# String Theory and Mathematical Physics

The group investigates a variety of topics, either related to current themes in string theory and gravity research, or to applications of mathematical methods in physics. The group interacts with a variety of other groups in Kumpula Campus with overlapping interests, and coordinates and develops teaching in the mathematics-physics interface.

The group has recently moved from Helsinki Institute of Physics (HIP), where it was created and operated for three years with the same title, String Theory and Mathematical Physics. (Link currently inactive.) It continues to collaborate closely with HIP projects Laws of Nature and Condensed Particle Matter @ LHC and Cosmophysics along with other research groups of the Elementary Particle Physics section of the University of Helsinki Physics Department. In mathematical physics, the group collaborates e.g. with the Mathematical Physics Research Group of the Centre of Excellence in Analysis and Dynamics at UH. Applications of mathematical methods to applied physics are currently being developed in collaboration with the Electronics laboratory.

The group has extensive international contacts and collaboration.

### String Theory Today

The name "String Theory" is today an umbrella term for many different research directions. Originally, the idea of elementary particles as different vibrational modes of an elastic string (like a short piece of a violin string) was conceived in 1970s as a principle to understand some properties of a class of new particles that had been discovered in particle accelerators. Subsequent work showed that strings could also form close loops, and then their vibrational states would include a fundamental quantum of gravitational force, the graviton. This lead to a reinterpretation of the string idea. Because gravity is a very weak force, the length scale of strings was more naturally close to the extremely small length scale of quantum gravity (the Planck length) rather than the much longer scales which can be directly studied in particle accelators.

Because the natural scale of string theory is not directly testable, research in string theory has followed at least two main roads. One direction of research tries to derive the experimentally tested Standard Model of particle physics at low energies (or known extensions of it), from string theory at Planck scale. The main way to do this is to understand what is the self-consistent geometry where the strings move. Mathematically, this becomes a question of "compactifying" six of the ten (mathematical) dimensions of string theory and arriving at the four space-time dimensions that are "real" and observable to us and particles in current colliders. The symmetry properties of the Standard Model would then have a geometrical explanation as the symmetries of the unseen space dimensions. More recently it has been understood that the (mathematically) ten dimensional string theory is a limit of a more general (mathematically) eleven dimensional M-theory, and this has given new ways to address the compactification problem. However, this whole line of work has turned out to be ultra-difficult, so progress in making detailed contact with low-energy particle physics, and new testable predictions, is slower than what was enthusiastically anticipated in the early years of string theory. There have been many cases in string theory where a sudden new insight has allowed rapid progress in problems that were previously thought to be intractable, and intense work continues. In the mean time, the study of the compactification problem continues to have an enormous impact in the development of mathematics in various areas, e.g. algebraic geometry.

Another central research theme in string theory emphasizes it as a candidate theory to unify quantum mechanics with General Relativity. Quantum gravitational effects become significant when the gravitational force becomes very strong, such as in the interior of black holes or in the Very Early Universe. Research in this direction has been strongly influenced by the desire to reconcile basic rules of quantum mechanics with gravity. For example, a famous puzzle for several decades has been to understand in detail how quantum mechanical information is preserved and redistributed in a process where a black hole forms and then evaporates because of Hawking radiation. A breakthrough step was to realize that in a quantum theory of gravity contains much less information than an ordinary quantum theory of matter. This idea was called the Holographic Principle of gravity, formulated by 't Hooft and Susskind in mid-90's. It was soon followed by two other breakthrough results: a matrix model of M-theory, and the so-called AdS/CFT correspondence, in the end of 90's. These two results have lead to detailed models of how the information in a gravitational theory can be recoded in a radically different form, as a quantum theory of matter without gravity in one less space dimension. An interesting spin-off of this new framework is that it gives powerful techniques to model difficult problems in other areas of physics. The study of such applications is one of the frontier directions of theoretical research at the moment.

The techniques of AdS/CFT correspondence are very powerful in analyzing transport problems in various physics contexts. In a sense it gives a new way to construct effective theories of strongly interacting quantum matter, based on the language of gravity. This approach has given new insights and viewpoints to the physics of hot and densed quark-gluon matter created in ultrarelativistic collisions of heavy ions in Brookhaven and CERN experiments, and to various condensed matter physics topics such as quantum phase transitions, hydrodynamics, strange fermion matter, superfluids and -conductors, striped phases, topological insulators, quantum entanglement, and far-from equilibrium dynamics. The String Theory and Mathematical Physics group has an active role in developing these frontier research directions.

### Mathematical Physics

The group has been studying various problems related to Mathematical Physics. Most notably, we have been studying Random Matrix Thoery as a technique to analyze the decay of D-branes in string theory, and have been deriving some new mathematical results in the process. These techniques can also be applied to the study of thermodynamics of log-gases, and most recently we have been applying them to study the structure of N=1 supersymmetric Yang-Mills theories.

As the most recent new direction, we have been studying applications of stochastic differential equations and processes to analyze data from some experiments conducted by the Electronics group at University of Helsinki. We hope to post more details about this interesting work here soon.

**Contact person:** Esko Keski-Vakkuri.

**Doctoral students and some recent collaborators:**

- MSc Janne Alanen, doctoral student

MSc Lasse Franti, doctoral student

MSc Ville Suur-Uski, doctoral student

Dr. Niko Jokela, U. Santiago de Compostela, Spain (former student)

Dr. Matti Järvinen, U. Crete, Greece

Dr. Shinsuke Kawai, Sungkyunkwan U., South Korea (former postdoc)

Dr. Ville Keränen, NORDITA, Sweden (former student)

Dr. Asad Naqvi, The Institute of Advanced Study, USA

Dr. Sean Nowling, NORDITA, Sweden(former postdoc)

Prof. K. "Patta" Yogendran, IISER Mohali, India (former postdoc)

Prof. Vijay Balasubramanian, U. Pennsylvania, USA

Prof. Ben Craps, Vrije Universiteit Brussel, Belgium

Prof. Jan de Boer, U. Amsterdam, The Netherlands

Prof. Per Kraus, UCLA, USA

Prof. Robert Leigh, U. Illinois at Urbana-Champaign, USA

Prof. Berndt Müller, Duke U., USA

Prof. Andreas Schäfer, U. Regensburg, Germany

Prof. Masaki Shigemori, Nagoya U., Japan

## Some links:

The history of string theory, Wikipedia