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Let $X$ be a compact connected strongly pseudoconvex CR manifold of dimension$2n+1, n\ge 1$ with a transversal CR $S^1$-action on $X$. In this paper weintroduce the Quillen metric on the determinant line of the Fourier componentsof the Kohn-Rossi cohomology on $X$ with respect to the $S^1$-action. We studythe behavior of the Quillen metric under the change of the metrics on themanifold $X$ and on the vector bundle over $X$. We obtain an anomaly formulafor the Quillen metric on $X$ with respect to the $S^1$-action.

The anomaly formula of the analytic torsion on CR manifolds with $S^1$ action