Chiral de Rham Complex on Riemannian Manifolds and Special Holonomy

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.

Recent advances and open questions on the susy structure of the chiral de Rham complex

“…We mention specifically the approach of de Boer, Naqvi and Shommer [14] that is close to our needs in the case of G 2 -manifolds. Another approach to seeing the two commuting N = 2 structures on the chiral de Rham complex of Calabi-Yau manifolds started in [29] and extended for for all special holonomy manifolds in [21] consists on using a Riemannian metric on M instead to produce two different and non-commuting embeddings replacing (3.6.1). Before we describe this we digress on the Boson-Fermion system.…”

“…Conjecture ( [21]). The vertex algebra C ∞ (M, Ω ch M ) carries two global sections X ± generating two commuting copies of SV Spin 7 of central charge c = 12 as in 2.19 in the sense of Remark 2.21.…”

“…s)! (r − s)!s!2 s x s , (3.14.2)and let T r,s := 0 when s < 0 or s > r. The following is the main technical result of[21] …”

Sheaves of N=2 supersymmetric vertex algebras on Poisson manifolds

“…For example, if M admits a generalized Calabi-Yau structure, then there exists an embedding of the N = 2 superconformal vertex algebra into global sections of this sheaf [4]. The reader may find more results along these lines in [2,3,5].…”

Lambda: A Mathematica package for operator product expansions in vertex algebras

“…Thus, insights from the sigma model could be used to further investigate symmetries of the CDR. In [7], the symmetries of the CDR on special holonomy manifolds were investigated. Here, the package Lambda was heavily used, and the OPEs between operators composed of up to five fields were calculated.…”