### 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) |
---|---|

Pages (from-to) | 467-486 |

Number of pages | 20 |

Journal | Nuclear Physics B |

Volume | 337 |

Issue number | 2 |

DOIs | |

State | Published - Jun 18 1990 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Nuclear and High Energy Physics

### Cite this

*Nuclear Physics B*,

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

**Quantum mechanics and the geometry of the Weyl character formula.** / Alvarez, Orlando; Singer, I. M.; Windey, Paul.

Research output: Contribution to journal › Article

*Nuclear Physics B*, vol. 337, no. 2, pp. 467-486. https://doi.org/10.1016/0550-3213(90)90278-L

}

TY - JOUR

T1 - Quantum mechanics and the geometry of the Weyl character formula

AU - Alvarez, Orlando

AU - Singer, I. M.

AU - Windey, Paul

PY - 1990/6/18

Y1 - 1990/6/18

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0011499358&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0011499358&partnerID=8YFLogxK

U2 - 10.1016/0550-3213(90)90278-L

DO - 10.1016/0550-3213(90)90278-L

M3 - Article

AN - SCOPUS:0011499358

VL - 337

SP - 467

EP - 486

JO - Nuclear Physics B

JF - Nuclear Physics B

SN - 0550-3213

IS - 2

ER -