# AdS/CFT and Kondo problem/ Ginzburg-Landau theory

1. "In the condensed matter context in two dimensions, one typically flows from an $AdS_4$ geometry near the boundary to an $AdS_2$ times $R_2$ one near the black hole horizon. The net result is local quantum criticality, since the spatial $R_2$ part has essentially decoupled ($AdS_2$ being dual to $CFT_1$, a conformal field theory in time). In that sense, it is similar to the Kondo problem, which is local in space and critical in time."

2. "Although this approach (AdS/CFT) is truly non-perturbative in nature, for most applications, the theory is essentially at the Ginzburg-Landau level. That is, one assumes a scalar field. Since it is a scalar, it should correspond to some charge 2e field, but since the theory does not explicitly invoke pairing, extra terms have to be added to the action to describe the coupling of fermions to the scalar field (i.e, to generate a Bogoliubov dispersion)."

I had hard time trying to understand the boldfaced statements, if someone can comment that'll be helpful. Thanks!

• 1) It can be seen from the correlators for the operators in the dual field theory of AdS2 times R2, e.g. arxiv.org/abs/1105.4621 – user9510 Feb 23 '16 at 11:31
• 2)The thermal phase transition in AdS/CFT is similar to GL theory, i.e. it is a mean field phase transition. However, AdS/CFT of course goes beyond GL theory, see e.g. arxiv.org/abs/1110.3814 – user9510 Feb 23 '16 at 11:33

2). The Ginzburg-Landau level means that the superconductivity is directly described by a charge-$2e$ scalar field, the order parameter, ignoring the electronic origin. A truly microscopic theory, like the BCS theory, starts from electrons and phonons and derive the Ginzburg-Landau theory from there.