## Final Activity Report Summary - SSTRING.SUGRA.VAC. (Superstring Vacua from Supergravity)

Since the start date of this grant, important results have been obtained towards the attainment of the objectives of the project. In order to illustrate them, let us give a brief overview of the scientific motivations behind them. Superstring theory in 10 dimensions or M-theory in 11 dimensions are regarded as the most promising candidates for a quantum theory in which gravity is naturally unified with the other fundamental interactions is the. This is a finite theory which describes the dynamics of one-dimensional strings propagating in a ten-dimensional Minkowskian space-time. A valuable framework in which to study the dynamics at low-energies around non-perturbative vacua in limits in which a consistent formulation of Superstring theory is missing has often been provided by Supergravity.

This field-theory describes the infinite tension (low-energy) limit of superstring theory and naturally encodes Einstein's gravity as a consequence of local supersymmetry. The right choice of vacuum should also account for the elementary fact that, at least at energy scales which are at present experimentally reachable, our universe is four-dimensional, in other words it should include a mechanism of dimensional-reduction (e.g. the Kaluza-Klein mechanism) on an internal compact space with very small size and suitable geometrical properties. An important role in understanding the non-perturbative aspects of string theory has been played by the dualities, behind which there is the idea that different string theories compactified on different backgrounds are alternative descriptions of the same microscopic degrees of freedom. Meanwhile, new non-perturbative vacua of superstring theory have been found which exhibit a non-vanishing v.e.v. of higher-order form field-strengths across cycles of the internal manifold (fluxes).

Fluxes seem to provide a valuable mechanism in order to reduce the number of massless scalar fields which naturally arise from dimensional reduction of Superstring theory and which are not desirable from a phenomenological point of view. The low-energy limit of flux compactifications is described by an effective four dimensional supergravity in which the internal fluxes define minimal couplings, mass terms and a scalar potential. These supergravity models are characterized by local symmetries and therefore are called gauged. Dr M. Trigiante and Prof. R. D'Auria have been involved in pioneering works on the application of gauged supergravity to flux compacitfications.

The main objective of the project was to apply the supergravity framework to the analysis of new superstring or M-theory flux vacua, with particular reference to the orientifold compactifications of Type II superstring theory. The results can be summarised as follows:

- The full low-energy gauged supergravity description of M-theory compactified on a twisted torus in the presence of form-fluxes was constructed and and its vacua and their duality relations were thoroughly studied.

- A generalised formulation of gauged supergravity theories was constructed, which include couplings to antisymmetric tensor fields and which are related by duality to ordinary formulations of the same theory.

- A manifestly duality covariant formulation of gauged supergravities was constructed. This allows to systematically study the effect of the known string dualities of fluxes and discover new flux vacua from dual gauged supergravities.

These results pave the way for the construction in string theory of stable vacua with positive cosmological constant, which are extremely relevant for cosmology and which have so far eluded all kinds of microscopic analysis.

This field-theory describes the infinite tension (low-energy) limit of superstring theory and naturally encodes Einstein's gravity as a consequence of local supersymmetry. The right choice of vacuum should also account for the elementary fact that, at least at energy scales which are at present experimentally reachable, our universe is four-dimensional, in other words it should include a mechanism of dimensional-reduction (e.g. the Kaluza-Klein mechanism) on an internal compact space with very small size and suitable geometrical properties. An important role in understanding the non-perturbative aspects of string theory has been played by the dualities, behind which there is the idea that different string theories compactified on different backgrounds are alternative descriptions of the same microscopic degrees of freedom. Meanwhile, new non-perturbative vacua of superstring theory have been found which exhibit a non-vanishing v.e.v. of higher-order form field-strengths across cycles of the internal manifold (fluxes).

Fluxes seem to provide a valuable mechanism in order to reduce the number of massless scalar fields which naturally arise from dimensional reduction of Superstring theory and which are not desirable from a phenomenological point of view. The low-energy limit of flux compactifications is described by an effective four dimensional supergravity in which the internal fluxes define minimal couplings, mass terms and a scalar potential. These supergravity models are characterized by local symmetries and therefore are called gauged. Dr M. Trigiante and Prof. R. D'Auria have been involved in pioneering works on the application of gauged supergravity to flux compacitfications.

The main objective of the project was to apply the supergravity framework to the analysis of new superstring or M-theory flux vacua, with particular reference to the orientifold compactifications of Type II superstring theory. The results can be summarised as follows:

- The full low-energy gauged supergravity description of M-theory compactified on a twisted torus in the presence of form-fluxes was constructed and and its vacua and their duality relations were thoroughly studied.

- A generalised formulation of gauged supergravity theories was constructed, which include couplings to antisymmetric tensor fields and which are related by duality to ordinary formulations of the same theory.

- A manifestly duality covariant formulation of gauged supergravities was constructed. This allows to systematically study the effect of the known string dualities of fluxes and discover new flux vacua from dual gauged supergravities.

These results pave the way for the construction in string theory of stable vacua with positive cosmological constant, which are extremely relevant for cosmology and which have so far eluded all kinds of microscopic analysis.