# Hamilton-Jacobi Approach to Holographic Renormalization of Massive Gravity

## TL;DR

This paper applies the Hamilton-Jacobi approach to holographic renormalization in massive gravity, deriving counterterms and revealing novel features like conformal anomalies in odd dimensions.

## Contribution

It extends the Hamilton-Jacobi holographic renormalization method to massive gravity and uncovers new phenomena such as conformal anomalies in odd dimensions.

## Key findings

- Counterterms derived up to boundary dimension d=4.
- Conformal anomaly can occur in odd dimensions.
- Counterterms cancel divergences in on-shell action.

## Abstract

Recently, a practical approach to holographic renormalization has been developed based on the Hamilton-Jacobi formulation. Using a simple Einstein-scalar theory, we clarify that this approach does not conflict with the Hamiltonian constraint as it seems. Then we apply it to the holographic renormalization of massive gravity. We assume that the shift vector is falling off fast enough asymptotically. We derive the counterterms up to the boundary dimension d=4. Interestingly, we find that the conformal anomaly can even occur in odd dimensions, which is different from the Einstein gravity. We check that the counterterms cancel the divergent part of the on-shell action at the background level. At the perturbation level, they are also applicable in several time-dependent cases.