This set-up occurs when the brane degrees of freedom are responsible for either breaking the symmetries of the larger system or describing an interesting isolated set of massless degrees of freedom whose interactions among themselves and with the background one is interested in studying. In the following, I very briefly describe how the first approach was used to introduce flavour in the AdS/CFT correspondence, and how the second one can be used to study physics reminiscent of certain phenomena in condensed-matter systems.

As an example, consider the addition of flavour in the standard AdS/CFT. It was argued in [333] that this could be achieved by adding D7-branes to a background of D3-branes. The D7-branes give rise to fundamental hypermultiplets arising from the lightest modes of the 3-7 and 7-3 strings, in the brane array

The mass of these dynamical quarks is given by , where is the distance between the D3- and the D7-branes in the 89-plane. If the D3-branes may be replaced (in the appropriate decoupling limit) by an AdS This logic can be extended to non-supersymmetric
scenarios^{44}.
For example, using the string theory realisation of four-dimensional QCD with colours and
flavours discussed in [499]. The latter involves D6-brane probes in the supergravity background dual
to D4-branes compactified on a circle with supersymmetry-breaking boundary conditions and
in the limit in which all the resulting Kaluza–Klein modes decouple. For and for
massless quarks, spontaneous chiral symmetry breaking by a quark condensate was exhibited
in [349] by working on the D6-brane effective action in the near horizon geometry of the
D4-branes.

Similar considerations apply at finite temperature by using appropriate black-hole backgrounds [499] in the relevant probe action calculations. This allows one to study phase transitions associated with the thermodynamic properties of the probe degrees of freedom as a function of the probe location. This can be done in different theories, with flavour [379], and for different ensembles [343, 378].

The amount of literature in this topic is enormous. I refer the reader to the reviews on the use of gauge-gravity duality to understand hot QCD and heavy ion collisions [137] and meson spectroscopy [207], and references therein. These explain the tools developed to apply the AdS/CFT correspondence in these set-ups.

One interesting possibility involving this mechanism consists on the emergence of gauge fields (“photons”) at the onset of such critical phases. For example, 2 + 1 Maxwell theory in the presence of a Fermi surface (chemical potential )

is supposed to describe at energies below , the interactions between gapless bosons (photons) with the fermionic excitations of the Fermi surfaces. The one-loop correction to the classical photon propagator at low energy and momenta is Due to the presence of the chemical potential, this result manifestly breaks Lorentz invariance, but there exists a non-trivial IR scaling symmetry (Lifshitz scale invariance) with dynamical exponent , replacing the UV scaling . Since these systems are believed to be strongly interacting, it is an extremely challenging theoretical task to provide a proper explanation for them. It is this strongly-coupled character and the knowledge of the relevant symmetris that suggest one search for similar behaviour in “holographic dual” descriptions.The general set-up, based on the discussions appearing, among others, in [334, 398, 286], is as follows. One considers a small set of charged degrees of freedom, provided by the probe “flavour” brane, interacting among themselves and with a larger set of neutral quantum critical degrees of freedom having Lifshitz scale invariance with dynamical critical exponent . As in previous applications, the latter is replaced by a gravitational holographic dual with Lifshitz asymptotics [324]

where will play the role of the holographic radial direction. Turning on non-trivial temperature corresponds to considering black holes having the above asymptotics [162, 370, 102, 34] where the function depends on the specific solution and characterises the thermal nature of the system.In practice, one embeds the probe “flavour” brane in the spacetime holographic dual, which may include some non-trivial cycle wrapping in internal dimensions when embedded in string theory, and turns on some non-trivial electric and magnetic fluxes on the brane

At low energies and in a quantum critical system, the only available scales are external, i.e., given by temperature , electric and magnetic fields and the density of charge carriers . Solving the classical equations of motion for the world volume gauge field, allows one to integrate , whose constant behaviour at infinity, i.e., at in the above coordinate system, defines the chemical potential of the system. Working in an ensemble of fixed charge carrier density , which is determined by computing the variation of the action with respect to , the free energy density is given by where stands for the volume of the non-compact 2-space spanned by and is the on-shell Dp-brane action. As in any thermodynamic system, observables such as specific heat or magnetic susceptibility can be computed from Eq. (469) by taking appropriate partial derivatives. Additionally, transport observables, such as DC, AC or DC Hall conductivities can also be computed and studied as a function of the background, probe embedding and the different constants controlling the world volume gauge field (468).More than the specific physics, which is nicely described in [334, 398, 286], what is important to stress, once more, is that using the appropriate backgrounds, exciting the relevant degrees of freedom and considering the adequate boundary conditions make the methods described in this review an extremely powerful tool to learn about physics in regimes of parameters that would otherwise be very difficult to handle, both analytically and conceptually.

Living Rev. Relativity 15, (2012), 3
http://www.livingreviews.org/lrr-2012-3 |
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