Intrinsic mirror symmetry

We associate a ring R R to a log Calabi-Yau pair ( X , D ) (X,D) or a degeneration of Calabi-Yau manifolds X → B X→ B . The vector space underlying R R is determined by the tropicalization of ( X , D ) (X,D) or X → B X→ B , while the product rule is defined using punctured Gromov-Witten invariants, defined in joint work with Abramovich and Chen. In the log Calabi-Yau case, if D D is maximally degenerate, then we propose that Spec ⁡ R Spec R is the mirror to X ∖ D X D , while in the Calabi-Yau degeneration case, if the degeneration is maximally unipotent, the mirror is expected to be Proj ⁡ R Proj R . The main result in this paper is that R R as defined is an associative, commutative ring with unit, with associativity the most difficult part.