Equal-mass sunset integrals and genus-zero local Gromov--Witten theory of Calabi--Yau $n$-folds

Abstract

We prove an exact identity, as convergent logarithmic germs at large Euclidean momentum, between the equal-mass two-dimensional $(n-1)$-loop sunset Feynman integral, with $n\geq4$, and the genus-zero one-point descendant invariants of the local Calabi--Yau $n$-fold $X_n=\operatorname{Tot}(K_{F_n})$, where $F_n\subset(\mathbb P^1)^{n}$ is a smooth $(1,\ldots,1)$ hypersurface. Writing $s$ for the momentum variable and $y=-1/s$, the identity reads \[ I^{(n-1)}_{\circleddash}(s)=(-1)^{n-1}yA\,\Bigl[nR^{n-1}-\theta_Q{\mathcal D}_0^\Delta\,R^{n-2} +\sum_{j=0}^{n-3}c_jR^j\Bigr], \] where $R=\log y+O(y)$ is the local mirror coordinate, $A=\theta_yR$, $Q=e^R$, ${\mathcal D}_0^\Delta(Q)$ is the primary one-point potential, and each $c_j$ has the form \[ c_j=c_j^{(0)}+\sum_{a=0}^{n-2}\bigl(c_{j,a}\,{\mathcal D}_a^\Delta(Q) +c'_{j,a}\,\theta_Q{\mathcal D}_a^\Delta(Q)\bigr) \] with the coefficients in $\mathbb Q[\zeta(3),\zeta(5),\ldots]$ the odd zeta values coming from the Gamma class. The descendant series ${\mathcal D}_a^\Delta(Q)$ are given explicitly by the ambient hypergeometric series and the inverse mirror map. We also prove global univalence of the principal mirror map, with explicit upper and lower bounds on the $Q$-Taylor radii, extract the full primary evaluation divisor and its integral cover transform, and reduce the ambient endpoint constants to a finite universal Gamma-class part and $\lfloor(n-1)/2\rfloor$ real parameters.