Intrinsic mirror symmetry and Frobenius structure theorem via Gromov-Witten theory of root stacks

Using recent results of Battistella, Nabijou, Ranganathan and the author, we compare candidate mirror algebras associated with certain log Calabi-Yau pairs constructed by Gross-Siebert using log Gromov-Witten theory and Tseng-You using orbifold Gromov- Witten theory of root stacks. Although the structure constants used to defined these mirror algebras do not typically agree, we show that any given structure constant involved in the construction the algebra of Gross and Siebert can be computed in terms of structure constants of the algebra of Tseng and You after a sequence of log blowups. Using this relation, we provide another proof of associativity of the log mirror algebra, and a proof of the weak Frobenius Structure Theorem in full generality. Along the way, we introduce a class of twisted punctured Gromov-Witten invariants of generalized root stacks induced by log \'etale modifications, and use this to study the behavior of log Gromov-Witten invariants under ramified base change.