## Final Activity and Management Report Summary - DUALDUAL (Dualities between gauge theory and string theory)

The holographic principle is a tool to study quantum gravity. It is realised in string theory as the gauge/gravity correspondence. It suggests that any theory with gravity can be related to a theory in one dimension less. Recently, significant progress has been made in the understanding of one particular version of this duality. The so-called ABJM model exhibits the previously unknown world-volume theory on M2-branes. This discovery has been called a mini-revolution, since previously all studies were confined to the gravity side. Now, also the gauge theory side can be studied. M-theory only contains two types of solitons: M2-branes and M5-branes. In lower dimensions and appropriate limits, M-theory gives rise to all known versions of string theory.

In this project, we have focused on studying various aspects of the string/gauge duality, and in particular properties which are unique to ABJM theory. We have constructed superspace and Lagrangian formulations of ABJM theory. This allowed us to make the full set of symmetries manifest, in particular the R-symmetry. This is a necessary condition for the theory to have the claimed amount of supersymmetry, and hence important for matching to the gravity side. We also took the first steps towards generalising the model in different ways, using techniques based on renormalisation group flows and orbifolding. Furthermore, we have calculated the free energy of ABJM theory to the next to leading order in perturbation theory.

This is an interesting problem, because ABJM theory has a holographic dual where the thermodynamics at strong coupling is determined by a system of black M2-branes. The zero coupling (black-body radiation) free energy disagrees with the strong coupling result. However, our computation confirms that the first correction goes in the right direction, indicating that the strong and weak coupling results will converge as loop corrections are taken into account. We have also investigated 3-point correlation functions of chiral primary operators in ABJM theory, which scale in the same characteristic way as the free energy at strong coupling. We have made progress on computing the leading quantum correction, to confirm this from the field theory side.

In this project, we have focused on studying various aspects of the string/gauge duality, and in particular properties which are unique to ABJM theory. We have constructed superspace and Lagrangian formulations of ABJM theory. This allowed us to make the full set of symmetries manifest, in particular the R-symmetry. This is a necessary condition for the theory to have the claimed amount of supersymmetry, and hence important for matching to the gravity side. We also took the first steps towards generalising the model in different ways, using techniques based on renormalisation group flows and orbifolding. Furthermore, we have calculated the free energy of ABJM theory to the next to leading order in perturbation theory.

This is an interesting problem, because ABJM theory has a holographic dual where the thermodynamics at strong coupling is determined by a system of black M2-branes. The zero coupling (black-body radiation) free energy disagrees with the strong coupling result. However, our computation confirms that the first correction goes in the right direction, indicating that the strong and weak coupling results will converge as loop corrections are taken into account. We have also investigated 3-point correlation functions of chiral primary operators in ABJM theory, which scale in the same characteristic way as the free energy at strong coupling. We have made progress on computing the leading quantum correction, to confirm this from the field theory side.