Chiral de Rham complex on Riemannian manifolds and special holonomy

TL;DR

This paper explores the structure of the chiral de Rham complex on manifolds with special holonomy, constructing algebraic structures related to supersymmetry and providing explicit examples on Calabi-Yau threefolds.

Contribution

It introduces a systematic method to construct global sections of the chiral de Rham complex on special holonomy manifolds and demonstrates the realization of extended superconformal algebras.

Findings

Constructed two commuting Odake algebras on Calabi-Yau threefolds.

Linked covariantly constant forms to algebraic structures in CDR.

Discussed quasi-classical limits of these algebras.

Abstract

Interpreting the chiral de Rham complex (CDR) as a formal Hamiltonian quantization of the supersymmetric non-linear sigma model, we suggest a setup for the study of CDR on manifolds with special holonomy. We show how to systematically construct global sections of CDR from differential forms, and investigate the algebra of the sections corresponding to the covariantly constant forms associated with the special holonomy. As a concrete example, we construct two commuting copies of the Odake algebra (an extension of the N=2 superconformal algebra) on the space of global sections of CDR of a Calabi-Yau threefold and conjecture similar results for G_2 manifolds. We also discuss quasi-classical limits of these algebras.