Gromov-Witten theory of the local model of the standard flop
We conjecture that the generating series of equivariant Gromov-Witten invariants of the local Calabi-Yau $(2n-1)$-fold $\operatorname{Tot} \mathcal{O}_{\mathbb{P}^{n-1}} (-1)^{\oplus n}$ at fixed genus can be presented as a linear combination of polylogarithms, generalising a well-known property of the resolved conifold. We prove this conjecture for two specialisations of the equivariant parameters. In the first, the parameters are specialised to roots of unity and the proof proceeds via Givental-Teleman reconstruction. In the second, we obtain a closed-form evaluation of the generating series from which the conjecture follows. For $n=3$, the resulting closed formula establishes a special case of a Gopakumar-Vafa-type conjecture for Calabi-Yau fivefolds.