Of McKay Correspondence, Non-linear Sigma-model and Conformal Field Theory

TL;DR

This paper explores a unifying framework connecting ADE classifications across mathematics and physics, linking quiver theory, McKay correspondence, modular invariants, and string theory models in two complex dimensions.

Contribution

It proposes a comprehensive scheme that unifies various ADE correspondences in mathematics and physics within a two-dimensional complex setting.

Findings

Constructs an intricate web of inter-relations among ADE correspondences.

Provides a potential pathway for new research directions in higher dimensions.

Suggests a unified framework for understanding mathematical and physical ADE phenomena.

Abstract

The ubiquitous ADE classification has induced many proposals of often mysterious correspondences both in mathematics and physics. The mathematics side includes quiver theory and the McKay Correspondence which relates finite group representation theory to Lie algebras as well as crepant resolutions of Gorenstein singularities. On the physics side, we have the graph-theoretic classification of the modular invariants of WZW models, as well as the relation between the string theory nonlinear $\sigma$-models and Landau-Ginzburg orbifolds. We here propose a unification scheme which naturally incorporates all these correspondences of the ADE type in two complex dimensions. An intricate web of inter-relations is constructed, providing a possible guideline to establish new directions of research or alternate pathways to the standing problems in higher dimensions.