# Hayward term The Hayward term (aka corner term) is a codimension-2 term in the action of a gravitational theory, needed to ensure that the full action has a good variational principle when the (codimension-1) boundary of a spacetime is non-smooth, i.e., has a kink. The original Hayward term was derived for [[0554 Einstein gravity|Einstein gravity]], where the total action is then $\begin{aligned}S&=S_{\mathrm{EH}}+S_{\mathrm{GHY}}+S_{\rm Hayward}\\&=\frac{1}{16\pi G} \int d^{D} x \sqrt{-g} R+\frac{1} {8\pi G}\int d^{D-1} x \sqrt{|h|} K+\frac{1}{8 \pi G} \int_{C}d^{D-2}x \sqrt{|\sigma|}\eta,\end{aligned}$where $\eta \equiv \operatorname{arccosh}\left(n_{0} \cdot n_{1}\right)$ for spacelike joints on a timelike boundary and $\eta \equiv \operatorname{arccosh}\left(-n_{0} \cdot n_{1}\right)$ for spacelike joints on a spacelike boundary. Dirichlet BC is imposed at both the codimension-1 surface and the codimension-2 corner to ensure a good variational principle. In Euclidean signature,$\begin{aligned} I= & -\frac{1}{16 \pi G_N} \int d^Dx \sqrt{g}(R-2 \Lambda)\\&-\frac{1}{8 \pi G_N} \int d^{D-1}x \sqrt{h} K\\ & -\frac{1}{8 \pi G_N} \int d^{D-2}x\sqrt{\gamma}(\pi-\theta) . \end{aligned}$ ## Methods - obtaining the corner term by taking the limit of a smooth codimension-1 surface - [[1993#Hayward (Geoff)]] - [[2018#Cano]] - via direct variation on a codimension-1 surface with a kink - [[2018#Jiang, Zhang]] - a comparison between the limiting procedure and direct variation - [[2017#Yang, Ruan]] ## Examples - [[0554 Einstein gravity|Einstein gravity]] - [[1993#Hayward (Geoff)]]: original - [[2019#Takayanagi, Tamaoka]]: uses it to understand [[0145 Generalised area|HEE]], [[0556 Edge mode|edge modes]], etc. - [[2016#Jubb, Samuel, Sorkin, Surya]]: a nice derivation, both in tetrad and in metric formalism; used in [[2020#Colin-Ellerin, Dong, Marolf, Rangamani, Wang]] - [[0341 Lovelock gravity|Lovelock]] - [[2018#Cano]]: obtains a formula that resembles the JM entropy density (see also [[0145 Generalised area|HEE]]) - f(Riemann) - [[2018#Jiang, Zhang]]: uses auxiliary fields; the resulting formula does not work in the non-generic case of Lovelock ## General refs - originals - [[HartleSorkin1981]] for Regge calculus - [[1993#Hayward (Geoff)]] - maths literature in the context of Gauss-Bonnet *theorem* (not [[0425 Gauss-Bonnet gravity|GB theory]]) - [[Jee1984]] - [[Law1992]] - [[BirmanNomizu1984]] - (\[8-10\] of [[2018#Cano]]) ## Related - [[0138 Variational principle]]