Hodge Structures of Complex Multiplication Type from Rational Conformal Field Theories

Under certain assumptions, we show that unitary rational $\mathcal{N}=(2,2)$ conformal field theories together with a certain generating set of Cardy boundary states in the associated boundary conformal field theories give rise to rational Hodge structures of complex multiplication type. We argue that these rational Hodge structures for such rational conformal field theories arising from infrared fixed points of $\mathcal{N}=(2,2)$ non-linear sigma models with Calabi-Yau target spaces coincide with the rational Hodge structures of the middle-dimensional cohomology of the target space geometry. This gives non-trivial evidence of the general expectation in the literature that rational $\mathcal{N}=(2,2)$ supersymmetric conformal field theories associated to Calabi-Yau target spaces yield middle dimensional cohomological rational Hodge structures with complex multiplication. We exemplify our general results with the $\mathcal{N}=2$ A-type minimal model series - which do not have a geometric origin as a non-linear sigma model - and with two explicit $\mathcal{N}=(2,2)$ Gepner models that correspond to particular non-linear sigma models with specific Calabi-Yau threefold target spaces.