Holomorphic anomaly conjecture for the Gromov–Witten potential of a Calabi–Yau A-infinity category

For a Calabi–Yau $A_\infty$ category, let the Gromov–Witten potential define a left ideal in the sheaf of Weyl algebras associated to its periodic cyclic chain complex, and let this sheaf carry the Gauss–Manin flat connection. After modifying the Gromov–Witten potential to take account of the unstable moduli spaces $(g,n)=(0,1)$ and $(0,2)$, the holomorphic anomaly conjecture. The ideal is preserved by this flat connection. This is the categorical analogue of the holomorphic anomaly for the B-model potential, whose associated line or ideal is expected to be flat under the Gauss–Manin connection. The statement is presented as a conjectural extension of the Calabi–Yau threefold picture to Calabi–Yau $A_\infty$ categories.