Effective action for conformal spins on spheres with the multiplicative and conformal anomalies

Two multiplicative anomalies are evaluated for the determinant of the conformal higher spin propagating operator on spheres given by Tseytlin. One holds for the decomposition of the higher derivative product into its individual second order factors and the other applies to its complete linear factorisation. Using this last factorisation, I also calculate the determinant explicitly in terms of the Riemann zeta function, for both even and odd dimensions. In the latter case there is no anomaly of course. The conformal anomaly is also found for arbitrary spin and dimension.

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