## Final Report Summary - ALADS-DS (New aspects of supergravity: the gravity dual of supersymmetric localization and string theory compactifications to de Sitter space)

During the fellowship period I focused on different aspects of quantum field theory (QFT) and supergravity in relation with string theory.

QFT plays a central role in theoretical physics. Besides being at the core of the Standard Model of elementary particles, it successfully describes phenomena at very different energy scales, ranging from condensed matter and hydrodynamics up to inflationary cosmology. It also has deep connections with pure mathematics. Despite this ubiquity, our understanding of QFT is far from complete. In fact, most QFTs can only be treated perturbatively at weak coupling or by lattice simulations. We thus have little analytical control on strongly coupled systems; for instance, proving confinement in Quantum Chromodynamics remains an open problem.

A major breakthrough in the last two decades has been the discovery that string theory provides non-perturbative access to QFT via the holographic principle. This is a conjectured equivalence between QFTs in d spacetime dimensions and gravity theories in d+1 dimensions. In its sharpest formulation, the holographic principle states that a theory of strings propagating in Anti de Sitter (AdS) space is equivalent to a conformal field theory (CFT - a QFT with additional conformal symmetry) living on the AdS boundary; this is called AdS/CFT correspondence. A main attractive feature of the AdS/CFT correspondence is that hard QFT problems at strong coupling are mapped into easier, weakly coupled computations in the dual gravity theory. This holographic approach has many applications, including QCD, superconductivity and other condensed matter phenomena.

When tackling a QFT question, it is often convenient to focus on situations where extra symmetry is present, so that analytic control is increased. A symmetry that is often invoked is supersymmetry. This is an invariance under exchanging bosons and fermions, which improves the behavior of QFTs at high energies and solves some difficulties of the Standard Model. Although it has not been detected in experiments so far, supersymmetry has very interesting mathematical properties and has proven an extremely successful theoretical tool, leading to important progress in our basic understanding of QFT. Another virtue of supersymmetry is that once this is imposed as a local symmetry, general relativity is incorporated automatically, and the resulting theory is called supergravity. Supergravity theories also naturally appear as the low energy limit of string theory, and as such they are very relevant for the AdS/CFT correspondence.

Despite having played a major role in theoretical physics for decades, supersymmetry continues to hold surprises, and much remains to be discovered. Recently, many new exact results have been derived for supersymmetric QFTs in curved space, using powerful techniques such as localization of the path integral.

During the fellowship period I gave significant contributions to the fundamental problems outlined above, using methods at the foreground of current investigations. This combined two complementary approaches: the first, purely field-theoretical, led to new exact results for supersymmetric QFT in curved space. The second investigated the holographic dual of this in supergravity, also using methods of generalised geometry.

A main achievement was to clarify the significance of the vacuum energy for supersymmetric QFT in curved space. I focused on a four-dimensional R x S^3 geometry, where R denotes the time line and S^3 a 3-sphere. In non-supersymmetric theories the vacuum energy on R x S^3 depends on the choice of renormalization scheme and is thus ambiguous. By constrast, I proved that in supersymmetric theories the ambiguities disappear, hence the vacuum energy acquires intrinsic meaning. Moreover, it can be computed explicitly, for any value of the coupling constant. This was proved in two ways. First, by classifying the finite supersymmetric counterterms that parameterise the possible ambiguities and showing that they all vanish in the relevant backgrounds. Second, by studying the problem in supersymmetric quantum mechanics upon reducing the four-dimensional QFT on the three-sphere. This also prompted a careful investigation of the issue of supersymmetric regularization in four-dimensional QFT.

I then explored the holographic dual of the QFT results above through a cutting-edge application of the AdS/CFT correspondence. According to the latter, the observables of a d-dimensional CFT at strong coupling may be computed via a (d+1)-dimensional solution of supergravity that is asymptotically AdS and satisfies suitable conditions at the d-dimensional boundary. In particular, the field theory partition function is obtained evaluating the gravity action on the solution.

