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

Title of host publication | Progress in Mathematics |

Publisher | Springer Basel |

Pages | 393-415 |

Number of pages | 23 |

Volume | 288 |

DOIs | |

State | Published - 2011 |

### Publication series

Name | Progress in Mathematics |
---|---|

Volume | 288 |

ISSN (Print) | 0743-1643 |

ISSN (Electronic) | 2296-505X |

### Fingerprint

### 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

**Heat kernel measures and critical limits.** / Pickrell, Douglas M.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

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

}

TY - CHAP

T1 - Heat kernel measures and critical limits

AU - Pickrell, Douglas M

PY - 2011

Y1 - 2011

N2 - 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 C0(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 = S1, 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.

AB - 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 C0(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 = S1, 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.

KW - Abstract Wiener group

KW - Abstract Wiener space

KW - Critical Sobolev exponent

KW - Feynman–Kac measure

KW - Heat kernel measure

KW - Loop group

KW - Sigma model

UR - http://www.scopus.com/inward/record.url?scp=85028052644&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85028052644&partnerID=8YFLogxK

U2 - 10.1007/978-0-8176-4741-4_12

DO - 10.1007/978-0-8176-4741-4_12

M3 - Chapter

VL - 288

T3 - Progress in Mathematics

SP - 393

EP - 415

BT - Progress in Mathematics

PB - Springer Basel

ER -