### Abstract

This article is an exposition of several questions linking heat kernel measures on infinite-dimensional Lie groups, limits associated with critical Sobolev exponents, and Feynman–Kac measures for sigma models. The first part of the article concerns existence and invariance issues for heat kernel measure classes. The main examples are heat kernel measures on groups of the form C^{0}(X,F), where X is a Riemannian manifold and F is a finite-dimensional Lie group. These measures depend on a smoothness parameter s > dim (X)/2. The second part of the article concerns the limit s ↓ dim(X)/2, especially dim(X) ≤ 2, and how this limit is related to issues arising in quantum field theory. In the case of X = S^{1}, we conjecture that heat kernel measures converge to measures which arise naturally from the Kac–Moody–Segal point of view on loop groups, as s ↓ 1/2.

Original language | English (US) |
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Title of host publication | Progress in Mathematics |

Publisher | Springer Basel |

Pages | 393-415 |

Number of pages | 23 |

Volume | 288 |

DOIs | |

Publication status | Published - 2011 |

### Publication series

Name | Progress in Mathematics |
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Volume | 288 |

ISSN (Print) | 0743-1643 |

ISSN (Electronic) | 2296-505X |

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### Keywords

- Abstract Wiener group
- Abstract Wiener space
- Critical Sobolev exponent
- Feynman–Kac measure
- Heat kernel measure
- Loop group
- Sigma model

### ASJC Scopus subject areas

- Algebra and Number Theory
- Analysis
- Geometry and Topology

### Cite this

*Progress in Mathematics*(Vol. 288, pp. 393-415). (Progress in Mathematics; Vol. 288). Springer Basel. https://doi.org/10.1007/978-0-8176-4741-4_12