A Comment on Deriving the Gibbons-Hawking-York Term From the String Worldsheet

TL;DR

This paper explores the connection between the boundary term derived from string worldsheet models and the Gibbons-Hawking-York boundary term in gravitational actions, highlighting its role in ensuring well-posed variational principles.

Contribution

It demonstrates that the boundary term from the nonlinear sigma model closely relates to the Einstein boundary term and has half its coefficient, clarifying its role in off-shell gravitational actions.

Findings

The boundary term from the sigma model is related to the Einstein boundary term.

The derived boundary term has half the coefficient of the Gibbons-Hawking-York term.

This boundary term ensures a well-posed variational principle with Dirichlet conditions.

Abstract

In this note, we show that the noncovariant metric boundary term obtained from the nonlinear sigma model worldsheet derivation of the bulk off-shell sphere partition function is closely related to the Einstein boundary term in the Gamma-Gamma noncovariant action. In fact, when expressed in terms of the trace of the extrinsic curvature tensor, we illustrate that this boundary term has one-half the coefficient of the Gibbons-Hawking-York boundary term required such that the total (bulk plus boundary) off-shell classical action has a well-posed variational principle with Dirichlet boundary conditions.