  The irreversibility of the renormalization group flow is conjectured to be
closely related to the concept of entropy. In this paper, the variation of
eigenvalues of the Laplacian in the Polyakov action under the renormalization
group flow will be studied. Based on the one-loop approximation to the
effective field theory, we will use the heat kernel method and zeta function
regularization. In even dimensions, the variation of eigenvalues is given by
the top heat kernel coefficient, and the conformal anomaly is relevant. In odd
dimensions, we will conjecture a formula for the variation of eigenvalues
through the holographic renormalization in the setting of geometric AdS/CFT
correspondence.
