Periodic Reporting for period 4 - Stringlandscape (Deconstructing the string landscape)
Reporting period: 2023-03-01 to 2024-06-30
String Theory is the most compelling candidate for a theory that encompasses all four interactions observed in nature, thus realizing Einstein's dream of unifying Quantum Mechanics and General Relativity. While the “first superstring revolution” proved the quantum consistency of five string theories, the second revolution showed that these were actually five different limits of a single theory.
Despite its uniqueness, string theory has many low-energy solutions (vacua), with very different observable physics: different cosmological constants, matter content and interactions. The multitude of string theory vacua (multiverse) is often invoked as supporting the anthropic principle, and string theory, with its believed “landscape” of vacua, appears to support it.
However, phenomenologically-relevant solutions in the landscape are not constructed directly in string theory, but are found using effective low-energy descriptions in four space-time dimensions. The effective four-dimensional theories are obtained from the low energy limit of ten-dimensional string theory in some particular background (a small six-dimensional “internal” curled-up space of size much less than any scale accessible to observations, typically supporting electromagnetic fluxes). In order to satisfy all experimental constraints from particle physics and cosmology, the effective theories require a number of intricate ingredients, whose string theory origin and consistency is unclear. Thus, the existence of a halfway top-down/bottom-up construction of an effective theory satisfying the constraints imposed by observations gives us absolutely no guarantee that string theory admits such a solution. At this stage, there is neither a known ultraviolet completion of the effective action, nor any guarantee of its existence or consistency. The objective of this proposal was to deconstruct the string theory landscape, and determine what part of it is made of honest-to-goodness stable string theory solutions and thus consistent, and what part of it is inconsistent.
The project was divided into three main themes. A first part deconstructs the known de Sitter (positive cosmological constant) landscape in string theory. In addition to a positive cosmological constant, the string theory solution describing our universe should also contain the Standard Model of particle physics. Deconstruction of the existent and exploration of new Standard Model landscapes was another main objective of the project. Finally, we aimed at constructing the non-supersymmetric and intriguing non-geometric landscapes (solutions where the internal space does not have a conventional geometric description, yet are perfectly allowed in string theory).
Despite its uniqueness, string theory has many low-energy solutions (vacua), with very different observable physics: different cosmological constants, matter content and interactions. The multitude of string theory vacua (multiverse) is often invoked as supporting the anthropic principle, and string theory, with its believed “landscape” of vacua, appears to support it.
However, phenomenologically-relevant solutions in the landscape are not constructed directly in string theory, but are found using effective low-energy descriptions in four space-time dimensions. The effective four-dimensional theories are obtained from the low energy limit of ten-dimensional string theory in some particular background (a small six-dimensional “internal” curled-up space of size much less than any scale accessible to observations, typically supporting electromagnetic fluxes). In order to satisfy all experimental constraints from particle physics and cosmology, the effective theories require a number of intricate ingredients, whose string theory origin and consistency is unclear. Thus, the existence of a halfway top-down/bottom-up construction of an effective theory satisfying the constraints imposed by observations gives us absolutely no guarantee that string theory admits such a solution. At this stage, there is neither a known ultraviolet completion of the effective action, nor any guarantee of its existence or consistency. The objective of this proposal was to deconstruct the string theory landscape, and determine what part of it is made of honest-to-goodness stable string theory solutions and thus consistent, and what part of it is inconsistent.
The project was divided into three main themes. A first part deconstructs the known de Sitter (positive cosmological constant) landscape in string theory. In addition to a positive cosmological constant, the string theory solution describing our universe should also contain the Standard Model of particle physics. Deconstruction of the existent and exploration of new Standard Model landscapes was another main objective of the project. Finally, we aimed at constructing the non-supersymmetric and intriguing non-geometric landscapes (solutions where the internal space does not have a conventional geometric description, yet are perfectly allowed in string theory).
We have determined that a significant portion of the known de Sitter landscape is phenomenologically inconsistent due either to instabilities or to the presence of massless scalar fields. These massless fields, corresponding to deformations of the internal space and ubiquitous to string theory compactifications, are ruled out as they would lead to long-range fifth forces.
Additionally, we have proposed a linear relationship between the number of scalar fields that acquire mass and the charge required to achieve this, which must remain below a specific threshold. This conjectured relationship has been verified in various scenarios, both by our team and by numerous other research groups. The recognition of this relationship represents a paradigm shift in the field, drastically reducing the phenomenologically-relevant landscape to a small subset of highly constrained geometries where the ratio between the total number of scalar fields and the charge threshold (in specific units) is below 4.
We have comprehensively mapped the landscape of vacua with half of the maximal supersymmetry possible, providing a complete list of the gauge groups that arise. We have also detailed their global topologies and the full massive spectra of the corresponding theories. Upon breaking of supersymmetry, many of these groups can accommodate the Standard Model.
