## Group Members

Often in physics it appears that the same physical system admit two (or even more) mathematical descriptions. Usually in this case we speak about the duality between different mathematical descriptions. AdS/CFT is one of this dualities which can help us to solve many problems in different fields of theoretical physics in the future.

In its original formulation[1] AdS/CFT is the duality between 4-dimensional = 4 supersymmetric Yang-Mills theory (SYM) and string theory that lives in 5-dimensional Anti-de Sitter space. This duality is also oftenly addressed as holographic duality, because one of the most striking things in this duality is correspondence between theories that lives in different number of dimensions. This resembles the usual Gabors holography, where people reproduce 3-dimensional image of the object using 2-dimensional plate that contains all information needed.

### What is N = 4 SYM theory?

This is one of the most interesting example of Quantum Field Theories (QFT) in the modern theoretical physics. This is Yang-Mills theory which makes it distinct relative of Quantum Chromodynamics (QCD) – the theory of strong interactions that physicists are trying to solve already for half a century.

Though being relative to QCD, = 4 SYM posses many nice properties that QCD doesn't have. First of this properties is supersymmetry which means that for each boson in theory we have a fermion related to this boson by a symmetry transformation. Second property is conformal symmetry. This means the theory is invariant under conformal transformations, i.e. transformations preserving angles (see example on the picture). In most cases in physics that just means that the theory is invariant under rescaling. Field theories invariant under conformal transformations are called CFTs. Finally the last interesting property is integrability, which will be discussed further.

All this nice properties change the behaviour of the theory a lot. The main difference between = 4 SYM and QCD is the absence of the confinement, which makes the theory much more simple. However = 4 SYM, being exactly solvable, still remains a good playground for the study of nonabelian gauge theories.

### What is Anti-de Sitter (AdS) space?

Anti-de Sitter space is just the space in which distance between different points is defined differently from our usual Minkowski space. On the picture here you can see the famous print by Dutch artist M. C. Escher, which visualise the tessellation of an AdS-like space by triangles and squares. All triangles and squares are of equal size if we take into account a proper definition of distance between points.

The important fact about d-dimensional AdS space is that it has a boundary and this boundary is (d-1)-dimensional Minkowski space.

To summarize, the original statement of AdS/CFT duality was considering string theory living in 5-dimensional AdS space and a particular 4-dimensional theory living on the boundary of AdS-space – in our usual 4-dimensional world. However during the last decade there was huge progress in the area and in our days people have constructed similar dualities for many different supersymmetric, conformal theories[2]-[6].

One of the most interesting and useful features of the duality is weak/strong coupling correspondence. Usually QFTs are well described in the regime of weak coupling, when we can express all observables in the theory as a series expansion in this small coupling (perturbation theory).

However if the coupling becomes strong perturbation theory doesn't make sense. This problem is found in many important examples in physics. In particular the problem of the description of nuclear matter is related to the strongly coupled regime of QCD. Another example of strongly coupled systems are high-temperature superconductors. Constructing a proper theory of high-temperature superconductivity is the “holy grail” problem of modern condensed matter physics. There are a lot of such examples and to resolve them we need to come up with some tool for the work with with strongly coupled theories.

Fortunately AdS/CFT correspondence provide us with such tool. It appears that the strong coupling limit of the boundary theory, which should describe some physical phenomenon, is dual to the solvable limit of string theory in AdS space. Recently there has been significant progress in applying AdS/CFT correspondence to nuclear physics[7]-[9], condensed matter physics[10]-[12] and even hydrodynamics[13].

## References

[1] Maldacena, Juan. “The Large N limit of superconformal field theories and supergravity”, (1998) (original paper on AdS/CFT)

[2] Aharony, Ofer; Bergman, Oren; Jafferis, Daniel Louis; Maldacena, Juan; “N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals” (2008) (example of AdS/CFT duality in 3 dimensions)

[3] Horatiu Nastase “Introduction to AdS-CFT” (2007) (review of AdS/CFT duality, basic level)

[3] Juan Maldacena “TASI 2003 Lectures on AdS/CFT” (2003) (review of AdS/CFT duality, intermediate level)

[4] F. Bigazzi; A. L. Cotrone; M. Petrini; A. Zaffaroni; “Supergravity duals of supersymmetric four dimensional gauge theories” (2003) (review of AdS/CFT duality, advanced level)

[5] Eric D'Hoker; Daniel Z. Freedman; “Supersymmetric Gauge Theories and the AdS/CFT Correspondence” (2002) (review of AdS/CFT duality, advanced level)

[6] O. Aharony; S.S. Gubser; J. Maldacena; H. Ooguri; Y. Oz; “Large N Field Theories, String Theory and Gravity” (1999) (review of AdS/CFT duality, advanced level)

[7] D. T. Son; A. O. Starinets; “Viscosity, Black Holes, and Quantum Field Theory” (2007) (original work on the viscosity of quark-gluon plasma)

[8] Johanna Erdmenger; Nick Evans; Ingo Kirsch; Ed Threlfall; “Mesons in Gauge/Gravity Duals – A Review” (2007) (review on holographic mesons)

[9] Youngman Kim; Ik Jae Shin; Takuya Tsukioka; “Holographic QCD: Past, Present, and Future” (2012) (review on the recent progress in AdS/QCD)

[10] Sean A. Hartnoll; “Lectures on holographic methods for condensed matter physics” (2009) (review on the holographic methods in condensed matter physics)

[11] John McGreevy; “Holographic duality with a view toward many-body physics” (2009) (review on the holographic methods in condensed matter physics)

[12] Subir Sachdev; “What can gauge-gravity duality teach us about condensed matter physics?” (2011) (review on the holographic methods in condensed matter physics)

[13] Mukund Rangamani; “Gravity & Hydrodynamics: Lectures on the fluid-gravity correspondence” (2009) (review on the holographic methods in hydrodynamics)