## Final Report Summary - OBSERVABLESTRING (The Low Energy Limit of String Theory and the Observable World)

The search for a unified theory of elementary particles and their interactions has culminated in the last three decades in the spectacular development of string theory. String theory reconciles general relativity with quantum mechanics, and contains the main ingredients of the Standard Model. To make direct contact with experiments we must, however, know the precise way in which the Standard Model and Einstein's gravity are embedded as low-energy limits in string theory. The goal of this project was to develop a new unified framework to formulate, compute and analyze this limit and its phenomenology.

String theory is consistently defined in ten dimensions, six out of which are curled up in some small “internal” compact manifold, whose geometry determines the physics we see in four dimensions. Because strings are one-dimensional objects, they have richer dynamics than point particles; in particular they can wind around circles in the internal geometry. This winding charge looks very much like momentum charge in a higher dimensional (a.k.a. double) space. Extending the space even further allows other charges in string theory (D-brane charges) appear as momentum charge. We have shown that the dimensional reduction of string theory (or more precisely a subsector of the ten-dimensional string theory that includes massless and light modes) on these higher-dimensional spaces precisely provides the low energy limit of string theory in four extended dimensions.

Establishing this required finding the four-dimensional low energy action resulting from compactifications in double and extended spaces and solving the mystery of the higher dimensional origin of all possible four-dimensional charges in theories with maximal or half-maximal supersymmetry. We have shown that the reduction on double spaces captures precisely the four-dimensional effective theory even for compactifications on manifolds that have the size of the string, where strings that wind around the space are very light and have to be included in the spectrum. We have also shown how to obtain a four-dimensional effective theory with less supersymmetry (minimal or next-to minimal) and its supersymmetric solutions from compactifications characterized by certain algebraic structures on the extended space. This allows on one hand to do a systematic search for solutions, and on the other hand understand their deformation spaces: solutions that can be deformed without any cost of energy lead to long-range forces in four-dimensions that have been ruled out experimentally, and therefore should be discarded. We have characterized particular deformations of the spaces and understood the topological conditions for them to be obstructed.

Our world is however not supersymmetric, and one of the main open problems in string theory is to understand how to break supersymmetry obtaining a tiny positive cosmological constant, such as the one of our current universe. Typical string theory solutions give four-dimensional spaces with zero or negative cosmological constant, and understanding how this vacuum cosmological constant is “uplifted” to a positive value remains a major challenge. The typical mechanism consists of placing branes of opposite charge as that of the background fluxes. Neglecting their back-reaction on the background geometry, these were shown to lead to meta-stable solutions with positive cosmological constant. We have shown that such solutions are actually not meta-stable but unstable, and thus as they stand cannot be used to uplift the cosmological constant. This invalidates the most generic known mechanism for uplifting the cosmological constant, and implies that string theory does not admit a landscape of vacua with a small positive cosmological constant. Most of the research done over the past few years on viable cosmological scenarios in string theory need to be seriously revisited, opening the door for new inventive paradigms, such as “T-branes” (certain vacuum configurations of branes), of we have found a new description valid at strong coupling.

We have studied quantum corrections in compactifications with minimal supersymmetry. On one hand we have found that compactifications in generalized geometries should be corrected in exactly the same fashion as happens in ordinary geometries, and on the other hand we have found a new quantum correction to the effective action. This correction is of the same order as a correction found many years ago that has been heavily used in alternative uplift mechanisms. Depending on the topology of the compactification space, this new correction can enhance, cancel or work against the previous one used for uplifting the cosmological constant, and thus these mechanisms have to be revisited as well.

String theory is consistently defined in ten dimensions, six out of which are curled up in some small “internal” compact manifold, whose geometry determines the physics we see in four dimensions. Because strings are one-dimensional objects, they have richer dynamics than point particles; in particular they can wind around circles in the internal geometry. This winding charge looks very much like momentum charge in a higher dimensional (a.k.a. double) space. Extending the space even further allows other charges in string theory (D-brane charges) appear as momentum charge. We have shown that the dimensional reduction of string theory (or more precisely a subsector of the ten-dimensional string theory that includes massless and light modes) on these higher-dimensional spaces precisely provides the low energy limit of string theory in four extended dimensions.

Establishing this required finding the four-dimensional low energy action resulting from compactifications in double and extended spaces and solving the mystery of the higher dimensional origin of all possible four-dimensional charges in theories with maximal or half-maximal supersymmetry. We have shown that the reduction on double spaces captures precisely the four-dimensional effective theory even for compactifications on manifolds that have the size of the string, where strings that wind around the space are very light and have to be included in the spectrum. We have also shown how to obtain a four-dimensional effective theory with less supersymmetry (minimal or next-to minimal) and its supersymmetric solutions from compactifications characterized by certain algebraic structures on the extended space. This allows on one hand to do a systematic search for solutions, and on the other hand understand their deformation spaces: solutions that can be deformed without any cost of energy lead to long-range forces in four-dimensions that have been ruled out experimentally, and therefore should be discarded. We have characterized particular deformations of the spaces and understood the topological conditions for them to be obstructed.

Our world is however not supersymmetric, and one of the main open problems in string theory is to understand how to break supersymmetry obtaining a tiny positive cosmological constant, such as the one of our current universe. Typical string theory solutions give four-dimensional spaces with zero or negative cosmological constant, and understanding how this vacuum cosmological constant is “uplifted” to a positive value remains a major challenge. The typical mechanism consists of placing branes of opposite charge as that of the background fluxes. Neglecting their back-reaction on the background geometry, these were shown to lead to meta-stable solutions with positive cosmological constant. We have shown that such solutions are actually not meta-stable but unstable, and thus as they stand cannot be used to uplift the cosmological constant. This invalidates the most generic known mechanism for uplifting the cosmological constant, and implies that string theory does not admit a landscape of vacua with a small positive cosmological constant. Most of the research done over the past few years on viable cosmological scenarios in string theory need to be seriously revisited, opening the door for new inventive paradigms, such as “T-branes” (certain vacuum configurations of branes), of we have found a new description valid at strong coupling.

We have studied quantum corrections in compactifications with minimal supersymmetry. On one hand we have found that compactifications in generalized geometries should be corrected in exactly the same fashion as happens in ordinary geometries, and on the other hand we have found a new quantum correction to the effective action. This correction is of the same order as a correction found many years ago that has been heavily used in alternative uplift mechanisms. Depending on the topology of the compactification space, this new correction can enhance, cancel or work against the previous one used for uplifting the cosmological constant, and thus these mechanisms have to be revisited as well.