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# Towards Hodge Theoretic Characterizations of 2d Rational SCFTs

May 2022

The study of rational conformal field theories is of particular interest inthe moduli space of conformal field theories since such rational pointscorrespond to points in moduli space where the algebraic and arithmeticstructure are usually richer, while also being points where non--trivialphysics occurs (such as in the study of attractor black holes and BPS states atrational points). This has led to various attempts to characterize and classifysuch rational points. In this paper, a conjectured characterization byGukov--Vafa of rational conformal field theories whose target space is a Ricciflat K\"ahler manifold is analyzed carefully for the case of toroidalcompactifications. We refine the conjectured statement as well as making aneffort to verify it, using $T^4$ compactification as a test case. Seven commonproperties in terms of Hodge theory (including complex multiplication) havebeen identified for $T^4$-target rational conformal field theories. By imposingthree properties out of the seven, however, there remain $\mathcal N = (1,1)$SCFTs that are not rational. Open questions, implications and future lines ofwork are discussed.

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