Abstract
Perturbative quantization of Yang-Mills theory with a gauge algebra given by the classical double of a semisimple Lie algebra is considered. The classical double of a real Lie algebra is a nonsemisimple real Lie algebra that admits a nonpositive definite invariant metric, the indefiniteness of the metric suggesting an apparent lack of unitarity. It is shown that the theory is UV divergent at one loop and that there are no radiative corrections at higher loops. One-loop UV divergences are removed through renormalization of the coupling constant, thus introducing a renormalization scale. The terms in the classical action that would spoil unitarity are proved to be cohomologically trivial with respect to the Slavnov-Taylor operator that controls gauge invariance for the quantum theory. Hence they do not contribute gauge invariant radiative corrections to the quantum effective action and the theory is unitary.
Highlights
Nonreductive metric Lie algebras are Lie algebras that (i) cannot be written as a direct product of semisimple and Abelian Lie algebras but (ii) admit a metric, where by a metric is meant a nondegenerate symmetric bilinear form that is invariant under the adjoint action
The pattern observed for the gauge invariant degrees of freedom in the quantum theory resembles very much that for the self-antiself dual instantons of the classical theory [16]
The number of collective coordinates of the G instantons is twice that of the embedded G instantons, yet ωab FTaμν FZbμν does not contribute to the instanton number
Summary
Nonreductive metric Lie algebras are Lie algebras that (i) cannot be written as a direct product of semisimple and Abelian Lie algebras but (ii) admit a metric, where by a metric is meant a nondegenerate symmetric bilinear form that is invariant under the adjoint action. We will be concerned with perturbative quantization of Yang–Mills theory for a particular class of such algebras, known as classical doubles. The classical double, we denote it as g , of any real Lie algebra g is a Lie algebra of dimension twice the dimension of g that admits a metric This metric determines a Yang– Mills Lagrangian for the field that results from gauging the algebra. Every g instanton has an embedded g instanton with the same instanton number and twice the number of collective coordinates This doubling of degrees of freedom and the simpler structure of classical doubles as compared to double extensions suggest considering perturbative quantization of Yang–Mills theory with gauge algebra g.
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