Semi-infinite Hodge structure and primitive forms for hyperbolic root systems of rank 2

Semi-infinite Hodge structure equipped with primitive forms is constructed for hyperbolic root systems of rank 2. As the consequences, we determine (1) the flat structure and the Frobenius manifold structure and (2) the ''virtual'' period maps for the primitive forms.

Hyperbolic root systems are the first case of root systems which do not have geometric origin as vanishing cycles (e.g. the monodromy actions are not quasi-unipotent and, hence, the Coxeter number and the exponents are  pure imaginary numbers), several new phenomena  appear.  First, the primitive derivation, whose inverse action determines the Hodge filtration and which itself gives the unit element in the Frobenius algebra, needs to be defined anew by using an equation on the polarization of the Hodge structure and solved by a suitable hypergeometric equation.

After fixing these structures, the constructions of the semi-infinite Hodge structure, primitive forms and the virtual period maps are achieved parallel to the classical period map theory except that the defining domain of the period map is no-longer the Frobenius manifold of the flat coordinates but a wrapping quotient space of its universal covering space. Finally, we  describe the  inversion maps to the virtual period maps, answering to the classical inversion problem, in case of virtual CY-dimension is equal to 2.