ be A-model partition functions that are mirrors to the same B-model. An upper-triangular Givental group element is an element $R$ wh…

Nonvanishing conjecture. The coefficient satisfies

Low-degree GW algorithm conjecture. Applying the localization vanishing to this data determines $N_{g,d}$ provided that

FJRW effective-algorithm conjecture. The polynomial relations among the FJRW invariants $\Theta_{g,k}$ obtained from these localization relations, with

GW effective-algorithm conjecture. By choosing $d_0$ and $d_\infty$ positive, these relations provide an effective algorithm to determine all genus GW invariants $N_{g,d}$, provide…

Initial-value conjecture. The initial-value requirement $N_{g,d'}$ for $d'<g$ in Theorem 1 can be dropped by using more of the localization relations.

LG mirror symmetry conjecture. Up to a change of variables, the generating function of the FJRW theory at all genera for $(W,G_W)$ can be identified with the generating function of…

Ruan's LG/CY correspondence conjecture. (1) There is a graded vector-space isomorphism