I studied very general supersymmetric solutions to five-dimensional supergravity, and discussed their relevance for the AdS/CFT correspondence. In particular, I focused on the dependence of the supergravity action on boundary data. Since according to the correspondence this must reproduce the field theory partition function, and since it has been proven that the latter only depends on few, specific background data, the same should be true for the supergravity action. One issue in this program is that the gravity action is divergent and needs to be renormalised. Crucially, this has to be done in a way compatible with supersymmetry. I realised that conventional holographic renormalisation corresponds to a scheme that breaks supersymmetry. In the recent work arXiv:1606.02724 (now submitted to a journal), I proved that supersymmetry can be restored by adding new, unconventional boundary terms. I demonstrated that only after this correction is made the supergravity action depends on the appropriate boundary data and reproduces the supersymmetric vacuum energy computed in QFT. This is a significant result emphasising the role of boundary terms beyond what is currently appreciated, and with potential consequences concerning the general set of data needed to specify the holographic dictionary.

A powerful tool for tackling many problems in the context of the AdS/CFT correspondence is provided by generalised geometry. This new mathematical formalism, developed both by mathematicians (mainly Hitchin) and physicists, introduces geometric structures on certain extensions of the ordinary tangent space to a manifold; it is tailor-made for describing dimensional reductions on compact spaces of the higher-dimensional supergravities that describe string theory at low energy. In particular, generalised geometry relates string theory to the embedding tensor formalism that classifies systematically all possible lower-dimensional supergravities. This is very useful as the latter are much more tractable than the original string theory.

In joint work with the Scientist in Charge M. Petrini, her student O. de Felice and others, I studied consistent dimensional reductions building on new developments in generalised geometry, that extend Hitchin’s original formalism on the sum of the tangent and cotangent bundle to a larger bundle which geometrises all degrees of freedom of higher-dimensional supergravity on the compactification manifold. In particular, I developed a novel generalised geometry formalism for massive type IIA supergravity by introducing a new deformation of an operator generalising the Lie derivative. Using this formalism I then explored consistent dimensional reductions preserving maximal supersymmetry on spheres in different dimensions. These results are interesting because they provide a number of supergravity models with rigorous embedding into string theory, that can be used to tackle different problems in the context of the AdS/CFT correspondence in a controllable setup. I will next pursue this line of research further by studying how a reduction of the structure group of the generalised bundle defines consistent truncations that partially break supersymmetry.

Overall, the results obtained and the experience acquired during the fellowship period will allow in the near future to tackle the very interesting problem of computing the path integral of quantum supergravity using technique of localization. This is exciting as it has the potential to shed new light on quantum gravity.

QFT plays a central role in theoretical physics. Besides being at the core of the Standard Model of elementary particles, it successfully describes phenomena at very different energy scales, ranging from condensed matter and hydrodynamics up to inflationary cosmology. It also has deep connections with pure mathematics. Despite this ubiquity, our understanding of QFT is far from complete. In fact, most QFTs can only be treated perturbatively at weak coupling or by lattice simulations. We thus have little analytical control on strongly coupled systems; for instance, proving confinement in Quantum Chromodynamics remains an open problem.

A major breakthrough in the last two decades has been the discovery that string theory provides non-perturbative access to QFT via the holographic principle. This is a conjectured equivalence between QFTs in d spacetime dimensions and gravity theories in d+1 dimensions. In its sharpest formulation, the holographic principle states that a theory of strings propagating in Anti de Sitter (AdS) space is equivalent to a conformal field theory (CFT - a QFT with additional conformal symmetry) living on the AdS boundary; this is called AdS/CFT correspondence. A main attractive feature of the AdS/CFT correspondence is that hard QFT problems at strong coupling are mapped into easier, weakly coupled computations in the dual gravity theory. This holographic approach has many applications, including QCD, superconductivity and other condensed matter phenomena.

When tackling a QFT question, it is often convenient to focus on situations where extra symmetry is present, so that analytic control is increased. A symmetry that is often invoked is supersymmetry. This is an invariance under exchanging bosons and fermions, which improves the behavior of QFTs at high energies and solves some difficulties of the Standard Model. Although it has not been detected in experiments so far, supersymmetry has very interesting mathematical properties and has proven an extremely successful theoretical tool, leading to important progress in our basic understanding of QFT. Another virtue of supersymmetry is that once this is imposed as a local symmetry, general relativity is incorporated automatically, and the resulting theory is called supergravity. Supergravity theories also naturally appear as the low energy limit of string theory, and as such they are very relevant for the AdS/CFT correspondence.

