Givental-type genus-zero mirror symmetry conjecture for the small FJRW I-function

Let (W,J)(W,\langle J\rangle) be an admissible LG pair, and let IFJRWsm(t,z)I_{\rm FJRW}^{\rm sm}(t,-z) be its small FJRW II-function. Let τ(t)\tau(t) be defined by the coefficient of z0z^0 in the small II-function, and let J(τ(t),z)J(\tau(t),-z) be the corresponding JJ-function. Genus-zero mirror symmetry conjecture. The small II-function IFJRWsm(t,z)I_{\rm FJRW}^{\rm sm}(t,-z) lies on the Lagrangian cone. In particular, there is a formal power series c(t,z)c(t,z) in tt and zz such that

tJ(τ(t),z)=IFJRWsm(t,z)+c(t,z)zIFJRWsm(t,z)t.tJ(\tau(t),-z)=I_{\rm FJRW}^{\rm sm}(t,-z)+c(t,z)z\frac{\partial I_{\rm FJRW}^{\rm sm}(t,-z)}{\partial t}.

This strengthens the preceding special-case mirror formula to a statement that the small II-function lies on the genus-zero FJRW Lagrangian cone; the source gives no resolution.

Sources & referencesView supporting material

Primary source

Yefeng Shen and Ming Zhang, “Quantum spectrum and Gamma structures for quasi-homogeneous polynomials of general type”, arXiv:2309.07446 (2025).

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