Kavli Affiliate: Taizan Watari
| Summary:
These notes combine results from two papers by the present authors viz., Part I (arXiv:2205.10299) and Part II (arXiv:2212.13028) into one streamlined version for better readability, along with a review on theory of complex multiplication for non-singular complex projective varieties and complex tori that is aimed at string theorists. We think that it is worth posting this edition as a separate entry in arXiv for those reasons, although this edition contains no essential progress beyond Part I and Part II.
S. Gukov and C. Vafa proposed a characterization of rational N=(1,1) superconformal field theories (SCFTs) on 1+1 dimensions with Ricci-flat Kahler target spaces in terms of the Hodge structure of the target space, extending an earlier observation by G. Moore. We refined this idea and obtained a conjectural statement on necessary and sufficient conditions for such SCFTs to be rational, which we indeed prove to be true in the case the target space is T^4. In the refined statement, the algebraicity of the geometric data of the target space turns out to be essential, and the Strominger–Yau–Zaslow fibration in the mirror correspondence also plays a vital role.
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