### Abstract

For a given gauge group and compact Riemannian two-manifold, it is known that the associated Yang-Mills measure can be defined directly as a finitely additive measure on the space of connections, and this finitely additive measure is invariant with respect to SDiff, the group of all area-preserving diffeomorphisms of the surface. The first question we address is whether this symmetry essentially characterizes the projection of the Yang-Mills measure to the space of gauge equivalence classes. The proper formulation of this question entails the construction of an SDiff-equivariant completion of the space of continuous connections, such that the projection of the Yang-Mills measure to the space of gauge equivalence classes has a countably additive extension. We also consider the coupling of the Yang-Mills measure to determinants of Dirac operators. The basic problems are to prove that the coupled measure is absolutely continuous with respect to the background Yang-Mills measure, to find a reasonable formula for the Radon-Nikodym derivative, and to analyze the action of SDiff.

Original language | English (US) |
---|---|

Pages (from-to) | 315-367 |

Number of pages | 53 |

Journal | Journal of Geometry and Physics |

Volume | 19 |

Issue number | 4 |

DOIs | |

State | Published - Aug 1996 |

### Fingerprint

### Keywords

- Sdiff
- Wiener measures
- Yang-mills measures
- Zeta determinants

### ASJC Scopus subject areas

- Geometry and Topology
- Mathematical Physics
- Physics and Astronomy(all)

### Cite this

**On Y M _{2} measures and area-preserving diffeomorphisms.** / Pickrell, Douglas M.

Research output: Contribution to journal › Article

_{2}measures and area-preserving diffeomorphisms',

*Journal of Geometry and Physics*, vol. 19, no. 4, pp. 315-367. https://doi.org/10.1016/0393-0440(95)00034-8

}

TY - JOUR

T1 - On Y M2 measures and area-preserving diffeomorphisms

AU - Pickrell, Douglas M

PY - 1996/8

Y1 - 1996/8

N2 - For a given gauge group and compact Riemannian two-manifold, it is known that the associated Yang-Mills measure can be defined directly as a finitely additive measure on the space of connections, and this finitely additive measure is invariant with respect to SDiff, the group of all area-preserving diffeomorphisms of the surface. The first question we address is whether this symmetry essentially characterizes the projection of the Yang-Mills measure to the space of gauge equivalence classes. The proper formulation of this question entails the construction of an SDiff-equivariant completion of the space of continuous connections, such that the projection of the Yang-Mills measure to the space of gauge equivalence classes has a countably additive extension. We also consider the coupling of the Yang-Mills measure to determinants of Dirac operators. The basic problems are to prove that the coupled measure is absolutely continuous with respect to the background Yang-Mills measure, to find a reasonable formula for the Radon-Nikodym derivative, and to analyze the action of SDiff.

AB - For a given gauge group and compact Riemannian two-manifold, it is known that the associated Yang-Mills measure can be defined directly as a finitely additive measure on the space of connections, and this finitely additive measure is invariant with respect to SDiff, the group of all area-preserving diffeomorphisms of the surface. The first question we address is whether this symmetry essentially characterizes the projection of the Yang-Mills measure to the space of gauge equivalence classes. The proper formulation of this question entails the construction of an SDiff-equivariant completion of the space of continuous connections, such that the projection of the Yang-Mills measure to the space of gauge equivalence classes has a countably additive extension. We also consider the coupling of the Yang-Mills measure to determinants of Dirac operators. The basic problems are to prove that the coupled measure is absolutely continuous with respect to the background Yang-Mills measure, to find a reasonable formula for the Radon-Nikodym derivative, and to analyze the action of SDiff.

KW - Sdiff

KW - Wiener measures

KW - Yang-mills measures

KW - Zeta determinants

UR - http://www.scopus.com/inward/record.url?scp=0030215279&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030215279&partnerID=8YFLogxK

U2 - 10.1016/0393-0440(95)00034-8

DO - 10.1016/0393-0440(95)00034-8

M3 - Article

AN - SCOPUS:0030215279

VL - 19

SP - 315

EP - 367

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

IS - 4

ER -