The techniques we developed to chart the landscape of supersymmetric solutions enabled us to also fully chart the non-supersymmetric landscape in nine dimensions. The absence of supersymmetry results in a cosmological constant generated by quantum effects, which we have calculated. Our findings reveal that, within this nine-dimensional landscape, the cosmological constant is universally positive, potentially giving rise to de Sitter vacua. However, upon examining all extremal points (which we have catalogued), we found that none constitutes a true minimum. Thus, these candidate de Sitter vacua are unstable.
We have also investigated a region of the non-geometric landscape where the concept of internal “space” volume is absent. This region is of particular interest in string phenomenology, as the problematic massless scalar fields that arise from volume deformations, which only gain mass through non-perturbative quantum effects, are absent. In this non-geometric context, we have not only confirmed the previously conjectured linear relationship with remarkable precision, but have also identified a set of solutions with zero cosmological constant, where no massless fields are present.
Additionally, we have proposed a linear relationship between the number of scalar fields that acquire mass and the charge required to achieve this, which must remain below a specific threshold. This conjectured relationship has been verified in various scenarios, both by our team and by numerous other research groups. The recognition of this relationship represents a paradigm shift in the field, drastically reducing the phenomenologically-relevant landscape to a small subset of highly constrained geometries where the ratio between the total number of scalar fields and the charge threshold (in specific units) is below 4.
We have comprehensively mapped the landscape of vacua with half of the maximal supersymmetry possible, providing a complete list of the gauge groups that arise. We have also detailed their global topologies and the full massive spectra of the corresponding theories. Upon breaking of supersymmetry, many of these groups can accommodate the Standard Model.
The techniques we developed to chart the landscape of supersymmetric solutions enabled us to also fully chart the non-supersymmetric landscape in nine dimensions. The absence of supersymmetry results in a cosmological constant generated by quantum effects, which we have calculated. Our findings reveal that, within this nine-dimensional landscape, the cosmological constant is universally positive, potentially giving rise to de Sitter vacua. However, upon examining all extremal points (which we have catalogued), we found that none constitutes a true minimum. Thus, these candidate de Sitter vacua are unstable.
We have also investigated a region of the non-geometric landscape where the concept of internal “space” volume is absent. This region is of particular interest in string phenomenology, as the problematic massless scalar fields that arise from volume deformations, which only gain mass through non-perturbative quantum effects, are absent. In this non-geometric context, we have not only confirmed the previously conjectured linear relationship with remarkable precision, but have also identified a set of solutions with zero cosmological constant, where no massless fields are present.
The results of this project represent a major milestone in exploring the string landscape. The techniques we developed enable the first comprehensive chart of string compactifications with half-maximal supersymmetry. This is not only the first but also the only exhaustive exploration of the landscape available so far. Moreover, the tools we built allowed the engineering of new theories consistent with quantum gravity, some without a known string realization. Investigating these theories is crucial for addressing the question of string universality (does any low-energy theory consistent with quantum gravity come from string theory?).
Our catalogue of the non-supersymmetric landscape in nine dimensions is also the first exhaustive map. The algorithms developed will allow to extend this catalogue to lower dimensions.
The conjecture we proposed relating the number of fields that can acquire mass and the charge necessary to achieve this, emerged from drawing an analogy with black hole physics. No one had previously entertained the idea that there could be any relationship between the two. After successfully passing numerous tests, the community has now incorporated this relationship, fundamentally altering the approach to string phenomenology, and reducing the phenomenologically-relevant portion of the landscape to a very small and constrained subset.
In our exploration of non-geometric vacua, we have shown not only that our conjecture applies to this part of the landscape, but we have also provided the first example of solutions with zero cosmological constant and no massless fields. These solutions are only possible in the absence of a true geometry. Their existence was completely unexpected, and invalidated a prior conjecture that such vacua did not occur.
Our catalogue of the non-supersymmetric landscape in nine dimensions is also the first exhaustive map. The algorithms developed will allow to extend this catalogue to lower dimensions.
The conjecture we proposed relating the number of fields that can acquire mass and the charge necessary to achieve this, emerged from drawing an analogy with black hole physics. No one had previously entertained the idea that there could be any relationship between the two. After successfully passing numerous tests, the community has now incorporated this relationship, fundamentally altering the approach to string phenomenology, and reducing the phenomenologically-relevant portion of the landscape to a very small and constrained subset.
In our exploration of non-geometric vacua, we have shown not only that our conjecture applies to this part of the landscape, but we have also provided the first example of solutions with zero cosmological constant and no massless fields. These solutions are only possible in the absence of a true geometry. Their existence was completely unexpected, and invalidated a prior conjecture that such vacua did not occur.