### Abstract

General field theoretic methods are developed which will allow a path integral derivation of the character formula for loop groups. The methods are introduced in the classical Weyl character case. The irreducible representations of a compact semi-simple Lie group G are realized as the ground states of a supersymmetric quantum mechanical system. The Hilbert space for the quantum mechanical system is the space of sections of a holomorphic line bundle L over the complex manifold G/T, where T is the maximal torus of G. The Weyl character formula is derived by an explicit path integral computation of the index of the Dolbeault operator ∂_{L}.

Original language | English (US) |
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Pages (from-to) | 467-486 |

Number of pages | 20 |

Journal | Nuclear Physics, Section B |

Volume | 337 |

Issue number | 2 |

DOIs | |

State | Published - Jun 18 1990 |

Externally published | Yes |

### ASJC Scopus subject areas

- Nuclear and High Energy Physics

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## Cite this

Alvarez, O., Singer, I. M., & Windey, P. (1990). Quantum mechanics and the geometry of the Weyl character formula.

*Nuclear Physics, Section B*,*337*(2), 467-486. https://doi.org/10.1016/0550-3213(90)90278-L