### Abstract

We obtain the superconformal algebra associated to a sigma model with target a manifold with G_{2} holonomy, i.e., the Shatashvili-Vafa G_{2} algebra as a quantum Hamiltonian reduction of the exceptional Lie superalgebra D(2, 1; α) for α = 1. We produce the complete family of W-algebras SW (3/2, 3/2, 2) (extensions of the N = 1 superconformal algebra by two primary supercurrents of conformal weight 3/2 and 2 respectively) as a quantum Hamiltonian reduction of D(2, 1; α). As a corollary we find a free field realization of the Shatashvili-Vafa G_{2} algebra, and an explicit description of the screening operators.

Original language | English (US) |
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Pages (from-to) | 331-351 |

Number of pages | 21 |

Journal | Bulletin of the Brazilian Mathematical Society |

Volume | 46 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 2015 |

### Keywords

- Shatashvili-Vafa G superconformal algebra
- W-algebras
- free field realization
- quantum Hamiltonian reduction

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Heluani, R., & Rodríguez Díaz, L. O. (2015). The Shatashvili-Vafa G

_{2}superconformal algebra as a quantum Hamiltonian reduction of D(2, 1; α).*Bulletin of the Brazilian Mathematical Society*,*46*(3), 331-351. https://doi.org/10.1007/s00574-015-0094-x