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I have encountered those terms in various places. As I understand it, "soft wall" can correspond to a smooth cutoff of some spacetime, while "hard wall" can be a sharp one, which can be described in terms of D-branes. Could somebody please explain the terminology, and in which context it can occur?

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You are right about your understanding of these terms. This terminology appears in extensions of the Randall-Sundrum type brane world models. The original model contains a single compact extra dimension bounded by two branes and is known as a hard wall model with the "hard wall" referring to the hard cutoff of space by the IR brane. With such a geometry it is found that the Kaluza Klein (KK) masses of particles that live in the bulk scale as $m_n^2 \sim n^2$ (like the energy levels of a particle in a box).

Attempts were made to use RS type setups to be dual to QCD in order to calculate meson masses etc. This is known as ADS/QCD. However the meson mass spectrum is what is called a Regge spectrum i.e. $m_n^2 \sim n$ and so the RS type model needed to be adapted. This paper first introduced the idea of a soft wall to solve this problem. One of the branes in the hard wall model is removed and a dilaton field $\Phi$ is introduced which dynamically cuts off the space-time $$S= \int d^5x \,\sqrt{g}\, e^{-\Phi}\mathcal{L}.$$ The profile of the dilaton in the extra dimension then determines the KK spectrum of bulk fields and for a quadratic dilaton profile ($\Phi(z) \sim z^2$) a Regge spectrum is produced.

The removal of one of the branes (hard spacetime cutoff) and replacement by a smooth dynamical cutoff coming from the dilaton coined the name "soft wall".

Following this idea, people decided to model electroweak physics with such a geometry (see e.g. here). All the standard model fields, including the Higgs must now propagate in the bulk. The new setup offered unique phenomenology and is far less constrained by electroweak precision observables and FCNCs which cause severe tensions in the original RS.

Note that since the dilaton field is not normally given a kintic term in such models, it is not a true dynamical field and one may simply consider the effect as being a different form of metric than RS. So essentially the difference between hard wall and soft wall is just a different geometry of the extra dimension which produces different phenomenology.

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