Mirrors of quotients and quotients of mirrors

Ideas from tropical geometry play an important role in Gross, Hacking, and Keel's construction of mirrors for log-Calabi-Yau varieties. In this talk, I'll present work in progress with Colin Diemer in which we study the structure of GHK-mirrors under the action of a finite group. When a log-CY variety X comes equipped with the left-action of a finite group G, its GHK-mirror family X* acquires a dual right-action by the same group. Using this dual action, we show that in many cases, the mirror family of the left-quotient G\X can be naturally realized as the pullback of a stratum inside the right-quotient X*/G of the original mirror family X* (up to a uniform re-scaling of the relevant instanton corrections). More concisely: the mirror family (G\X)* is "almost" a canonically defined piece of the quotient family X*/G. In invariant theoretic terms: the broken line counts that give the structure constants for the coordinate ring of the mirror (G\X)* already appear as structure constants in the ring of G-invariants on the mirror X* (again up to uniform re-scaling).