Quantum periods, toric degenerations and intrinsic mirror symmetry

Given a Fano variety $X$, and $U$ an affine log Calabi-Yau variety given as the complement of an anticanonical divisor $D \subset X$, we prove that for any snc compactification $Y$ of $U$ dominating $X$ with $D' = Y\setminus U$, there exists an element $W_D \in R_{(Y,D')}$ of the intrinsic mirror algebra whose classical periods give the regularized quantum periods of $X$. Using this result, we deduce various corollaries regarding Fano mirror symmetry, in particular integrality of regularized quantum periods in large generality and the existence of Laurent mirrors to all Fano varieties whose mirrors contain a dense torus. When $U$ is an affine cluster variety satisfying the Fock-Goncharov conjecture, we use this result to produce a family of polytopes indexed by seeds of $U$ determined by enumerative invariants of the pair $(X,D)$ which give a family of Newton-Okounkov bodies and toric degenerations of $X$. Moreover, we give an explicit description of the superpotential in the Grassmanian setting, in particular recovering the Pl\"ucker coordinate mirror discovered by Marsh and Rietsch. Finally, we use the main result to show that the quantum period sequence is equivalent to all theta function structure constants for $R_{(X,D)}$ when $D$ is a smooth anticanonical divisor.