Period domain of primitive forms for uni-modular exceptional singularities of Arnold

We give a conjectural description of the flat structure on the ring of automorphic forms on a one dimensionally extended domain of a bounded symmetric domain of type IV. The structure appears in the study of period map associated with primitive forms for the 14 exceptional unimodular singularities of Arnold. In physics terminology, the one dimensional extension corresponds to a deformation to an irrelevant direction in the renormalization group action. Mathematically, the extended domain appears mirror symmetrically as the space of stability conditions on certain K3-surfaces (Bridgeland), which has one dimensional negatively graded parameter.