Hodge Structures of Complex Multiplication Type from Rational Conformal Field Theories

TL;DR

The paper develops a precise link between unitary rational N=(2,2) superconformal field theories and rational Hodge structures of complex multiplication type. By constructing A- and B-type Hodge structures from the (c,c) and (c,a) rings and using boundary Cardy data, the authors obtain rational Hodge structures whose endomorphisms are controlled by Galois actions on modular S-matrices. Under a compatibility condition ensuring the Galois action closes on the RR sector, these Hodge structures exhibit CM-type behavior, with explicit CM fields arising in minimal and Gepner models. The work provides nontrivial evidence that rational N=(2,2) RCFTs linked to Calabi–Yau geometries encode CM-type arithmetic data, and offers explicit examples (e.g., Fermat quintic and octic) where the middle cohomology CM structure matches CFT-derived data, suggesting deep ties between worldsheet physics and arithmetic geometry.

Abstract

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.

Figures (1)

• Figure 1: Relations between different field extensions of $\mathbb{Q}$ relevant for the study of the Galois symmetry of minimal models of an odd level $k$.