## Final Report Summary - APPSG (Holographic applications of supergravity)

The overall aim of this project is to develop supergravity tools for the anti-de-Sitter/Conformal Field Theory (AdS/CFT) correspondence.

The description of strongly interating systems is usually challenging. At strong coupling, perturbative methods break down and other strategies to describe the strongly interacting behaviour of physical systems must be devised. A powerful method to deal with this problem is available when an equivalence (so-called duality) exists between the strong and weak coupling sectors of two different theories. Calculations can then be performed in the weak coupling sector of one of them to draw conclusions about strongly coupled phenomena in the other.

This type of duality has been argued to exist for a theory of quatum gravity such as string theory, on the one hand, and certain quantum field theories, involving no gravity, on the other hand. More precisely, the gauge/gravity duality, or AdS/CFT correspondence conjectures the equivalence between IIB string theory on an Anti-de-Sitter (AdS) supergravity background, and a precise conformal field theory (CFT): the maximally supersymmetric Yang-Mills theory. This correspondence is usually referred to as `holographic', since it maps physics of the higher-dimensional spacetime where string theory is formulated, to that of its lower-dimensional boundary, where the CFT lives. The original formulation of AdS/CFT involved highly supersymmetric CFTs with unsuitable features for phenomenological applications. Consequently, a lot of effort has been devoted over the last decade to extend AdS/CFT to account for quantum field theories close to those used in nuclear physics, high energy physics (especially Quantum Chromodynamics, QCD) and, much more recently, condensed matter physics.

Despite significant headway, progress has been hampered by the lack of appropriate lower-dimensional supergravity theories needed to formulate the correspondence in precise terms. For AdS/CFT to have actual predictive power, those supergravity models need to be exactly embeddable into string theory and, the problem is, such models are far and between.

Consistent Kaluza-Klein (KK) truncation is the technique that allows, or forbids, the embedding of lower-dimensional, effective theories of (super)gravity into the ten or eleven-dimensional supergravity theories, themselves directly related to string/M-theory. Even though the subject of consistent truncation has been around since the advent of supergravity and string theory, due to the higher dimensional nature of string/M-theory and the need to extract low energy physics out of it, the subject has recently experienced a dramatic revitalisation and increased attention, fueled by holographic applications in condensed matter and nuclear physics. This is because KK truncation 1) allows one to deal with a reduced set of supergravity and dual field theory degrees of freedom in a consistent fashion with the equations of motion; and 2) it is a powerful solution generating technique for higher-dimensional supergravity backgrounds. At a more fundamental level, KK consistency will prove instrumental to achieve a classification of all lower-dimensional supergravities (along with their matter content, preserved supersymmetry, moduli spaces, gaugings, vacua, spectrum and possible instabilities) that can be consistently embedded into string/M-theory.

It had been traditionally thought that consistency in a KK truncation was rather exceptional, with a only a handful of cases known to be consistent, most notably the spherical reductions of the D=10 and D=11 supergravities to the maximally supersymmetric gauged supergravities in four and five dimensions. This understanding has considerably changed in recent years: we now know that very general classes of compactification spaces, equipped with non-trivial G-structures, allow for consistent truncations, both to the massless sector, and to matter-coupled gauged supergravities. Building on these existence results, we have now found new families of gauged supergravities that derive from string theory, by studying the string/M-theory truncations on constant-torsion G-structure manifolds. We have extended this classification to the lower-dimensional gauged supergravities that arise from string/M-theory truncation on homogeneous spaces, thereby producing many more new gauged supergravities. We have also gone beyond the homogeneous class and have investigated examples of consistent truncations on non-homogeneous G-structure spaces. Finally, we have given a string theory realisation of the recently discovered dyonic gaugings of maximal supergravity.

These results provide decisive steps towards the longer term goal to achieve a full classification of lower-dimensional supergravities that admit a string theory realisation.

The description of strongly interating systems is usually challenging. At strong coupling, perturbative methods break down and other strategies to describe the strongly interacting behaviour of physical systems must be devised. A powerful method to deal with this problem is available when an equivalence (so-called duality) exists between the strong and weak coupling sectors of two different theories. Calculations can then be performed in the weak coupling sector of one of them to draw conclusions about strongly coupled phenomena in the other.

This type of duality has been argued to exist for a theory of quatum gravity such as string theory, on the one hand, and certain quantum field theories, involving no gravity, on the other hand. More precisely, the gauge/gravity duality, or AdS/CFT correspondence conjectures the equivalence between IIB string theory on an Anti-de-Sitter (AdS) supergravity background, and a precise conformal field theory (CFT): the maximally supersymmetric Yang-Mills theory. This correspondence is usually referred to as `holographic', since it maps physics of the higher-dimensional spacetime where string theory is formulated, to that of its lower-dimensional boundary, where the CFT lives. The original formulation of AdS/CFT involved highly supersymmetric CFTs with unsuitable features for phenomenological applications. Consequently, a lot of effort has been devoted over the last decade to extend AdS/CFT to account for quantum field theories close to those used in nuclear physics, high energy physics (especially Quantum Chromodynamics, QCD) and, much more recently, condensed matter physics.

Despite significant headway, progress has been hampered by the lack of appropriate lower-dimensional supergravity theories needed to formulate the correspondence in precise terms. For AdS/CFT to have actual predictive power, those supergravity models need to be exactly embeddable into string theory and, the problem is, such models are far and between.

Consistent Kaluza-Klein (KK) truncation is the technique that allows, or forbids, the embedding of lower-dimensional, effective theories of (super)gravity into the ten or eleven-dimensional supergravity theories, themselves directly related to string/M-theory. Even though the subject of consistent truncation has been around since the advent of supergravity and string theory, due to the higher dimensional nature of string/M-theory and the need to extract low energy physics out of it, the subject has recently experienced a dramatic revitalisation and increased attention, fueled by holographic applications in condensed matter and nuclear physics. This is because KK truncation 1) allows one to deal with a reduced set of supergravity and dual field theory degrees of freedom in a consistent fashion with the equations of motion; and 2) it is a powerful solution generating technique for higher-dimensional supergravity backgrounds. At a more fundamental level, KK consistency will prove instrumental to achieve a classification of all lower-dimensional supergravities (along with their matter content, preserved supersymmetry, moduli spaces, gaugings, vacua, spectrum and possible instabilities) that can be consistently embedded into string/M-theory.

It had been traditionally thought that consistency in a KK truncation was rather exceptional, with a only a handful of cases known to be consistent, most notably the spherical reductions of the D=10 and D=11 supergravities to the maximally supersymmetric gauged supergravities in four and five dimensions. This understanding has considerably changed in recent years: we now know that very general classes of compactification spaces, equipped with non-trivial G-structures, allow for consistent truncations, both to the massless sector, and to matter-coupled gauged supergravities. Building on these existence results, we have now found new families of gauged supergravities that derive from string theory, by studying the string/M-theory truncations on constant-torsion G-structure manifolds. We have extended this classification to the lower-dimensional gauged supergravities that arise from string/M-theory truncation on homogeneous spaces, thereby producing many more new gauged supergravities. We have also gone beyond the homogeneous class and have investigated examples of consistent truncations on non-homogeneous G-structure spaces. Finally, we have given a string theory realisation of the recently discovered dyonic gaugings of maximal supergravity.

These results provide decisive steps towards the longer term goal to achieve a full classification of lower-dimensional supergravities that admit a string theory realisation.