# Susy preserving branes and equation of motion

1. Dec 10, 2012

### switch_df

Hello,

I've thought about it a few times and this question interests me. If a brane (M2 in my case) in euclidean signature preserves some supersymmetry, does it automatically solve the equation of motion?

In order to be more precise here is what I mean:

Take 11D SUGRA with an AdS_4xS^7 background solution. Now embed an M2 brane in this background such that its Lagrangian is simply given by the volume of the brane since the metric has an euclidean signature.

Imagine you were able to prove that this brane preserves some supersymmetry, i.e. the Killing spinors on AdS_4 and S^7 satisfy the usual projection equation. Does it automatically follows that the brane satisfies the equation of motion, i.e. has minimal volume?

This seams to be true because any susy preserving brane in the literature is also solution to the equation of motion. On top of that I have heard that a proof exists in minkowskian signature by using a Hamiltonian formalism so this might as well be true in euclidean signature. One could wick rotate the metric into an minkowskian one and use the Hamiltonian proof but I find it a bit ugly. Has someone ever seen a proof of this?

2. Dec 15, 2012

### mitchell porter

I am happy to discuss this problem, but I would like you to specify conventions and other formalism, e.g. that you would prefer to work according to notations and equations appearing in a particular textbook or paper. Meanwhile see these notes by Joan Simon.

3. Dec 16, 2012

### switch_df

Hello,

Thank you for you reply and the great review you attached to it. The conventions used in the review are the one I am working with. Let's stick to that if it ok for you.

I can more precisely rephrase my question now:

According to equation (214) in the review, the condition for susy for an M2 brane is (1-Gamma)epsilon=0.

In euclidean signature this condition is modified to (1-iGamma)epsilon=0 where Gamma is the same operator as before, given in table 5 (just pick the one for the M2 brane).

The question is now the following: imagine this equation is satisfied for a given Killing spinor epsilon, does this imply that the M2 brane solves the equation of motion, i.e. has minimal volume?

In the review, they go on to show this equivalence in Minkowski signature using a hamiltonian formalism, as I mentionned in my previous post. Since this procedure is not well-defined in euclidean signature, how can we show that this still holds?

Thanks for your help. Hope it's clearer now.

D