No coin nor oath required. For personal study only.
containing possible obstructions to the renormalization of the transformation laws, and of noting the absence of obstructions when M has finite fundamental group and nonabelian semi-simple isometry group. The renormalization group equation for the metric coupling is A-i(8/&-')gy= -bg(g) &tT-'d = -ET-k,+ R,+~~Rpq,J&r+ WT*J. &,, is the curvature tensor and RU= Ripjp the Ricci tensor of the metric gy. The b-function fi,,&) is a vector held on the iniinite dimensional space of Riemannian metrics on M. Two results on global properties of /I are obtained. When M is a homogeneous space G/H, the b-function is shown to be a gradient on the tinite dimensional space of Ginvariant metric couplings on M. And, when ,%f is a two-dimensional compact manifold, the b-function is shown to be a gradient on the intinite dimensional space of metrics on M. The rest of the results are concerned with fixed points. The fixed points are shown to correspond to the metrics satisfying a generalized Einstein equation Rj, -og,, = ~,a, + r,r,, c = & 1 or 0 for ~1' some vector held on M. Known solutions to these equations are discussed and some of their general properties described. In particular, it is shown that infrared instability occurs in at most a mute number of directions in the iminite dimensional space of metric couplings.
📜 SIMILAR VOLUMES
We review and develop geometrical gauging involving the sequence: Lie group/Principal Bundle, for an Internal symmetry group/Soft Group Manifold, for Non-Internal groups. In the Internal case, we rederive the Ghost-fields and the BRS transformations and equations geometrically. In the case of a Soft