Notes on Characterizations of 2d Rational SCFTs: Algebraicity, Mirror Symmetry, and Complex Multiplication

Abstract

S. Gukov and C. Vafa proposed a characterization of rational superconformal field theories (SCFTs) in dimensions with Ricci‐flat Kähler target spaces in terms of the Hodge structure of the target space, extending an earlier observation by G. Moore. The idea is refined, and a conjectural statement on necessary and sufficient conditions for such SCFTs to be rational is obtained, which is indeed proven to be true in the case the target space is . 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.