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

TL;DR

This paper explores the algebraic and geometric conditions, including mirror symmetry and complex multiplication, that characterize rational 2D superconformal field theories with Ricci-flat Kähler target spaces, providing new proofs and insights.

Contribution

It refines the characterization of rational N=(1,1) SCFTs by establishing necessary and sufficient algebraic conditions, especially for T^4 target spaces, linking geometric data to rationality.

Findings

Proved the conjecture for T^4 target spaces.

Identified algebraicity of geometric data as essential for rationality.

Highlighted the role of mirror symmetry and complex multiplication in SCFT characterization.

Abstract

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.