By holography we usually mean the connection between conformal quantum field theories in d-spacetime dimensions with classical solutions of gravitational theories on the (d+1)-dimensional spacetime with negative curvature. They represent weak-strong dualities, and are so very useful in modelling otherwise inaccessible strong
coupling field theories. Conformal theories do not possess a mass scale and are quite common: quantum field theories are in fact essentially flows between UV and IR fixed points. Such fixed points exhibit more-conformal-symmetries. Due to them correlation functions are additionally restricted and can be sometimes determined exactly. In such cases the correspondence (called AdS/CFT) has been tested, but in most cases it is postulated, i.e. the strongly coupled theory essentially defined through it.
Correlation functions of some conformal theories are somewhat surprisingly related to a completely different type of theories, supersymmetric ones. There has been a huge amount of work in supersymmetric theories,
both in particle physics phenomenology and formal theory. While from the phenomenological point of view only the so called N=1 supersymmetries in 4 spacetime dimensions are chiral and thus interesting, from the formal point of view the N=2 theories are mathematically even more interesting. The large symmetry involved permits an exact, the so-called Seiberg-Witten solution of the low-energy theory. Its generalisation is particularly useful to describe some conformal theories in two dimensions.
These techniques can be used in various contexts. For example, in describing a scalar propagator in a conformal theory at non-zero temperature, one is often using a holographic approach. This turns the problem into a solution of the Heun equation. It has been shown that in the limit of zero energy the low temperature expansion of the propagator can be obtained directly from such deformed Seiberg-Witten solution. It would be interesting to generalise this solution to non-zero energies. Another interesting aspect of the Heun equation is its relation through isomonodromic deformations to the Painleve VI equation, which solution, the tau function, is known exactly in terms of the conformal correlation functions mentioned above. It would be interesting to understand better here the interplay of the correlation functions of different central charges. Another phenomenon which can be studied using holography is the spectral form factor. Its behaviour in various regimes
like the slope, ramp and plateau can be explicitly reproduced in some toy models. Although the plateau is very difficult to reproduce, the calculation of the ramp boils down to obtain the quasi-normal modes of the asymptotically AdS black-hole.