Renormalization group flow, Entropy and Eigenvalues

TL;DR

This paper explores the relationship between renormalization group flow, entropy, and eigenvalues of the Laplacian, using heat kernel methods and holographic principles to analyze eigenvalue variations across dimensions.

Contribution

It introduces a conjecture for eigenvalue variation in odd dimensions via holographic renormalization within the AdS/CFT framework.

Findings

Eigenvalue variation in even dimensions linked to heat kernel coefficients

Conjectured formula for eigenvalue variation in odd dimensions

Connections between RG flow irreversibility and entropy concepts

Abstract

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.