The first edtition of the Mathematica Summer School on Theoretical Physics was devoted to the topic of **Integrability in Gauge/String Dualities** and took place in Porto, from the 15th to the 20th June 2009.

Lecturers:

Miguel Costa

(Porto University)

**Nikolay Gromov**

(St. Petersburg INP & Hamburg University)

Lev Lipatov

(St. Petersburg INP & Hamburg University)

Christoph Sieg

(Niels Bohr Institute)

Matthias Staudacher

(Potsdam, Max-Planck-Institut)

Pedro Vieira

(Potsdam, Max-Planck-Institut)

Dmytro Volin

(CEA-Saclay & Bogolyubov ITP)

**Scientific Advisors:**

Vladimir Kazakov

(Ecole Normale Superieure, Paris)

**Arkady Tseytlin**

(Imperial College, London)

**Why Integrability in Gauge/String dualities?**

The school first edition was dedicated to the theme of Integrability and the Gauge/String duality. This duality, commonly referred as the AdS/CFT duality, is an equivalence between a string theory in a curved space and a gauge theory. The study of these gauge theories is of particular importance both to theoretical and high-energy physics, since our current understanding of the strong interactions is based on a gauge theory known as QCD. Quite remarkably, on the other side of the equivalence there is a theory with gravity, so that the AdS/CFT duality became in the last decade an incredibly rich field for the interplay between experts in gauge theory, string theory and gravitational physics. Very recently this subject became even more fascinating, since experimental results at particle accelerators (RHIC and the forthcoming LHC) require the knowledge of the dynamical behavior of QCD that is not within reach of standard computational techniques. However, such dynamical regime can be analyzed by using the duality, which is now also seen as an important computational tool. Moreover in recent years it was understood that the most known examples of the AdS/CFT duality relate integrable theories, thus amenable of exact solution. Since this discovery this research field has observed a huge growth and has already allowed to access strongly coupled regimes of particles physics and quantum gravity which had never been explored before.

In sum, the subject of Integrability in AdS/CFT is actually one of the hottest and most interesting topics in theoretical physics.This research program envisages solving, for the very first time, a nontrivial four dimensional gauge theory. If such goal is accomplished, it would also teach us a lot about the nature of Quantum Gravity. Moreover Integrability is an extremely beautiful subject per se, with many connections to other areas of physics such as Condensed Matter and Mathematical Physics.

Finally, Integrability is an excellent subject where the power and usefulness of Mathematica can be mostly appreciated. This subject is a very rich and technical one, with many examples where the Mathematica tool is of great importance. To name a few:

• Algebraic manipulations involving the N=4 spin chain S-matrix such as checking its symmetries, the triangle Yang-Baxter relations and understanding the scattering of bound states can take huge advantage of Mathematica.

• The computation of effective spin chain Hamiltonians, such as that following from the Hubbard model at half-filling (an example where this subject touches the condensed matter literature), can also benefit a lot from Mathematica.

• Higher loop computations, such as the remarkable recent four-loop computation of the anomalous dimension of the Konishi operator, could never be done without the use of computer symbolic manipulation and Mathematica is an excellent choice to do that.

• The numeric solution to Bethe equations involving often hundreds of variables can be found with the help of this software.

• High precision expansions checking the transcendentally nature of the N=4 spectrum from the known Bethe equations can be done using Mathematica.

• The computation of classical string solutions and the computation of their semiclassical spectrum using algebraic curve techniques is another example where Mathematica can be of great use.

• The computation of the scaling function, a first example of a non-BPS quantity whose interpolation from weak to strong coupling in this gauge theory is believed to be under control, can be also optimized using Mathematica.