### Locally stable SUSY-broken flat vacua in SUGRA

What a long and criptic title! Actually it's a partly abbreviated version of the real one: "Locally stable non-supersymmetric Minkowski vacua in supergravity". This is the paper on which last week's seminar (by Marta) was based, and about which I had promised to write.

The basic idea is to say: observations tell us that we live in a (almost) flat universe with no supersymmetry (for sure not in low energies) and we'd like to see if this two conditions

1) flatness

2) broken supersymmetry

combined with the requirement of (at least local) stability, give some restrictions on the kind of theory we start with. Practically the usual thing to do is to work with a low energy effective field theory, which in the case of string theory is supergravity. The part of the supergravity action that determines these features is the scalar potential, so that's the part that one should study.

These conditions for the vacuum translate for the potential in it having a (local) minimum equal to zero. A minimum means that dV/dx^I=0 for all the chiral superfields x^I and that the matrix V_{IJ} = d^2 V /dx^I dx^J is positive definite. Studying the positive definiteness of a 2n x 2n matrix can be quite difficult, but a necessary and sufficient condition is that every upper-left submatrix should also be positive definite. Then asking a specific upper-left submatrix to be positive definite is a necessary (though not sufficient) condition for the whole matrix to be positive definite. So the authors carried out this analysis for a specific submatrix and found that, using the conditions we started with, but with no assumption on the superpotential, there is a simple condition on the Kähler curvatures (which I won't attempt to write). This simplifies even more with the additional assumption that the Kähler potential is separable for each chiral multiplet.

This is an interesting result because it gives a direct a simple way to check if, for instance, some given Calabi-Yau compactification (with or without fluxes) of string theory could give vacua with these properties (locally stable, flat and broken susy). Actually they go even a little further in their analysis, and find some conditions on the range of F-terms, which in turn are relevant for computing soft terms, but I didn't quite get that part and now they are working on an extension to the case with also vector superfields.

The basic idea is to say: observations tell us that we live in a (almost) flat universe with no supersymmetry (for sure not in low energies) and we'd like to see if this two conditions

1) flatness

2) broken supersymmetry

combined with the requirement of (at least local) stability, give some restrictions on the kind of theory we start with. Practically the usual thing to do is to work with a low energy effective field theory, which in the case of string theory is supergravity. The part of the supergravity action that determines these features is the scalar potential, so that's the part that one should study.

These conditions for the vacuum translate for the potential in it having a (local) minimum equal to zero. A minimum means that dV/dx^I=0 for all the chiral superfields x^I and that the matrix V_{IJ} = d^2 V /dx^I dx^J is positive definite. Studying the positive definiteness of a 2n x 2n matrix can be quite difficult, but a necessary and sufficient condition is that every upper-left submatrix should also be positive definite. Then asking a specific upper-left submatrix to be positive definite is a necessary (though not sufficient) condition for the whole matrix to be positive definite. So the authors carried out this analysis for a specific submatrix and found that, using the conditions we started with, but with no assumption on the superpotential, there is a simple condition on the Kähler curvatures (which I won't attempt to write). This simplifies even more with the additional assumption that the Kähler potential is separable for each chiral multiplet.

This is an interesting result because it gives a direct a simple way to check if, for instance, some given Calabi-Yau compactification (with or without fluxes) of string theory could give vacua with these properties (locally stable, flat and broken susy). Actually they go even a little further in their analysis, and find some conditions on the range of F-terms, which in turn are relevant for computing soft terms, but I didn't quite get that part and now they are working on an extension to the case with also vector superfields.

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