Research
Application to condensed matter systems
Field theories are used whenever many body physics is required, including condensed matter. In some cases one can write solvable (or manageable) field theories that capture much of the essential physics of some such systems. Another tool in our disposal is the gauge/gravity duality which allows us to describe such systems, when strongly coupled, in terms of weakly coupled gravity. In the group we are applying these techniques to the study of Kondo impurities, electrons interacting with arrays of impurities and new phases of strongly coupled electron systems.
Supersymmetry
Supersymmetry is a symmetry that maintains Bose-Fermi degeneracy, so the spectrum of bosons and fermions has to be identical. Supersymmetry also imposes severe constraints on the allowed interactions between the various particle species. These models are special, and lead to powerful tools that are applicable even for strongly coupled theories. Traditionally, the power of supersymmetry was mostly limited to studying the vacuum structure (more precisely, the chiral ring). Recently, it was realized that supersymmetry leads to remarkable simplifications also on curved spaces, and it allows to extract information that extends well beyond the chiral ring of the theory. These recent developments can also be used to evaluate the expectation values of non-local observables, such as the entanglement entropy of the vacuum. These observables are of exceeding interest in particle physics and also in condensed matter physics. Which additional quantities can be evaluated non-perturbatively in supersymmetric field theories? What can we learn about general aspects of quantum field theory such as dualities and anomalies? What do we learn about the vacuum of quantumfield theory? What is the mathematical interpretation of supersymmetric theories on curved spaces?
Quantum field theories in higher dimensions
Weakly coupled interacting quantum field theories exist only in four or less space-time dimensions. However, there are arguments that consistent quantum field theories exist also in five and six space-time dimensions. In particular there are indirect arguments for the existence of local superconformal field theories in five and six dimensions, and of non-local field theories (called "little string theories") in six dimensions. Can we provide direct constructions of these higher dimensional theories, and understand their properties ? What can we learn about lower dimensional field theories by compactifying these theories on various manifolds ? Are there any consistent field theories above six space-time dimensions, and are there any consistent non-supersymmetric field theories above four space-time dimensions ? What are the rules for dealing with non-local (but non-gravitational) theories like "little string theories" ?
Singularities in quantum gravity
Many classical solutions of gravity (namely, of Einstein's theory of general relativity) have singularities. For some types of singularities it has been found that going to a quantum theory of gravity, such as string theory, "resolves" the singularity and enables computations to be performed. However, other types of singularities, including the "big bang" singularity and the similar singularity that may hide in the interior of black holes, are still not understood, even within string theory. Can we find ways to resolve "big bang"-like singularities in string theory ? Does this lead to a specific initial state (or a set of possible initial states) for the universe ? Is there any observational way to detect quantum gravitational effects related to the "big bang" singularity ?
Dualities in quantum field theory
Quantum field theories can easily be studied in perturbation theory when they are weakly coupled, but very little is known about them when they are strongly coupled, even though many interesting field theories (including quantum chromodynamics at low energies) are strongly coupled. For some quantum field theories, it has been found that the strongly coupled theory has an alternative description in terms of some different quantum field theory, that is sometimes weakly coupled. In two space-time dimensions this phenomenon has been known for a long time and is reasonably well-understood. In higher dimensions we have a list of examples of this phenomenon, mostly in supersymmetric field theories, but no general understanding of when and how it happens. Can we find dual descriptions for more strongly coupled field theories in three and four space-time dimensions, in particular for more non-supersymmetric theories ? Can we relate the dualities of different theories (including theories in different space-time dimensions) ? Can we understand duality mappings in general, and obtain rules that will tell us when some theory has a dual description, and what it is ?
String theory and gauge/gravity duality
String theory is a consistent theory of quantum gravity - the only one known so far. However, so far we only have a non-perturbative formulation of this theory on a limited class of space-times which have specific types of boundaries, using the gauge/gravity duality and its generalizations. And even this duality has not yet been derived (though there is substantial evidence that it is true). Can we generalize the gauge/gravity duality to more general situations, for instance to weakly coupled field theories which should be dual to backgrounds with light high-spin fields ("high-spin gravities") ? Can this help us to derive the gauge/gravity duality, namely construct the mapping between the field theory degrees of freedom and those of the gravitational theory (such as string theory) ? Can we find a non-perturbative formulation for string theory on more general backgrounds, such as flat space or de Sitter space ?
Conformal field theory
Some of the most interesting Quantum Field Theories are conformal. What makes Conformal Field Theories (CFTs) interesting is that they describe second order phase transitions, they are the end-points of renormalization group flows, and they also describe quantum gravity in Anti-de Sitter spaces. A better understanding of CFTs would thus shed light on many interesting branches of physics, ranging from statistical physics to quantum gravity. CFTs above two space-time dimensions remain elusive, and the full power of the conformal group is not yet fully appreciated. In which dimensions do non-trivial CFTs exist ? What can be said about critical exponents at second-order phase transitions ? What can we learn from CFTs about quantum gravity ? Which CFTs can be connected by Renormalization Group flows ? Is the field theory which describes boiling water at the second order phase transition solvable?