Quantum flips I: local model

We study analytic continuations of quantum cohomology under simple flips $f: X \dashrightarrow X’$alongtheextremalrayquantumvariable$q^\ell$. The inverse correspondence $\mathrm{\Psi }=[{\mathrm{\Gamma }}_f{]}^{\ast }$ by the graph closure gives an embedding of Chow motives $[{\widehat{X}}^{\mathrm{\prime }}]\hookrightarrow [\widehat{X}]$ which preserves the Poincar'e pairing. We construct a deformation $\widehat{\mathrm{\Psi }}$ of $\mathrm{\Psi }=[{\mathrm{\Gamma }}_f{]}^{\ast }$ which induces a non-linear embedding $$QH(X^{\mathrm{\prime }})\hookrightarrow QH(X)$$ in the category of $F$-manifolds into the regular integrable loci of $QH(X)$ near $q^{\mathrm{\ell }}=\mathrm{\infty }$. This provides examples of functoriality of quantum cohomology beyond $K$-equivalent transformations. In this paper, we focus on the case when $X$ and $X^{\mathrm{\prime }}$ are (projective) local models.