# Treatment of boundary terms when applying the variational principle

One of the main sources of subtlety in the AdS/CFT correspondence is the role played by boundary terms in the action. For example, for a scalar field in AdS there is range of masses just above the Breitenlohner-Freedman bound where there are two possible quantizations and which one you get depends on what boundary terms you add to the action. Boundary terms are also essential in the treatment of first-order Lagrangians for fermions and self-dual tensor fields. These all involve the "UV" boundary as $z \rightarrow 0$ in Poincare coordinates. Then there are dual models of QCD like the hard-wall model where one imposes an IR cutoff and imposes boundary conditions at the IR boundary and/or adds IR boundary terms to the action. My question is a bit vague, but basically I would like references to reviews, books or papers that give a good general treatment of the variational principle when one has to be careful about boundary terms. It would help if they clearly distinguish the requirements that follow from mathematical consistency from those that are imposed because of a desire to model the physics in a certain way.

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UV boundary terms for AdS at conformal infinity aren't problematic. But in Randall-Sundrum models with a Planck brane, things can get a little bit tricky after quantization. – QGR Feb 4 '11 at 17:08
OK, but I don't find this too helpful without some explanation of what the trickiness is or a reference to the literature. – pho Feb 4 '11 at 18:06

## 2 Answers

I'm not sure how far back you want to go, but one of the earliest "careful" treatments (in the Hamiltonian formalism) is in the paper by Regge and Teitelboim:

• T. Regge and C. Teitelboim, “Role Of Surface Integrals In The Hamiltonian Formulation Of General Relativity,” Annals Phys. 88, 286 (1974)

Analogous work in the Lagrangian formalism didn't happen until much more recently. See the paper by Mann and Marolf, or the treatment in this relevant but shameless self-promotion:

• Mann, Marolf, McNees, and Virmani, "On the Stress Tensor for Asymptotically Flat Gravity," (http://arxiv.org/abs/0804.2079)

As far as calculations specific to AAdS spacetimes there is the paper by Skenderis and Papadimitriou

• Papadimitriou and Skenderis, "Thermodynamics of Asymptotically Locally AdS Spacetimes," (http://arxiv.org/abs/hep-th/0505190)

which attempts to formulate things in a language similar to the Wald paper referenced by Moshe. All of these papers are concerned with the consistent treatment of the variational problem on spacetimes with a spatial infinity. The references of the last two will turn up additional relevant works.

You should immediately discount any paper which claims that the inclusion of the Gibbons-Hawking-York surface term gives a good variational problem for a gravitational theory.

This topic -- the proper variational formulation of gravitational theories -- is one of my main interests. I'm trying to take the Red Line from Loyola down to U of C for seminars this term (as teaching allows), so perhaps we can talk about it some time.

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Thanks Robert. Let me know next time you come down to U of C. – pho Feb 4 '11 at 13:27
I'll let you know next time I head down, which will hopefully be soon. – Robert McNees Feb 4 '11 at 13:45
@Robert what is the deficiency with the Gibbons-Hawking-York term? – user346 Feb 4 '11 at 17:19
@Robert, yes I'm curious about your comment on G-H-Y as well. That approach seems to be what Wald does in Appendix E and he is usually quite careful about getting things right. – pho Feb 4 '11 at 18:05
The problem isn't with GHY. It's the claim that adding GHY to the EH action leads to a "well-defined variational problem". Take the EH action and add the GHY term. The Schwarzschild solution should be a stationary point of this action, right? Now consider the change in the action due to a small variation of the metric. You will find that the surface terms in the first-order change in the action only vanish if $\delta g$ falls off faster than 1/r. You have an 'action' that allows Dirichlet boundary conditions, but it can't tell between Schwarzschild and other metrics with 1/r asymptotics. – Robert McNees Feb 4 '11 at 19:32
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Here is a general treatment, Ishibashi and Wald, in a slightly more formal language than normal in this type of discussion. Not sure if this is what you are looking for though.

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