Despite having played a major role in theoretical physics for decades, supersymmetry continues to hold surprises, and much remains to be discovered. Recently, many new exact results have been derived for supersymmetric QFTs in curved space, using powerful techniques such as localization of the path integral.

During the fellowship period I gave significant contributions to the fundamental problems outlined above, using methods at the foreground of current investigations. This combined two complementary approaches: the first, purely field-theoretical, led to new exact results for supersymmetric QFT in curved space. The second investigated the holographic dual of this in supergravity, also using methods of generalised geometry.

A main achievement was to clarify the significance of the vacuum energy for supersymmetric QFT in curved space. I focused on a four-dimensional R x S^3 geometry, where R denotes the time line and S^3 a 3-sphere. In non-supersymmetric theories the vacuum energy on R x S^3 depends on the choice of renormalization scheme and is thus ambiguous. By constrast, I proved that in supersymmetric theories the ambiguities disappear, hence the vacuum energy acquires intrinsic meaning. Moreover, it can be computed explicitly, for any value of the coupling constant. This was proved in two ways. First, by classifying the finite supersymmetric counterterms that parameterise the possible ambiguities and showing that they all vanish in the relevant backgrounds. Second, by studying the problem in supersymmetric quantum mechanics upon reducing the four-dimensional QFT on the three-sphere. This also prompted a careful investigation of the issue of supersymmetric regularization in four-dimensional QFT.

I then explored the holographic dual of the QFT results above through a cutting-edge application of the AdS/CFT correspondence. According to the latter, the observables of a d-dimensional CFT at strong coupling may be computed via a (d+1)-dimensional solution of supergravity that is asymptotically AdS and satisfies suitable conditions at the d-dimensional boundary. In particular, the field theory partition function is obtained evaluating the gravity action on the solution.

I studied very general supersymmetric solutions to five-dimensional supergravity, and discussed their relevance for the AdS/CFT correspondence. In particular, I focused on the dependence of the supergravity action on boundary data. Since according to the correspondence this must reproduce the field theory partition function, and since it has been proven that the latter only depends on few, specific background data, the same should be true for the supergravity action. One issue in this program is that the gravity action is divergent and needs to be renormalised. Crucially, this has to be done in a way compatible with supersymmetry. I realised that conventional holographic renormalisation corresponds to a scheme that breaks supersymmetry. In the recent work arXiv:1606.02724 (now submitted to a journal), I proved that supersymmetry can be restored by adding new, unconventional boundary terms. I demonstrated that only after this correction is made the supergravity action depends on the appropriate boundary data and reproduces the supersymmetric vacuum energy computed in QFT. This is a significant result emphasising the role of boundary terms beyond what is currently appreciated, and with potential consequences concerning the general set of data needed to specify the holographic dictionary.

A powerful tool for tackling many problems in the context of the AdS/CFT correspondence is provided by generalised geometry. This new mathematical formalism, developed both by mathematicians (mainly Hitchin) and physicists, introduces geometric structures on certain extensions of the ordinary tangent space to a manifold; it is tailor-made for describing dimensional reductions on compact spaces of the higher-dimensional supergravities that describe string theory at low energy. In particular, generalised geometry relates string theory to the embedding tensor formalism that classifies systematically all possible lower-dimensional supergravities. This is very useful as the latter are much more tractable than the original string theory.

In joint work with the Scientist in Charge M. Petrini, her student O. de Felice and others, I studied consistent dimensional reductions building on new developments in generalised geometry, that extend Hitchin’s original formalism on the sum of the tangent and cotangent bundle to a larger bundle which geometrises all degrees of freedom of higher-dimensional supergravity on the compactification manifold. In particular, I developed a novel generalised geometry formalism for massive type IIA supergravity by introducing a new deformation of an operator generalising the Lie derivative. Using this formalism I then explored consistent dimensional reductions preserving maximal supersymmetry on spheres in different dimensions. These results are interesting because they provide a number of supergravity models with rigorous embedding into string theory, that can be used to tackle different problems in the context of the AdS/CFT correspondence in a controllable setup. I will next pursue this line of research further by studying how a reduction of the structure group of the generalised bundle defines consistent truncations that partially break supersymmetry.

Overall, the results obtained and the experience acquired during the fellowship period will allow in the near future to tackle the very interesting problem of computing the path integral of quantum supergravity using technique of localization. This is exciting as it has the potential to shed new light on quantum gravity.