Landau-Ginzburg skeletons
Ian C. Davenport, Ilarion V. Melnikov
TL;DR
We address the problem of classifying indecomposable two-dimensional (2,2) Landau-Ginzburg theories with central charge $c<6$ by developing a quasi-homogeneous combinatorial framework that encodes each theory with skeleton graphs, roots, pointers, and linking structures. The central tool is the Kreuzer-Skarke correspondence between GC families and quasi-homogeneous isolated singularities, complemented by Frobenius-coin feasibility constraints for the allowed exponents. The main result is a near-complete catalog: 38 infinite families and 418 sporadic indecomposable theories for $c<6$, with rigorous results proven for $n\le 3$ fields and extensive numerical evidence up to $n\le 5$. The findings reveal overlooked singularities and demonstrate a deep interplay between singularity theory and 2D $(2,2)$ SCFTs, providing a foundation for extending to $(0,2)$ LG theories and informing future rigorous proofs of completeness.
Abstract
We study the class of indecomposable two-dimensional Landau-Ginzburg theories with (2,2) supersymmetry and central charge c < 6 with the aim of classifying all such theories up to marginal deformations. Our results include cases overlooked in previous classifications. The results are rigorous for three or fewer fields and more generally are rigorous if we assume an extra bound. Numerics suggest that we have the complete set of indecomposable Landau-Ginzburg families with c<6. This set consists of 38 infinite families and a finite list of 418 sporadic cases. The basic tools are classic results of Kreuzer and Skarke on quasi-homogeneous isolated singularities and solutions to certain feasibility integer programming problems.