## Final Activity Report Summary - INTEGR-QHE-STRING (New integrable dynamics: from condensed matter systems to gauge and string theories)

There exist lots of models in theoretical particle physics that assume existence of extra spatial dimensions, in addition to the 'conventional' three spatial and one temporal dimension. The string theory is the most known, but not the only, example of such models. These models may be very attractive and rich from a theoretical viewpoint, yet their experimental verification is difficult.

Indeed, the higher-dimensional nature of space and time explicitly exhibits itself only at energies above a certain characteristic scale. Thus, until modern experiments reach this energy scale, the direct test of the presence of the extra dimensions is not possible and one should look for other ways to explore this fascinating possibility. This means that theoreticians should strive to find low-energy signatures' of various higher-dimensional models, i.e. to find effects which are unique for the theory in question and can be tested in the modern day laboratories.

Why do we believe that such effects exist? Surprisingly, a hint for their existence came from a branch of physics very distant from particle physics, namely the Quantum hall (QH) effect. QH effect is one of the most remarkable condensed matter phenomena. Its experimental observation is a quantisation of the hall conductance of a two-dimensional electron gas in a strong magnetic field, a quantisation which is universal and independent of all microscopic details, such as the type of material, the purity of sample, the precise value of the magnetic field etc. It was recently understood that there was a deep connection between the physics underlying the QH system and the theories with extra dimensions. To give an analogy, back in the 1970s the theory of phase transitions was applied to the physics in the early Universe. As a result, the theory, which was extremely fruitful in the realm of statistical and condensed matter physics, notably in the description of superconductivity and superfluidity, turned out to become a big success in cosmology.

The interplay and interrelation between QH effect and theories with extra dimensions, including string theory, was the main motivation for this project. Its main objective was to explore their relation deeper and obtain new results, using this analogy. This goal was achieved, as can be derived by the following results:

1. New signature effects of theories with extra dimensions were found in our works, their experimental consequences were explored and new experiments were suggested. It was shown that special interactions might exist between our world and extra dimensions. These interactions could not be effectively tested in the modern day accelerators, but could be detected in 'table-top' precision measurement experiment. What was even better, such experiments required only slight modifications of already existing ones.

2. Right when these results were published a new experiment, called 'Polarizzazione del vuoto con laser' (PVLAS), announced its results. PVLAS collaboration studied vacuum polarisation in strong magnetic field and observed changes to the light propagation in such a medium. The numbers obtained by these experiments were in stark contradiction with all conventional models predicting such effects. My collaborators and I then proposed a new model, explaining the PVLAS data. The motivation for this model originated from our previous research and used new ingredients, dictated by the understanding of common structure underlying the QH systems and the theories with extra dimensions. The proposed model was, by the time of the project completion, among the most popular ones.

By this time, I was working on the interpretation of the PVLAS results as a signal from the theory with extra dimensions. Similar signals were also anticipated by my previous works, however detailed studies, their relation to the actual experiment etc. was the subject of this specific research.

Indeed, the higher-dimensional nature of space and time explicitly exhibits itself only at energies above a certain characteristic scale. Thus, until modern experiments reach this energy scale, the direct test of the presence of the extra dimensions is not possible and one should look for other ways to explore this fascinating possibility. This means that theoreticians should strive to find low-energy signatures' of various higher-dimensional models, i.e. to find effects which are unique for the theory in question and can be tested in the modern day laboratories.

Why do we believe that such effects exist? Surprisingly, a hint for their existence came from a branch of physics very distant from particle physics, namely the Quantum hall (QH) effect. QH effect is one of the most remarkable condensed matter phenomena. Its experimental observation is a quantisation of the hall conductance of a two-dimensional electron gas in a strong magnetic field, a quantisation which is universal and independent of all microscopic details, such as the type of material, the purity of sample, the precise value of the magnetic field etc. It was recently understood that there was a deep connection between the physics underlying the QH system and the theories with extra dimensions. To give an analogy, back in the 1970s the theory of phase transitions was applied to the physics in the early Universe. As a result, the theory, which was extremely fruitful in the realm of statistical and condensed matter physics, notably in the description of superconductivity and superfluidity, turned out to become a big success in cosmology.

The interplay and interrelation between QH effect and theories with extra dimensions, including string theory, was the main motivation for this project. Its main objective was to explore their relation deeper and obtain new results, using this analogy. This goal was achieved, as can be derived by the following results:

1. New signature effects of theories with extra dimensions were found in our works, their experimental consequences were explored and new experiments were suggested. It was shown that special interactions might exist between our world and extra dimensions. These interactions could not be effectively tested in the modern day accelerators, but could be detected in 'table-top' precision measurement experiment. What was even better, such experiments required only slight modifications of already existing ones.

2. Right when these results were published a new experiment, called 'Polarizzazione del vuoto con laser' (PVLAS), announced its results. PVLAS collaboration studied vacuum polarisation in strong magnetic field and observed changes to the light propagation in such a medium. The numbers obtained by these experiments were in stark contradiction with all conventional models predicting such effects. My collaborators and I then proposed a new model, explaining the PVLAS data. The motivation for this model originated from our previous research and used new ingredients, dictated by the understanding of common structure underlying the QH systems and the theories with extra dimensions. The proposed model was, by the time of the project completion, among the most popular ones.

By this time, I was working on the interpretation of the PVLAS results as a signal from the theory with extra dimensions. Similar signals were also anticipated by my previous works, however detailed studies, their relation to the actual experiment etc. was the subject of this specific research.