Difference Equations for Local Gromov-Witten Potentials of Threefold Flops

Abstract

We study the information encoded by difference equations for the local Gromov-Witten theory of contractible rational curves in Calabi-Yau threefolds. Starting from the Bryan-Katz-Leung multiple-cover decomposition, we organize all nonzero primitive degrees on an exceptional ray by a single central-difference operator associated with the greatest common divisor of their support. The resulting equation is governed by explicit Fej\'er-type Laurent kernels. Our main structural result shows that, once the exceptional ray and its lattice normalization are fixed, the full quantum forcing term, its classical specialization at $\lambda=0$, and the local genus-zero Gopakumar-Vafa spectrum mutually determine one another. In particular, the classical forcing already determines every local GV multiplicity through an explicit divisor inversion. Thus the quantum difference equation carries a nontrivial shift profile, but this additional $\lambda$-dependence contains no further information about the genus-zero GV spectrum. For finite support, we further introduce the finest common shift lattice and determine the minimal Laurent-polynomial denominator-clearing operator with respect to divisibility. Its cyclotomic factorization defines a strictly coarser shadow of the local enumerative theory. Explicit length-two, $E_6$, $E_7$, and $E_8$ flop models exhibit three distinct failures of reconstruction: the same Koll\'ar length can give different forcing terms, different GV supports can have the same cyclotomic skeleton, and identical local difference data need not determine the analytic type of the flop. These results give a precise hierarchy of the geometric and enumerative information retained by local Gromov-Witten difference equations.