Superconformal field theories and cyclic homology

TL;DR

This paper explores the connection between superconformal field theories and cyclic homology, revealing how the spectrum of protected operators relates to mathematical structures and string theory dualities.

Contribution

It introduces a novel approach linking the spectrum of protected operators in superconformal theories to cyclic homology, utilizing the Hochschild-Kostant-Rosenberg theorem.

Findings

Spectrum of protected operators corresponds to cyclic homology groups.

Cyclic homology relates to deRham cohomology via Hochschild-Kostant-Rosenberg theorem.

Mathematical map models open-closed string transition in AdS/CFT.

Abstract

One of the predictions of the AdS/CFT correspondence is the matching of protected operators between a superconformal field theory and its holographic dual. We review the spectrum of protected operators in quiver gauge theories that flow to superconformal field theories at low energies. The spectrum is determined by the cyclic homology of an algebra associated to the quiver gauge theory. Identifying the spectrum of operators with cyclic homology allows us to apply the Hochschild-Kostant-Rosenberg theorem to relate the cyclic homology groups to deRham cohomology groups. The map from cyclic homology to deRham cohomology can be viewed as a mathematical avatar of the passage from open to closed strings under the AdS/CFT correspondence.