Towards Hodge Theoretic Characterizations of 2d Rational SCFTs

TL;DR

This paper investigates the Hodge theoretic properties of 2D rational superconformal field theories, refining conjectures and identifying key Hodge-theoretic features that characterize rational points in the moduli space, especially for toroidal compactifications.

Contribution

It refines the Gukov--Vafa conjecture on characterizing rational conformal field theories via Hodge theory and verifies these properties in the case of $T^4$ compactifications.

Findings

Seven Hodge-theoretic properties are identified for $T^4$-target rational CFTs.

Imposing three of these properties does not exclude some non-rational $ ext{SCFTs}$.

The study provides a refined conjecture and highlights open questions for future research.

Abstract

The study of rational conformal field theories in the moduli space is of particular interest since these theories correspond to points in moduli space where the algebraic and arithmetic structure are usually richer, while also being points where non--trivial physics occurs (such as in the study of attractor black holes and BPS states at rational points). This has led to various attempts to characterize and classify such rational points. In this paper, a conjectured characterization by Gukov--Vafa of rational conformal field theories whose target space is a Ricci flat K\"ahler manifold is analyzed carefully for the case of toroidal compactifications. We refine the conjectured statement as well as making an effort to verify it, using $T^4$ compactification as a test case. Seven common properties in terms of Hodge theory (including complex multiplication) have been identified for $T^4$-target rational conformal field theories. By imposing three properties out of the seven, however, there remain $\mathcal N = (1,1)$ SCFTs that are not rational. Open questions, implications and future lines of work are discussed.