The GHY boundary term from the string worldsheet to linear order

TL;DR

This paper derives the boundary term of the Einstein-$ Gamma^2$ action in half-space using the method of images, including the Gibbons-Hawking-York term and additional metric-dependent terms, ensuring a well-posed variational principle.

Contribution

It provides the first derivation of the boundary term for the Einstein-$ Gamma^2$ action in half-space to linear order in metric perturbation and first order in $\alpha'$.

Findings

Derived the boundary term of the Einstein-$ Gamma^2$ action in half-space.

Included Gibbons-Hawking-York and additional metric-dependent boundary terms.

Ensured a well-posed variational principle for Dirichlet boundary conditions.

Abstract

Using the method of images we derive the boundary term of the Einstein-$\Gamma^2$ action in half-space from the spherical worldsheet to first order in $\alpha'$ and to linear order in the metric perturbation around flat half-space. The $\Gamma^2$ action, written down by Einstein more than 100 years ago, includes a boundary term that consists of the Gibbons-Hawking-York action along with two additional terms that are functions of the metric, normal vector, and tangential derivatives. With this boundary term, the total (bulk + boundary) sphere effective action has a well-posed variational principle for Dirichlet boundary conditions.