Periodic Reporting for period 2 - ML4SFT (Machine Learning for String Field Theory and for the String- and F-Theory landscapes)
Periodo di rendicontazione: 2022-11-01 al 2023-10-31
String theory is one of the leading theory for quantum gravity and unification. As such, it provides a complete description of the fundamental aspects of our Universe. Because it predicts a 10-dimensional Universe, one must compactify 6 dimensions into a tiny volume such that they are invisible to usual experimental scales. The objective of this project is to investigate two aspects of string theory: first, understanding the compactification geometries using machine learning, second to study the quantum field theory of strings and obtain interactions more explicitly. The latter part of the project includes both formal and machine learning techniques. This will push the boundary of our field by providing new and efficient techniques for bridging the gap between string theory and phenomenology.
The project has succeeded in developing neural networks to compute topological properties of the compactification geometries and of the string interactions. Moreover, it has been demonstrated that, in some instances, string field theory can be rewritten in a simpler fashion by introducing extra auxiliary degrees of freedom. These results provide major conceptual and technical progress in string theory and will provide a strong basis for future developments.
The project has succeeded in developing neural networks to compute topological properties of the compactification geometries and of the string interactions. Moreover, it has been demonstrated that, in some instances, string field theory can be rewritten in a simpler fashion by introducing extra auxiliary degrees of freedom. These results provide major conceptual and technical progress in string theory and will provide a strong basis for future developments.
For the first part of the project, I proposed a neural network architecture to compute the Hodge numbers (some topological data describing the shape of the surface, and related to the number of fermions in the low-energy theory) of complete intersection Calabi-Yau 4-folds (which are the manifolds needed to describe compactifications in F-theory), taking inspiration from my previous work on 3-folds (used for compactifications in string theory). In another work, I used this architecture to study again the 3-folds and add new analyses, such that the robustness of the predictions under extrapolation. In both cases, we achieved state-of-the-art accuracy. Moreover, this exemplified for the first time that neural networks can handle data from algebraic topology, which is also interesting for the mathematical and machine learning communities.
For the second part, I provided an algorithm incorporating deep learning to construct string field interactions, and we showed that it reproduces known results at the lowest order. The main building blocks correspond to functions on and subspaces of the moduli spaces of Riemann surfaces: our approach consists in parametrizing these functions by neural networks, which are trained by solving some mathematical constraint. Hence, my results are also useful for mathematicians and the method can be applied to more general problems. In a second work, I explored the Hubbard-Stratonovich method to reduce the maximum interaction order (how many particles/strings can interact at the same time) in a Lagrangian. In particular, we demonstrated that open string field theory with stubs, which is originally non-polynomial (i.e. it has an infinite number of interactions), can be written in a cubic form with a single auxiliary field. This is a major result which opens new possibilities for closed string field theory, whose action is particularly difficult to manipulate. We have also applied these ideas to a scalar field theory, and studied the relation with renormalization. We have proved that particle field theories enjoy a geometric BV algebra similar to the one found in string theory. We have also explored these ideas in the context of the neural network / field theory, where the same structures appear. Finally, we have explored how to formulate closed string field theory with a string field without any constraint: indeed, there are some reasons indicating that the action could be made cubic in this case.
These results have been published as 3 preprints (all submitted to journals, plus 1 to appear), 4 journal articles, 1 book chapter, 2 proceedings, 1 habilitation thesis. They have also been presented at several conferences and institute seminars.
For the second part, I provided an algorithm incorporating deep learning to construct string field interactions, and we showed that it reproduces known results at the lowest order. The main building blocks correspond to functions on and subspaces of the moduli spaces of Riemann surfaces: our approach consists in parametrizing these functions by neural networks, which are trained by solving some mathematical constraint. Hence, my results are also useful for mathematicians and the method can be applied to more general problems. In a second work, I explored the Hubbard-Stratonovich method to reduce the maximum interaction order (how many particles/strings can interact at the same time) in a Lagrangian. In particular, we demonstrated that open string field theory with stubs, which is originally non-polynomial (i.e. it has an infinite number of interactions), can be written in a cubic form with a single auxiliary field. This is a major result which opens new possibilities for closed string field theory, whose action is particularly difficult to manipulate. We have also applied these ideas to a scalar field theory, and studied the relation with renormalization. We have proved that particle field theories enjoy a geometric BV algebra similar to the one found in string theory. We have also explored these ideas in the context of the neural network / field theory, where the same structures appear. Finally, we have explored how to formulate closed string field theory with a string field without any constraint: indeed, there are some reasons indicating that the action could be made cubic in this case.
These results have been published as 3 preprints (all submitted to journals, plus 1 to appear), 4 journal articles, 1 book chapter, 2 proceedings, 1 habilitation thesis. They have also been presented at several conferences and institute seminars.
My neural networks for Calabi-Yau manifolds have largely outperformed existing works and still provide the state of the art of this kind of problem. Similarly, my new approach to build string vertices, while less precise compared to previous works, is much faster, efficient, and can be generalized to higher orders. It is also the first application of machine learning to the theory of Riemann surfaces. Moreover, the method I developed can be applied to very general problems (anything where we want to construct a function which extremizes some properties), making it particularly adapted to mathematics and physics. Finally, using auxiliary fields to reduce the order of interactions while preserving the homotopy algebra structure is completely new development with far-reaching consequences.
The results of the project will be used to implement higher-order string interactions and provide a usable framework, which would allow answering important questions in string theory, such as the vacuum for the closed string tachyon.
Overall, the project has delivered new decisive results for all objectives.
As a project in theoretical physics, the main societal impact is to increase our knowledge about our Universe. In the long term, knowing the fundamental theory of everything could help develop new technologies.
The results of the project will be used to implement higher-order string interactions and provide a usable framework, which would allow answering important questions in string theory, such as the vacuum for the closed string tachyon.
Overall, the project has delivered new decisive results for all objectives.
As a project in theoretical physics, the main societal impact is to increase our knowledge about our Universe. In the long term, knowing the fundamental theory of everything could help develop new technologies.