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Weyl versus conformal invariance in quantum field theory. (English) Zbl 1383.81209
Summary: We argue that conformal invariance in flat spacetime implies Weyl invariance in a general curved background metric for all unitary theories in spacetime dimensions $$d \leq 10$$. We also study possible curvature corrections to the Weyl transformations of operators, and show that these are absent for operators of sufficiently low dimensionality and spin. We find possible ‘anomalous’ Weyl transformations proportional to the Weyl (Cotton) tensor for $$d > 3$$ ($$d=3$$). The arguments are based on algebraic consistency conditions similar to the Wess-Zumino consistency conditions that classify possible local anomalies. The arguments can be straightforwardly extended to larger operator dimensions and higher $$d$$ with additional algebraic complexity.

##### MSC:
 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 53Z05 Applications of differential geometry to physics 81T50 Anomalies in quantum field theory
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