String field theory motivates Kodaira-Spencer gravity whose classical physics encodes variation of Hodge structures on Calabi-Yau manifolds. It is "obstructed" to write down action functional in such theory in the usual sense and this causes a big trouble for quantization. In this talk, I will explain the homotopic idea that L-infinity transformation arising from period maps turns classical equation of motions to a form that we can write down action functional. Its quantization in the Batalin-Vilkovisky formalism leads to the higher genus B-model that is conjectured to be mirror to Gromov-Witten counting of holomorphic curves. Integrable hierarchy arises naturally from the background symmetry of Kodaira-Spencer gravity and is also encoded into the period map.

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