A Hiker's Guide to K3 - Aspects of N=(4,4) Superconformal Field Theory with central charge c=6

TL;DR

This work advances a detailed, conformal-field-theory–based map of the moduli space $\mathcal{M}$ for $N=(4,4)$ SCFTs at $c=6$, clarifying the torus and $K3$ components and locating a rich web of orbifold and Gepner-type models within it. By combining Narain-type lattice constructions, Teichmüller theory, and lattice automorphism results (notably for Kummer surfaces), the authors provide a purely CFT derivation of the $K3$ moduli structure and explicitly realize several Gepner constructions as nonlinear sigma models on concrete geometric targets such as the Fermat quartic. Key results include the precise embedding of $\mathcal{M}^{tori}$ into $\mathcal{M}^{K3}$ via $\mathbb{Z}_2$ and $\mathbb{Z}_4$ orbifolds, the determination of B-field values on exceptional divisors (notably $B=\tfrac{1}{2}$ for orbifolds), and the identification of Gepner model $(2)^4$ with the Fermat quartic target space. The paper also demonstrates dualities in this setting (T-duality and Fourier–Mukai transforms) and establishes explicit correspondences between Gepner-type theories and algebraic automorphisms of $K3$ surfaces, yielding a cohesive panorama of the landscape near common meeting points of torus and $K3$ moduli. Overall, the results illuminate how CFT data encode geometric information about $K3$ surfaces and their orbifolds, enabling exact identifications of several important models in the moduli space and their geometric realizations.

Abstract

We study the moduli space ${\cal M}$ of N=(4,4) superconformal field theories with central charge c=6. After a slight emendation of its global description we find the locations of various known models in the component of ${\cal M}$ associated to K3 surfaces. Among them are the Z_2 and Z_4 orbifold theories obtained from the torus component of ${\cal M}$. Here, SO(4,4) triality is found to play a dominant role. We obtain the B-field values in direction of the exceptional divisors which arise from orbifolding. We prove T-duality for the Z_2 orbifolds and use it to derive the form of ${\cal M}$ purely within conformal field theory. For the Gepner model (2)^4 and some of its orbifolds we find the locations in ${\cal M}$ and prove isomorphisms to nonlinear sigma models. In particular we prove that the Gepner model (2)^4 has a geometric interpretation with Fermat quartic target space.

Figures (3)

• Figure 2.1 : Action of the algebraic automorphisms $r_{12}$ (left) and $r_{13}$ (right) on $\Pi$.
• Figure 3.1 : Action of the algebraic automorphism $t_{1111}$ on the Kummer lattice $\Pi$.
• Figure 4.1 : Strata of the moduli space.